Talk:Lion Capital of Ashoka

Shape of the spokes
You make above an interesting remark about the spoke fragments and their thickness in the Sarnath Museum photograph from Huntington p.90 (you say: "The three spoke fragments are too thick for tapering down from where they are placed to where they are destined to end at the inner wheel; if not that, they are too thick for expanding upward and fitting between the little knobs.") Although the precise positioning of the fragments in the Sarnath Museum is debatable, my understanding is that the spokes were probably diamond-shaped (or kite-shaped to be more precise), and positionned quite close to one another, as shown in various reliefs, on in Agrawala's reconstruction, which is then broadly coherent with the thickness of the spoke fragments in the museum. The fragments don't fit if you hypothesize that the spokes were wedge-shaped (oblong triangular). It seems also that the spoke holes along the inner periphery of the rims were quite small, which too tends to confirm a kite shape. पाटलिपुत्र Pat  (talk) 11:32, 26 August 2022 (UTC)


 * पाटलिपुत्र That's a very good point.
 * I have thought about it, but it is still not adding up. Was the wheel carved from a single block of sandstone such as  the late Gupta age image seems to be (and shows some signs of the rhombus shaped spokes you posit), or were the rim, the hub, and the spokes carved separately and were fitted together to form the wheel? If the former, then it would be near impossible for none of the stubs&mdash;the broken off spokes' ends&mdash;to remain on the rim, i.e. some jagged ends, or not so jagged ends, should be visible on the inner perimeter (as are visible on the hub in the late Gupta image). The inner perimeter, however, seems to be smooth in the pictures. ( I had first thought those little "buttons" were the spokes' ends, but they are some form of ritual or conventional design to serve as markers between which the spokes fit. )  If the latter, then the same inner perimeter should have evenly spaced holes in which the spokes fit. But no one mentions any holes. Its surprising that there are no pictures.
 * Fowler&amp;fowler «Talk»  14:57, 26 August 2022 (UTC)


 * Agrawala does draw the holes of the inner periphery, but not the knobs, which is a bit disapointing. But if you look at the Sarnath Museum photograph very closely (you'll have to download) , you will see the neat rounded and regular intercalary knobs all around very clearly, and on the most exposed rim portion (the one at the bottom), you can almost see some of the holes between the knobs. In all probability the intercalary knobs and the rim were carved from one block, but the spokes were carved individually and then inserted, which indeed explains why (almost) no spoke fragments remain on the rim, and why the wheel was easily disassembled when broken. I think I read this somewhere too, but can't find where at this point. पाटलिपुत्र  Pat  (talk) 15:10, 26 August 2022 (UTC)
 * Well, Agrawala does draw the intercalary knobs in profile (together with the outwardly tapering section of the rim) in 7c, but 7b is clumsily drawn really... पाटलिपुत्र  Pat  (talk) 15:26, 26 August 2022 (UTC)
 * Thank you पाटलिपुत्र for that.
 * In my version, I do see what appears to be a clear hole, 1/6 inch across (if the diameter of the wheel is 2 ft 8 in as Agrawala states, and there is no reason to doubt him as he measured it in 1946 as superintendent of the ASI attached to the Central Asian Museum in New Delhi). So, the spaces between the intercalary knobs are approx 1 inch apart and the two filled in spaces to the right of the hole are approximately 3/4 inch wide. If these are the end of the spokes, how were they fitted into the holes?  Was the securing pin carved out of the ends of the spokes? This could explain why there is no hole visible in the two spaces to the immediate right of the clear hole; otherwise a metal pin and/or a hole would have been visible, as the stubs barely rise above the rim.
 * But if that is the case, then how were the spokes fitted in a one-piece rim with pre-made holes? It would be impossible. In the absence of a hub, this is becoming too conjectural. Fowler&amp;fowler  «Talk»  16:54, 26 August 2022 (UTC)
 * More generally, पाटलिपुत्र, Agrawala 1964b's Figure 6a showing thin chariot-like spokes and a small hub does not look realistic as the spokes would need to be pinned at the other end as well and they would have very little room if the holes were 1/6 inch wide; and if they weren't what stability would they have at 47 feet height? We are talking 100 degrees summer days, followed by monsoon storms, followed by winter cold waves from Nepal, year in and year out, not to mention earth tremors if not occasional full-blown earthquakes. Agree, all this is conjecture in turn, but questions obviously remain.  We are talking about stability for a period of 1500 years if we are going to look for the common reasons (Muslim iconoclasm) for the fall.
 * Also, I am not seeing why one asymmetric direction would be favored in Agrawala's or for that matter Irwin's reconstruction. If Agrawala can write pages and pages about the meanings of the horse (which is the same as a rhino) and the lion (the same as a tiger) and so forth&mdash;which it is possible the culture of 2250 years ago, without access to modern genetics might have identified in some fashion&mdash;why is there nothing about why one of the two orientations (i.e showing bull and horse together on one side of the wheel and the lion and elephant on the other, or the bull and elephant on one side and the lion and horse on the other) have been favored. Surely in all the speculative writings on symbolism someone should have worried about this if the reconstruction was widely accepted.  Agrawala 1964b is too full of wild conjectures and unreasoned assertions (per Irwin and others) for it to be reliable much beyond the dimensions of the capital.  Fowler&amp;fowler  «Talk»  17:18, 26 August 2022 (UTC)
 * In all probability there is one hole between each pair of knobs, as drawn by Agrawala, some holes having filled up with concretions during their nearly 1000 or so years buried in the ground. The rim was likely in one piece, and the spokes were inserted one by one from the inside, gathering neetly at the level of the hub. Then the hub would only have to be in two parts, the cover being secured by a pin. It would be all very neat and beautiful, and rather sturdy actually if manufactured precisely, except in the case of a catastrophic fall, which would project all the pieces far and wide. As for orientation, I think nobody really knows, except for the near-contemporary reliefs (more or less 2 centuries after the erection of the pillar) showing examples of wheels on top of animal pillars. पाटलिपुत्र  Pat  (talk) 17:39, 26 August 2022 (UTC)
 * पाटलिपुत्र That's good construction, but that will explain the stability in the plane of the wheel. It will not explain stability in the direction of the normal (perpendicular) direction at the hub.  The spokes experience torque at the rim end from the perpendicular pressure at the hub and vice-versa.  If a child had to take such a wheel apart, they would alternately pull and push the hub and loosen the arrangement. That is because the hub is like a doorknob, at the farthest end of each spoke where the moment is the maximum and only a little pressure is needed to loosen (or to open the door) With even ordinary winds blowing in the face of a wheel 32 inches wide with 32 spokes secured in such fashion would feel a large torque, let alone in storms. And there is no reason that monkeys would not be tempted to do the same. Are there examples of wheels secured in such fashion that have survived in the open air for a long time?  Fowler&amp;fowler  «Talk»  18:39, 26 August 2022 (UTC)
 * Just my personal opinion, but I think tight and precise handcrafting of hard sandstone, with the spokes finely and deeply inserted inside the rim on the outside, and strongly anchored through the hub at the center, could hold any kind of natural torsion. The shaft too would have to be sturdy. But Mauryans have shown how precisely they can handle hard stone (or even granite: Barabar Caves!), if necessary through ultra-fine grinding and polishing, so a very high level of precision in assembly is indeed possible... Of course, the lions had much more inherent sturdiness then the wheel itself, and would have lasted much longer in any circumstance. Quite impressive nonetheless... पाटलिपुत्र  Pat  (talk) 18:53, 26 August 2022 (UTC)
 * पाटलिपुत्र I have no doubt they were expert stone cutters and arrangers, but there are some mechanical limitations we are dealing with. The spokes at the rim end are 3/4 inch wide. At the hub end, they can't be much narrower than half an inch (1.25 cm) for a 1/6 inch hole to be drilled safely.  At that dimension, and assuming the tightest fit, you will need a hub of 16-inch circumference, which means a diameter of 5 inches.  If it is a disk without holes, it will experience quite a bit of normal pressure.  You could assume holes, or make the hub much bigger, say 9 inches in diameter, or even a foot, then although there would be more pressure, it would also be more stable.  But all the reconstructions show a smallish hub. Perhaps if some structural-, mechanical-, material-, and chemical engineers at one of India's institutes of technology were rounded up they could shed more certain light on it, working with archaeologists and art historians.  New methods are needed.  I did find one book chapter on spokes and their significance in Thai Buddhist art.  It cites Irwin for the usual pre-Buddhist influences.  Perhaps you might find some nuggets there:
 * Best, Fowler&amp;fowler  «Talk»  20:13, 26 August 2022 (UTC)
 * In my personal opinion, the hub should only work as a "cover plate", and actually the smaller the better. In this case, the resistance to the wind only has to be considered individually for each spoke (imagine there is no hub and the spokes are independent from one another, simply inserted into the rim). The surface of one spoke was roughly 2.5cmx30cm at worst (0.0075 m2). Even in winds of 50 meters per second (about 110 miles per hour, which generate around 170kg of pressure per square meter ), the pressure on a single spoke would be around 1 kilogram only (the weight of a bottle of water!), spread on the whole length of the spoke. This is a very manageable for a thickish hardstone spoke firmly inserted into the rim. If the hardstone spoke is structurally sound, I don't think it would risk breaking until at the very least 20kg of pressure (a small kid sitting on it. Even the wooden spokes of a pole ladder will handle much more.)... At 111 miles per hour, the whole wheel taken globally would have to sustain around 70kg of wind pressure (32x1kg for the spokes + 40kg for the rim), something a properly set 8" sandstone shaft would also be able to handle ... पाटलिपुत्र  Pat  (talk) 05:46, 27 August 2022 (UTC)
 * पाटलिपुत्र That's a clever suggestion, but it brings up new issues. As you've seen at the other end, we can see that the spokes are 3/4 inch thick. Assuming they narrow to 1/2 inch, they would need a hub of diameter 5 inches, as I have explained.  If you suppose that the spokes can be narrowed further somehow and the hub is correspondingly smaller across and in the limiting case it is a point, the spokes will fit so tightly, that there will be a virtual disk of a diameter of five inches, and there will be turbulence inside the disk.   You can't assume that a spoke exists independently of all the others, and then sum the pressures. Fluid dynamics is highly nonlinear. It will be easier to see this if you imagine a metal boat with a one-meter-wide hole in its flat bottom.  If you attach one metal strip 1/2 inch wide across the hole and set the boat afloat, it will sink like a rock.  If you take a large number of 1/2 wide strips and space them 1/16 of an inch apart, the boat will take a while to sink, and you will see turbulence as the water wells up.  Fowler&amp;fowler  «Talk»  11:57, 27 August 2022 (UTC)
 * Yes, viscosity/turbulence is harder to evaluate, but I don't think it changes the equation fundamentally. To further the approximation, even if you took maximum viscosity for the air (no air manages to get through the spokes of the wheel), the whole surface of the wheel would still be 0.78 m2, hence a global wind pressure of about 132 kilograms maximum (winds of 110 miles per hour). If you take out the pressure on the rim and divide by 32, you would still be at 3kg per spoke maximum (maximum viscosity, or no space for air between them). I think what is happening is that we have a relatively small object actually (about 1 meter in diameter), manufactured with a very strong, rigid and heavy material (sandstone), and handcrafted and assembled to high precision (the rim/spoke assembly alone can be very sturdy). It was probably capable of sustaining very tough weather conditions, and for a long period, since this material does not deteriorate with time (contrary to wood, limestone or even metals). But beyond that, I'm afraid only specialists could tell us! Best पाटलिपुत्र  Pat  (talk) 12:34, 27 August 2022 (UTC)
 * I've magnified Huntington's image five times, but I don't see any substantive traces of the spokes' ends or of holes. Fowler&amp;fowler  «Talk»  15:10, 26 August 2022 (UTC)
 * Sorry that was an ec. Will look at the museum image next.  Fowler&amp;fowler  «Talk»  15:12, 26 August 2022 (UTC)

Measurements
user:पाटलिपुत्र I respond to your post above later, but we first need to establish that this was a high-precision construction. The wheel, I mean. I'm not convinced that the wheel and the capital were crafted by the same craftsmen. In any case, The measurements according to Agrawala are:
 * height of the chakra = 2ft 8 in.
 * internal diameter of the chakra = 2 ft 5 inches to 2 ft 1 inch.

What do you make of this measurement? As far as I can tell, the internal rim was not a perfect circle, and it is unclear whether the external was either. The inner arcs formed by the different wheel fragments did not define circles of the same radii. I can actually check this later this weekend on Agrawala's own image. Fowler&amp;fowler «Talk»  13:23, 27 August 2022 (UTC) Updated  Fowler&amp;fowler  «Talk»  13:44, 27 August 2022 (UTC)  Fowler&amp;fowler  «Talk»  13:47, 27 August 2022 (UTC)


 * I'm attaching a rough reconstruction of what we've been talking about (a reconstruction based on the exact archaeological remains, and the kite-shaped spokes accomodating the exact size of the 3 spoke remains. You can click on it to see it without the red circle). All reconstructions by Agrawala also show a perfect circle, so I'm not sure what he means by the two internal diameters of the chakra...Here's a possible explanation. 2ft 8 inches is indeed for him the diameter of the complete wheel (outside perimeter). 2ft 5 inches does correspond to the diameter of what we would call the inner perimeter of the rim (the one where the intercalary knobs are positionned). But 2ft 1 inch would correspond closely to the diameter of the circle defined by the lateral summits of the kite-shaped spokes (red circle on my drawing, attached)... a kind of geometrical inner diameter which would be useful for any reconstruction (it actually defines the reconstruction to a large extent). It seems very improbable that there would be so much variation in the inner diameter otherwise, as it would vastly contradict all his reconstructions. This notion would also be completely incompatible with the quite perfectly regular arc of the four large rim fragments of our archaeological remains. In all likelihood the editor of his book did not really understand what this second inner diameter was about... पाटलिपुत्र  Pat  (talk) 14:24, 27 August 2022 (UTC)
 * user: पाटलिपुत्र Please acknowledge by signing anew here that you had made an error above, that 2 ft one inch is where the rim is. Best,  Fowler&amp;fowler  «Talk»  18:08, 29 August 2022 (UTC) Updated  Fowler&amp;fowler  «Talk»  18:10, 29 August 2022 (UTC)
 * Yes you're right, the 2.1feet circle is too small, I don't know how I obtained that (probably got the feet/cm conversions wrong). Here's a proper layout of the 2ft8ins, 2ft5ins and 2ft1in circles. Thanks for pointing this out! पाटलिपुत्र  Pataliputra  (talk) 18:32, 29 August 2022 (UTC)
 * [[File:Schematics of the remains of the topmost wheel of the Sarnath Lion Capital of Ashoka, in the Sarnath Museum showing radii.jpg|thumb|right|Circles formed by the inner and outer arcs of the wheel segments of the upper wheel of the [[Lion capital of Ashoka]]]]
 * I just checked all three (museum, agrawala, and your schematics) using the perfect circle tool. All three arcs form circles of different curvature. The perfect circles are fitted to the upper rim. There may be some distortion in these pictures depending on the camera angle, but these are quite a bit more than that. You might not be able to see the upper rim as the colored circle cover them entirely.  Fowler&amp;fowler  «Talk»  15:14, 27 August 2022 (UTC)
 * Your inner circle for example, dotted on the right, fits the upper rims of two with the lower rim of one. Fowler&amp;fowler  «Talk»  15:24, 27 August 2022 (UTC)
 * That's Agrawala's own book. It has an older picture different from the Sarnath museum's.  He made the measurements when the was the Superintendent of the Archaeological survey of India attached to the Central Asian Antiquities Museum in New Delhi in 1946. He would have received all the help available. Also, it says inner diameter from 2 ft 5 in to 2 ft 1 in.  That's not the kind of error an editor or a typesetter would make. 29 inches/25 inches = 1.16
 * The blue and the green circles are about the same radius. But the bottom one (red) is bigger. If you assume some distortion in the camera by the way in which the rims are configured, the ratio in my image is not much different.  Fowler&amp;fowler  «Talk»  15:47, 27 August 2022 (UTC)
 * I have to go now. I'll check late tonight or tomorrow AM. Best,  Fowler&amp;fowler  «Talk»  16:01, 27 August 2022 (UTC)
 * Interesting... I did the same exercise on the Sarnath Museum photograph of the original archaeological remains, and indeed the top left and top right rims appear to have an identical, slightly smaller, curvature, while the bottom rim appears to have a slighly wider curvature, giving a diameter about 10% larger. But I am afraid this is due to the lighting: there is a very strong top-down lighting in the photograph, which pushes the shadow of the rim's inner edge downward for the portions closest to the light, and produces a characteristic visual deformation, giving at the top (on both sides) the impression of an unnatural, extra-curvature towards the inside (in other words, the over-exposed, bright surfaces at the top of both left and right rims seem to "bend" inside)... but this is an optical illusion (shadow displacement on a slightly curved surface). A concomitant side-effect is that the inside band of the rim also gives the impression of being thinner in these top areas. Once you take this into account, you will see that the inside curvatures are identical. You can also refer to the photograph of the reconstruction by Agrawala , because he worked on casts of the archaeological remains, and visibly had no problem at all adjusting the curvatures, which are clearly all identical. पाटलिपुत्र  Pat  (talk) 19:15, 27 August 2022 (UTC)
 * user:पाटलिपुत्र The reason that the inner rim is narrower above than below is that it is closer to eye (camera) level which is a little above the point where the blue and the green circles intersect. Its the foreshortening I was talking about upstairs, for the same reason that the space between the dark tiles in File:HK Hung Hum Station Corridor.jpg keeps decreasing as the tiles move away to the horizon or eye level.
 * Edges in an images are discontinuities of the image intensity, or the zero crossings of David Marr. Lighting does not substantially affect edges, but viewpoint or camera angle does.
 * (More technically you can convolve the image to offset the lighting disparities in a nonlinear fashion, but the discontinuities don't change&mdash;the old edges are still discontinuous but at different values of image intensity)
 * But there ís some foreshortening as you insightfully point out (similar to holding up a coffee mug and turning it so you can look in. At eye level, the (b)rim and the bottom will both be circles, concentric ones, i.e. have the same center.  If you lower the mug a little, the two circles will no longer be concentric; there will be more room between them below than above.
 * But these effects affect the (b)rim as well. If you keep lowering it, it will look more like an oval or ellipse than a circle. The problem here is that the outer rim of all three pieces, more or less, fit in a proper circle (the brown one I have added), which means that the perspective/foreshortening effects are not significant.
 * That is why Agrawala has one diameter for the outer rim, but two for the inner. I suspect that of these three pieces, either one (the bottom) did not belong to the same wheel, i.e., to another wheel (bespeaking previous mishaps) or the wheel itself had this feature/defect. A fuller discussion will have to await the appearance of another picture, with the camera held lower. Fowler&amp;fowler  «Talk»  10:37, 28 August 2022 (UTC)
 * Fowler&amp;fowler «Talk»  10:54, 28 August 2022 (UTC)


 * Deformation due to directional lighting.jpg to see you back! I think, it is too strong and sudden to be foreshortening only. There is also a strong effect due to the intense top-light, which warps the shadow line and induces you to mesure a shorter curvature. I am attaching an explanatory drawing. Additional care should also be taken to the fact that the top left and right rims are in effect reconstructed from 2 pieces each (5 rim pieces in all per Agrawala), and their alignement is visibly not perfect, tending to further bend inside. Best पाटलिपुत्र  Pat  (talk) 11:02, 28 August 2022 (UTC)
 * You are describing shading effects. If you change the lighting on Venus de Milo, yes the edges you see will change, but the edges here are ridges. Lighting will not create new ridges. Agrawalla, btw, in his 1964 book has only two pieces.  Fowler&amp;fowler  «Talk»  11:11, 28 August 2022 (UTC)
 * Even if you light the edge lower, i.e. with the bulb, right next to the edge, you will still see the old edge (or an edge detector will in software). This is because the faces on either side of the edge, being at different angles with respect to the light, will have different gradients of intensity. Good simulation by the way.  Fowler&amp;fowler  «Talk»  11:19, 28 August 2022 (UTC)
 * I mean there are more pieces in the museum picture than in Agrawala's 1964 book. There is a third piece in the museum picture.  Fowler&amp;fowler  «Talk»  11:25, 28 August 2022 (UTC)


 * Yes, but the ridges are visibly not sharp but rounded, which is sufficient to create a shading effect. As to Agrawala's reconstruction, he definitely used the top left rim block (made of 2 pieces) and the top right one (made of 2 pieces) (positions as seen in the Sarnath Museum photograph), but they are reversed, which means we are seeing the back of the wheel (or that the photograph in his book was reversed). There is a third fragment at the top in his photograph, which can only be our bottom rim fragment (the last remaining one, the 5th rim fragment), but probably seen from the back, ie partially, if it was damaged on that side. Hard to tell for sure... पाटलिपुत्र  Pat  (talk) 11:33, 28 August 2022 (UTC)
 * Only parts are rounded. On the two top pieces, only the one on the right is rounded toward the top, not the lower half. You can fit the circle to just that and the problem remains.  In Agrawala 1964, btw, the pieces together do not make a semi-circle.  In the museum picture they do.  Fowler&amp;fowler  «Talk»  11:52, 28 August 2022 (UTC)
 * It is unlikely Agrawala would not show a big piece if most of it was damaged on one side. He would indicate what its extent was.  Fowler&amp;fowler  «Talk»  11:55, 28 August 2022 (UTC)
 * Listen, you can keep arguing ad infinitum, but the bottom line is
 * Agrawala's measurement:
 * height of the chakra = 2ft 8 in.
 * internal diameter of the chakra = 2 ft 5 inches to 2 ft 1 inch.
 * It is impossible that an editor or typesetter could have made an error such as this. Fowler&amp;fowler  «Talk»  11:58, 28 August 2022 (UTC)
 * Sure, I know what the books says, and I know it contradicts Agrawala's own reconstruction, and, in my opinion, the pieces in the Sarnath Museum. Just trying to reconcile the facts. Thanks for the exchange... पाटलिपुत्र  Pat  (talk) 12:05, 28 August 2022 (UTC)

Apologies, user:पाटलिपुत्र for the impatient tone. I have now fitted circles to the farther of the two inner rim edges, i.e. the ones closer to the wall. There the shading effect would be minimal. But you still get the same disparity. Fowler&amp;fowler «Talk»  12:21, 28 August 2022 (UTC)
 * OK. Just to make sure: for your circles, are you working from my schematic drawing, or are you working from the Sarnath Museum photograph? Because I am working on the museum photograph, and I do not see anything near your variations (after adjustment for shading effect -about one thickness of the circle worth- and slight positioning issues, especially the top right rim fragments which has to be redressed)... Here are roughly the variations I am seeing after these adjustments (attached). It's basically all within the drawing margin of error... पाटलिपुत्र  Pat  (talk) 12:31, 28 August 2022 (UTC)


 * I have worked with both images. In the museum picture. I first applied the Canny edge detector to it and then fitted circles to all three edges (avoiding the top right, i.e. fitting the third circle to only the lower portion of the fragment on the right. The fitted circles for the left fragment and the right, which are similar, easily fit inside the circle for the lowest fragment, i.e. with lots of room to spare. Later today, I'd be happy to upload a non-free image to the article for showing this.  Fowler&amp;fowler  «Talk»  13:12, 28 August 2022 (UTC)
 * I'm sending you my image by e-mail. पाटलिपुत्र  Pat  (talk) 13:16, 28 August 2022 (UTC)
 * Sure. The problem is not the upper two fragments, but the lower one.  Its inner and out rims do not form concentric circles, not even close. The outer rim matches the outer rims of the other two, more or less.  The inner does not.  Fowler&amp;fowler  «Talk»  13:33, 28 August 2022 (UTC)
 * Doesn't seem to be working. Actually, you can just use my drawing above and superpose it to the Museum image. It should even be the same dimension. "Its inner and out rims do not form concentric circles" I do not see that either... never mind. I have to go. Please check my post above about WP:BALANCE. Best  पाटलिपुत्र  Pat  (talk) 13:51, 28 August 2022 (UTC)
 * If you are using the Canny edge detector for your circles, I think it probably gets thrown off by the over-exposure of the inside of the lower rim fragment, again due to the intense light from above. You would indeed get a significantly wider inner rim than for the other rim fragments (just as a result of the light pushing the shadow of the roundish ridge downward), at least by 3-4mm I would guess. Maybe you should work manually instead, allowing for these light effects.... From a lighting standpoint, it is a very asymetrical photograph, and a program wouldn't see that. पाटलिपुत्र  Pat  (talk) 14:21, 28 August 2022 (UTC)
 * Modern edge-detection in machine vision, based on the work of John Canny, David Marr, Berthold K.P. Horn, Jan Koenderink, David Mumford and many others has been used on complex images for many years. I have now fitted to the circles to the raw image at the museum you refer to and they show the same result as Canny.  I'm happy to upload the image to this article using the non-free format.  It shows the same disparity between the lowest segment and the other two.  You don't even need the top right.  The remaining two themselves point to it.   Fowler&amp;fowler  «Talk»  15:35, 28 August 2022 (UTC)
 * Fowler&amp;fowler «Talk»  15:38, 28 August 2022 (UTC) Updated.  Fowler&amp;fowler  «Talk»  22:20, 28 August 2022 (UTC)
 * Thanks User:Fowler&fowler. For my part, I do not see issues with the various rim parts, after adjustments for positions and lighting are made, and Agrawala doesn't seem either when doing his reconstructions, neither has this ever been discussed on the literature I'm afraid. The maximum you could do is mention his statement about "Internal diameter of the chakra = 2 ft 5 inches to 2 ft 1 inch" if you wish, but this should be WP:BALANCEd with his own reconstructions (and, frankly, the whole literature on the subject), which never takes into account nor mentions any such variations in inner diameter (as far as I'm aware...). In effect, that specific statement by Agrawala seems a bit fringe, and not even supported by the rest of his own research, nor anything that has been published on the subject: this is why I think it is probably some sort of typo or misunderstanding by the editor. It would be really interesting if there were Secondary sources discussing this issue, I'll keep my eyes open. Best पाटलिपुत्र  Pat  (talk) 05:10, 29 August 2022 (UTC)
 * Wheels of various inside diameter.jpg me address a few contradictions in that model too... You state that the rim fragments all have roughly the same outside diameter (2.8 feet per Agrawala, which we can see in the photograph, I agree to that), but you suggest that the two dimensions given by Agarawala for the inner diameter (2.5 and 2.1 feet) would point to the fragments belonging to two different wheels. The problem is that this is not logically possible: the two wheels under such specifications would have to be vastly different objects (see attached diagram), which is not at all what we are seeing in the photograph of the remains . Also the width of the rims would have to be vastly different (3 and 7 inches respectively), which not only is not visible at all in the photograph, but contradicts Agrawala's precise statement that the rim is 3.65 in width, and of course his reconstruction . Again I cannot see this statement ("internal diameter 2.5 to 2.1 feet") as anything other than a typo or a misunderstanding of some sort... I'm afraid it cannot be taken at face value, given all the contradictions that it, alone, generates. Best पाटलिपुत्र  Pataliputra  (talk) 08:03, 29 August 2022 (UTC)

Arbitrary break (continuing the "Measurements" thread)

 * Scaled diameters (Sarnath Wheel) per Agrawala.jpgAnother hypotheses. I tried to scale the three measurements by Agrawala (2ft8ins, 2ft5ins, 2ft1in) on top of my schematic drawing of the archaeological remains in Sarnath Museum (attached), and actually the circle with the 2ft1in diameter exactly defines the inner perimeter of the Sarnath reconstruction. The second inside diameter (at 2ft5ins) remains a bit puzzling although it could correspond to the depth to which the spokes were inserted... it would be rather exactly the diameter of the circle defined by the spokes seen alone, without the rim (a "Dharma-Chakra" making abstraction of the rim, the "radiating sun" seen alone... definitely another important "internal diameter"). पाटलिपुत्र  Pataliputra  (talk) 09:26, 29 August 2022 (UTC)

User:पाटलिपुत्र There is no need for a second hypothesis. The pictures you have displayed are incorrect. All Agrawala, or anyone, has are three wheel fragments. They all have the same external curvature, positing the same external radius of curvature They have the similar thickness, but different internal curvature, positing different internal radii of curvatures. How the wheel was shaped when it was carved we don't know. If the fragments are indeed from the same wheel, the defect/feature had to be corrected farther along. Agrawala can't tell us how it was corrected. In reality, the "circles" are not really circles in any picture in which the camera is held above the center of the wheel, but ellipses defined by the projective geometry of the picture. But those issues are small as the size of the wheel is small and the observer is at some distance from the wheel. For the larger circular recessed window or alcove in which the wheel is set, you can see the elliptical effect more readily. But for the "clips," to which the fragments are affixed in the recess, and which can be seen in the places where there are no fragments, the elliptical distortion in the picture (which you can check) is minor. If you fit a confocal ellipse (i.e. with the same two foci&mdash;the generalization of concentric circles) to the inner rim of the lower fragment, it will not fit inside the ellipse defined by the ends of "clips." Inded it reaches more than half way up the clips. In other words, as I stated, the elliptical effects due to foreshortening are not significant enough to explain the disparity in the curvatures. The thickness 3.65 in Agrawala is easy to explain: the lower curvature of the inner perimeter of bottom fragment will not register an appreciable change in thickness over a small stretch of perimeter.

As for the literature, there aren't too many people who have studied it at any depth. Irwin has not; Asher has not; Tomor has not. Only Agrawala had the wheel and he measured it. Oertel did as well but did not mention the diameter. Our hands are tied. In the description section body and the picture caption, we will need to mention both the external and (the two) internal diameters of the wheel. Best, Fowler&amp;fowler  «Talk»  10:30, 29 August 2022 (UTC)


 * PS Mentioning anything more from Agrawala would be WP:FALSEBALANCE Agrawala has the statistics on page 1.  There is no illustrative plate in the back except the cross-sections. He mentions a reconstruction with plaster cast on page 2. In relation to that he has a picture.  Besides, by a prior agreement, we cannot have reconstructions in the article body, and we already have a reconstruction of Agrawala (not substantially different) in the Reconstructions gallery.  Fowler&amp;fowler  «Talk»  11:00, 29 August 2022 (UTC)


 * Diameters of an indented circle.jpg only point here (your initial question at the top) is to try to understand what Agrawala means with his measurements. Actually, in terms of technical draughtsmanship, when describing an indented circle (or an indented wheel), the only way is to give one external diameter (here 2ft8ins) and two internal diameters: one internal diameter taken from the depth of the motise holes (here 2ft5ins), and one internal diameter taken outside of it (here 2ft1in). This is not WP:OR, this is standard, universal, practice. Agrawala is even extremely accurate when he says "internal diameter of the chakra = 2 ft 5 inches to 2 ft 1 inch", because, since the mortise holes are slanted internally, there is indeed a technical progression from 2 ft 5 inches to 2 ft 1 inch (you could technically take any measurement in between). There can be no other explanation, and this is the only reading which is both logical and 100% coherent with all the reliably sourced evidence. Best पाटलिपुत्र  Pataliputra  (talk) 11:41, 29 August 2022 (UTC)
 * user: पाटलिपुत्र  To "indent," is "to make an incision in (a board, etc.), for the purpose of mortising or dovetailing; to join or joint together by this method." (OED) That is not the situation with the Sarnath fragments.  There is an indented gutter which runs along the internal perimeter but it is of inconsequential depth as is obvious in Figure 7(b) of Agrawala The depth of the gutter is no more than 1/8 of an inch.  In the gutter, spaced approximately 2 inches apart 32 holes of depth 2 inches and radius  approximately 1/4 inch had been drilled. You can see one hole in the bottom fragment in the museum picture (as you pointed out) and I later verified.  The hole is tiny.
 * Sorry, but this not what Agrawala means. Fowler&amp;fowler  «Talk»  12:10, 29 August 2022 (UTC)
 * Of course it is, I'm afraid. Everything fits: the "32 holes have a depth 2 inches": yes, the proper way is to take measurement from the bottom of one hole (mortise) to the bottom of the facing hole (mortise), to get the maximum internal diameter (2 ft 5 inches). And then to measure from the rims to get the minimal internal diameter (2 ft 1 inch). Even the arithmetics match perfectly: 2 ft 5 inches = 2 ft 1 inch + the two holes (2 x 2 inches). It can be nothing else... पाटलिपुत्र  Pataliputra  (talk) 12:20, 29 August 2022 (UTC)
 * No one in their right minds would give two diameters when they can simply say, "Holes 2 inches deep had been drilled along the inside of the rim fragments." Fowler&amp;fowler  «Talk»  12:41, 29 August 2022 (UTC)
 * That is, "Holes 2 inches deep, 2 inches apart, and 1/4 inch (diameter) wide, had been drilled along the inside of the rim. Who would paraphrase that as, "The wheel has two internal diameters ...?"  Fowler&amp;fowler  «Talk»  12:55, 29 August 2022 (UTC)
 * Internal diameters.JPG "No one in their right minds..." In technical parlance, it seems that the proper way to define a wheel with internal indentations in indeed to give its two internal diameters: the " Inside diameter " (between the rim walls) and the " Root diameter " (from the bottom of one mortise hole to the bottom of the mortise on the other side). See a diagram found on Commons. It seems Agrawala did know his technical geometry actually, oblivious of how following generations would be troubled by his assertions... पाटलिपुत्र  Pataliputra  (talk) 16:25, 29 August 2022 (UTC)
 * user: पाटलिपुत्र picture is wildly inaccurate. Your picture shows a surface, the internal rim of which zigzags up and down across its width. Such a surface will not have a clear internal diameter but it will shift from the ridges to the valley floors. Our wheels, have tiny holes, as is obvious in the museum picture. The holes are 1/4 inch wide. The internal perimeter at 2 ft 1 inch diameter is 78 1/2 inches.  The area of the internal strip is 208 sq inches. We have 32 holes of diameter 1/4 inch (it can't be much more as the spaces between the holes is only a little over 1 inch; please look at Agrawala's picture of the cast.
 * Even if we assume that the holes are 1/2 wide, i.e. of radius 1/4 inch, the area of the combined wheels is 6 1/4 square inches, which is 3% of the area. It would be highly unusual for anyone to posit two diameters on the basis of such a small percentage of the area, let alone to do that when the diameter is 1/4 inches.
 * Example: Across a six and half feet (78 inches) long two by four (i.e. A length of sawn wood of cross section approximately 2 inches by 4 inches), if we drill 32 evenly spaced 1/4 inch holes of depth 2 inches, we don't say a two by four (internally two by two).
 * Also, I am troubled by your changing arguments. You first had a picture of the kite-like spokes, which you conjectured to achieve their maximal thickness at length 12.5 inches (half of 2 ft 1 inch) when that was where the rim is. Please acknowledge that you made an error in that argument and picture.  This exchange is for the benefit of this page which other Wikipedians will read.  So again please acknowledge by signing anew that you made an error there.  Fowler&amp;fowler  «Talk»  18:03, 29 August 2022 (UTC)
 * Diameters of an indented circle (Sarnath Pillar wheel).jpg My brown drawing above ("Diameters of an indented circle") is not meant to be an accurate depiction, it is just devised to show clearly the measuring method of the Inside diameter and the Root diameter. I can make an accurate one, but first I would like to make sure we agree about the measurements of the holes: per Agrawala's drawing they seem to be about 1 inch in diameter . I think the photograph cannot be relied on for this, since many of the holes would have filled up with concretions. So I'm planning to make indents 1 inch in circumference and 2 inches deep. Is that OK with you? Can you give me the source where they say 2 inches for the depth of the mortisses? Thanks पाटलिपुत्र  Pataliputra  (talk) 18:38, 29 August 2022 (UTC)
 * I am attaching my drawing of the section of the wheel (intended to reflect the configuration of the Sarnath Capital wheel) as explained in my previous post. Best पाटलिपुत्र  Pataliputra  (talk) 19:07, 29 August 2022 (UTC)
 * It is still wildly incorrect. There were 32 holes of width 1/4 inch drilled in a rim of width 2.65 inches and perimeter 78 inches, not cuts or notches in the rim, i.e. indentations, as you have drawn. Wildly incorrect. Fowler&amp;fowler  «Talk»  20:09, 29 August 2022 (UTC)
 * Where do you get a 1/4 inch diameter for the holes from? Agrawala's are roughly 1 inch in diameter if you look at the drawing and its measurements.... Drawing 7b And this is what would be necessary to accomodate the spokes anyway... पाटलिपुत्र  Pataliputra  (talk) 20:13, 29 August 2022 (UTC)
 * From the museum picture. They can't be one inch in diameter.
 * The Agrawala picture is horizontally exaggerated,as the centers are placed 2.2 inches apart in order for 32 to cover the perimeter ( pi =3.1415 times 25 inches/32 = 2.18 inches. But that distance in Agrawala is 1.5 times 2.65 (the thickness) Fowler&amp;fowler  «Talk»  20:29, 29 August 2022 (UTC)
 * I will post something that explains this better using the elliptical distortion resulting from the camera placement above the center of the wheel in the museum. The problem is mainly that the inner rim of the lower fragment has less curvature, and when you reconstruct the radius of that fragment is approximately 29 inches as opposed to the others which are 25 inches.  Fowler&amp;fowler  «Talk»  20:36, 29 August 2022 (UTC)
 * I'm afraid we can't use your personal calculations based on the museum's picture, that's really unreliable: the holes do not appear clearly and most of them were probably filled-up with concretions. Agrawala's drawing may not be perfectly accurate, but it's the best we have, and it does give a ball-park figure (about 1 inch). And all his reconstructions are roughly based on holes around 1 inch in diameter. There's no way these holes were 1/4 inch in diameter when you look at the sources... पाटलिपुत्र  Pataliputra  (talk) 20:39, 29 August 2022 (UTC)
 * here if figure 7b (for anyone who cares to investigate). The wheel has diameter 25 inches. Its perimeter of pi times 25 = 78.5 inches. If 32 evenly placed holes are made, they are 78.5/32 = 2.45 inches.
 * (I made an error, not 2.18 inches.) But in Agrawala the distance between them seems to be 1.5 times 2.65 inches. Fowler&amp;fowler  «Talk»  21:10, 29 August 2022 (UTC)
 * Apologies 1.4 times 2.65 inches. That means everything will need to be scaled down by 0.66  Fowler&amp;fowler  «Talk»  21:17, 29 August 2022 (UTC)

Instead of faciley Wikilawyering, please tell me, are the holes 2.18 inches apart in Agrawala if the thickness is 2.65 inches? Let's forget about 1/4 for a minute.
 * I'm afraid your calculation above is mistaken. If the holes are 1 inch in diameter, then the space between the holes has to be (25xpi-32)/32= 1.45 inches. Graphically, the intercalary space between the spokes therefore has to be about 1.45xholes in width, which is about what we're seing in Agrawala (roughly, +-10%). Agrawala's image is indeed slightly deformed horizontally, but the horizontal proportions and vertical proportions remain correct nonetheless (deformation does not affect proportions, but don't compare vertical and horizontal measurements). So we're still at roughly 1 inch for the diameter of the hole, as also shown in all reconstructions. I could decrease a bit to 0.8 inches for example, but going below that would contradict all the sources we have. पाटलिपुत्र  Pataliputra  (talk) 21:14, 29 August 2022 (UTC)


 * I'm talking about the distance between the centers in Agrawala. What is it according to your calculations?  Fowler&amp;fowler  «Talk»  21:19, 29 August 2022 (UTC)
 * Sorry, I have to go for now. पाटलिपुत्र  Pataliputra  (talk) 21:21, 29 August 2022 (UTC)
 * Of course, that is the standard response any time you are in a tight spot  Fowler&amp;fowler  «Talk»  21:24, 29 August 2022 (UTC)
 * Anyway, the holes are 2.45 inches apart. In Agrawala's picture they are shown to be 1.4 times 2.65 = 3.71 inches apart. That means they need to be scaled down by a factor of 2.45/3.71 = 0.66.
 * In Agrawala's picture, they are 0.883 inches wide. 0.66 times 0.883 = 0.583. That is the diameter.  Therefore the radius of each hole is 0.2915 inches. Its area is pi times radius squared = 0.27 square inches. There are 32 holes. Their total area is 8.644 square inches. The total area of the rim is 208 square inches.
 * So, we are saying 4% of the total area has depth 2 inches, 96% doesn't. What form of mechanical drawing describes that as "The wheel has two internal diameters). Let's say for a minute, I bump up the diameter to 1 inch; the radius is 0.5 inch.  The radius squared is 0.25 inches. The area of one hole is 0.25 times pi = 0.78 sq inches. The area of 32 holes is 25 square inches.  It is still only 12% of the total rim area.  Who will give two internal diameters for that? Doesn't make any sense.  Fowler&amp;fowler  «Talk»  22:00, 29 August 2022 (UTC)


 * Thank you for your patience... Please look at the schematics of the actual drawing by Agrawala (black-and-white drawing of the rim fragment, attached). The important point is that it defines the proportions between the holes for the spokes and the intercalary space in between: for 1 hole you have approximately a space corresponding to about 1.7 holes in between. Since we know the circumference (78.6 inches) and the number of holes (32), this simple fact gives us the diameter of one hole: 78.6/((1+1.7)x32)=0.91 inches. It is also coherent with all the reliable reconstructions we have (and coherent with my drawing above, but I will tweak it a bit more later to be perfectly precise). In term of global surface, you are right that the holes only represent a tiny surface of the inside of the rim: (32xpix(0.91/2)x(0.91/2))/78.6x2.65=9.8%, but that's irrelevant (although good news for the sturdiness of the rim): the transversal section of the rim is nonetheless quite perfectly similar to the brown "gear" drawing I made above. So, I'm afraid there's no way the holes are 1/4 inch in diameter (they are around 0.91 inches), my technical drawing is broadly correct, and I think someone versed in technical drawings would definitely define it as an internally indented wheel with an Inside diameter of 2ft1in and a Root diameter of 2ft5in, for an outside diameter of 2ft8in. So I think that sums it up for the "gear hypotheses", and answers your initial question here ... I will look into your "elliptical distortions" a bit later... Best पाटलिपुत्र  Pataliputra  (talk) 03:17, 30 August 2022 (UTC)
 * I've finished with the fine-tweaking of the "gear drawing" above and updated the file (the holes correspond exactly to the layout defined by Agrawala). पाटलिपुत्र  Pataliputra  (talk) 04:14, 30 August 2022 (UTC)
 * Spoke holes (Sarnath Lion Capital).jpg actually I think we can also see the traces of the "big" spoke holes in the photograph (bottom rim ), but they are somehow filled up, slightly protruding above the surface of the rim, but do appear faintly. पाटलिपुत्र  Pataliputra  (talk) 10:17, 30 August 2022 (UTC)

Comment User:पाटलिपुत्र Please tell me once here when you are done with pushing this POV. You have made 18 edits, pinging me several times, even after claiming you have made the final tweaks. Tell me definitively when you are done, so I can reply. And once you are done do not make any edits for an hour. Fowler&amp;fowler «Talk»  10:29, 30 August 2022 (UTC)
 * I'm done. Looking forward to your feedback. पाटलिपुत्र  Pataliputra  (talk) 10:33, 30 August 2022 (UTC)
 * user:पाटलिपुत्र Please view Agrawala's reconstruction of the wheel Figure 6a.  Examine the dark gray wheel fragment (the actual one) at the bottom left. What is the ratio of the spoke diameters there to the distance between the centers?  So again: if  D = the width of the spokes when they meet the wheel at the bottom left and S = the average distance between their centers.  What is the ratio D/S?  Please do not tell me what the ratio is elsewhere on the wheel or what it is in the museum picture.  Only on the fragment on the bottom left in Figure 6(a).  What is the ratio D/S?  Fowler&amp;fowler  «Talk»  10:46, 30 August 2022 (UTC)
 * [[User:पाटलिपुत्र Please do not draw any more pictures. Just tell me what average D/S do you detect there?  Fowler&amp;fowler  «Talk»  10:58, 30 August 2022 (UTC)
 * (Perhaps I was too aggressive there by boldfacing some words. Apologies user:पाटलिपुत्र  for that.) First let me say: hat's off to you for making the painstaking illustrations. Not too many people on Wikipedia can do that on a moment's notice. So my compliments.
 * OK, in the absence of a response, let me just say that in my measurements the ratio of D/S in that fragment on the lower left was 1/4. But we know that S is 2.45 inches from the fact that the perimeter is 78.5 inches and there are 32 spokes; i.e. S = 78.5/32= 2.45. So if D/S = 1/4 and S = 2.45, D = 2.45 x 1/4 = 0.6121.  This means that the radius of these holes in 0.6121/2 = 0.304 inches.
 * In the top right fragment, D/S is more like 1/3, i.e. D = 2.45 x 1/3 = 0.815. In one instance, D/S = 2/5 = 0.4 and D = 2.45 x 0.4 = 0.98 which is close to your estimate.  But regardless, the total area of these holes does not seem to be much more than 10% of the area of the internal rim. That is really too little for it to be considered a  cog and wheel scenario. There is a much simpler explanation in the elliptical distortions section below.  Best,  Fowler&amp;fowler  «Talk»  12:42, 30 August 2022 (UTC)
 * Thanks User:Fowler&fowler. I see what you mean, but the black-and-white image is so poor quality and so "eaten up" by shadows that large portions of the spokes disappear in the dark, especially in the bottom left corner. You can see that by how off-centered they are, compared to their intercalary knobs. Also, I am afraid we cannot assume that Agrawala used vastly different spokes in the same wheel for his reconstruction: all the spokes in the photograph are the same... So your higher number 0.98 (the one you see on the clearest spokes) is most probably the right one. I have found a much sharper photograph of the Sarnath Museum remains (below), and it gives a much better sense of the mortise holes between the intercalary knobs, among other potential new data. Best पाटलिपुत्र  Pataliputra  (talk) 13:22, 30 August 2022 (UTC)

Here is the reason. I can't go into all the details, but the picture is the result of geometry applied to the picture File:Schematics of the remains of the topmost wheel of the Sarnath Lion Capital of Ashoka, in the Sarnath Museum.jpg That in turn is based on  this flickr image. The reference "clips" is to the flickr image. The point is that circles of curvature become ellipes when an object is viewed at an angle. The image is an example as the camera was held well above the center of the wheel. It explains why one radius D is 1.538 times 25 inches = 28.845 inches, i.e. approximately equal to 29 inches = 2 ft 5 inches. Fowler&amp;fowler «Talk»  22:19, 29 August 2022 (UTC)
 * The elliptical distortion
 * So you're saying that the differences in curvature you were seeing are simply a result of the position of the camera? It would make sense: the higher the camera, the flatter the curvature, becoming a straight line at the vertical... This would be good news and finally solve our riddle. Or is your conclusion different? पाटलिपुत्र  Pataliputra  (talk) 04:31, 30 August 2022 (UTC)
 * user:पाटलिपुत्र You have the insight. It needs a little tweaking. I am saying (as you are saying as well) that we can't really fit perfect circles to the rim diameters in our images.   This is because circles made in the museum when viewed at angle (because the cameraman is tall) will appear as ellipses in the images.  But which ellipse do we fit? Well, there is one ready-made for us.  It is the one formed by the "clips" in the Sarnath Museum picture.  These are the strips of metal (or some other material) that the fragments are affixed to.  If we then draw  a confocal ellipse (i.e. of the same shape) and fit it to the lower rim, its inner radius of curvature is 1.538 times 25 inches = 28.845 inches = approximately = 29 inches.  (Earlier I had been drawing perfect circles to fit the rims.  One such is the red one in the image.  It radius was turning out to be higher.
 * I think what Agrawala means is simply that the when he sat down with a his instruments to draw circles to fit the various rims, he found that the outer radii were all the same and = 2 ft 8 inches = 32 inches, but the inner radii were 2ft 1 inch in one fragment and 2 ft 5 inches in the other(s). There is really not much more he can say after that.  Whether they were parts of the same wheel and this feature was deliberate (to accommodate something or other) or unwitting (by a stone mason temporarily veering off the pattern) or whether they were parts of different wheels, neither he nor we can say.  So, it is not really a reconstruction, but a wheel that best fits the fragments found at Sarnath.  Fowler&amp;fowler  «Talk»  11:53, 30 August 2022 (UTC)

A new photograph of the wheel
Here's a new, free, and much sharper image of the wheel, with a slightly different angle. Hopefully, this can help us further move forward! Best पाटलिपुत्र  Pataliputra  (talk) 13:09, 30 August 2022 (UTC)
 * user:पाटलिपुत्र Wow. Fabulous.  How did you find it? My compliments.  You are right the camera is held a little lower.  So, I doubt that the elliptical/projective distortions will be substantial.  Let me do a little experimentation and I'll get back soon.  Thanks again.   Fowler&amp;fowler  «Talk»  14:31, 30 August 2022 (UTC)
 * Food for your Canny edge detector :) ... It was a bit of a long shot, but sometimes it works... पाटलिपुत्र  Pataliputra  (talk) 15:17, 30 August 2022 (UTC)
 * [[File:Buddhist Wheel of the Law surmounting the Lion Capital of Ashoka-- best fit of fragments in a circle at the Museum of Archaeology Sarnath.jpg|thumb|right|Buddhist Wheel of the Moral Law surmounting the Lion Capital of Ashoka--best fit of fragments in a circle at the Museum of Archaeology Sarnath, annotated by [[User:Fowler&amp;fowler| Fowler&amp;fowler ]] «Talk» 15:27, 30 August 2022 (UTC)]]
 * So here are the results. According to Agrawala, the inner diameter of the wheel is 2 ft 1 inches to 2 ft 5 inches.  2 ft 1 inch = 25 in; 2 ft 5 in = 29 in.
 * 29/25 = 1.16.
 * This picture is still not centered, ie. the camara was not as the same height as the center of the wheel, but it was only a little higher.
 * On my screen, the results are given. They are quite close.  The fragment on the right is in two pieces, they each have a slightly different curvature.  If you fit the top fragment with yellow, it low balls the bottom; if you fit the bottom fragments with the green it high balls the upper.  But otherwise, the figure explains why Agrawala says the internal diameters are between 29 inches and 25 inches, ie their ratio is 1.16 and above.  Fowler&amp;fowler  «Talk»  15:27, 30 August 2022 (UTC)
 * Can you confirm the meaning of your green and yellow circles?
 * Hummm, with this photographic definition it seems that the bottom fragment doesn't belong: different surface texture, different width?, different curvature? even different shape? Could this be the reason why we seem to see only the two top fragment groups in Agrawala's reconstruction ? Could a case be made that the two top fragment groups, only, belong to the same wheel?
 * For the top two fragment groups, the edges clearly seem to be rounded. We probably have to take "shading distortion" Deformation_due_to_directional_lighting.jpg into account (powerful top light) to determine the most probable curvatures (from a topographical standpoint, ie points of curvature at the same elevation). For example, the "edge line/shadow line" at the top of the right group probably cannot be taken at face value (nor the top left, possibly to a lesser extent)... पाटलिपुत्र  Pataliputra  (talk) 16:32, 30 August 2022 (UTC)
 * The top right fragment is in two parts. It has a more significant fracture than the bottom fragment or the top left fragment. If you fit a circle to the top right fragment's lower half (the circle is in green) it rides above any reasonable edge for the upper half.  If you fit a circle to the edge of the upper half (in yellow), it lies a fraction below the lower half's edge.  If you had to approximate a circle, its radius would be the mean of those two. But the main thing is that the radius of curvature of the lower fragment's inner rim, is lower than the other two.  I don't know why that would be the case. The lower could be from different wheels, or maybe it was designed that way for some reason.  Another possibility is that this is not the configuration.  Perhaps it was on the side and the other two were on the top and bottom.  That would made the wheel oblong like an egg, but from that great height it would appear as a circle to people on the ground level nearby. Maybe the wheels were temporary and they were changed for various reasons.
 * As for the shading difference, it is one reason why I have two circles (green and yellow). It is clear from the green that whatever is the edge, it lies below it. Same with the yellow, whatever is the edge, it lies above it.  Neither are anywhere near the red in curvature.  I have other things to do, but I feel more confident that this is what Agrawala meant. Best,  Fowler&amp;fowler  «Talk»  16:48, 30 August 2022 (UTC)
 * [[File:Edge detector applied to Sarnath Museum image showing different inner curvature of the three fragments of the Lion Capital of Ashoka.jpg|thumb|right|An edge detector applied to an image of the surmounting wheel atop the Lion Capital of Ashoka in the Sarnath museum. It shows that the lower fragment is less curved in its inner rim than the other two fragments. I made a slight error in the annotation. It should have been < 1.16.]]
 * And here is the edge data. user:पाटलिपुत्र you are cknowledged in the file description. Best  Fowler&amp;fowler  «Talk»  16:55, 30 August 2022 (UTC)
 * And here is the edge data. user:पाटलिपुत्र you are cknowledged in the file description. Best  Fowler&amp;fowler  «Talk»  16:55, 30 August 2022 (UTC)

External diameters
Now that we have a sharper image, I have the impression that the external diameters are different too: bottom rim (around 15cm external diameter on my screen) and the top rims (around 13 cm on my screen), that is a 15% variation: see tracings. It would be rather logical, since the internal diameters do seem to differ (above), while we cannot see significant variations in width of the rim on any fragment (nor would variations in width of the rim be likely in normal wheel designs).... User:Fowler&fowler, do you have a different analysis of these measurements? पाटलिपुत्र Pataliputra  (talk) 20:01, 30 August 2022 (UTC)


 * [[File:Buddhist Wheel of the Law surmounting the Lion Capital of Ashoka-- best fit of fragments in a circle at the Museum of Archaeology Sarnath-outer and internal diameters.jpg|thumb|right|This camera is too close to the fragments. As a result, the real outer boundary, which is defined by the "clips" to which the fragments are affixed is occluded by the fragment edges.]] (continued from the caption) If the camera was held even closer, it would eventually even hide the circle defined by the boundary between the brown and the beige/tan in the recessed space. So, in terms of thickness, the lowest fragment is also the least thick, and the one that varies the least in thickness along its length; then the one on the top left (defined by the purple circle), whose thickness varies quite a bit, and finally the top right (green circle) which is the thickest, and whose thickness varies the most.  The differences account for what we see, and what Agrawala was very likely looking it.
 * He must have had all three fragments; otherwise, he would not have said, "the internal diameters vary from 21 inches to 25 inches" or words to that effect. He would have simply said, "The internal diameters are 21 and 25." Why the three don't show as much as they do in the museum picture, I can't tell.  Perhaps as you said, the museum picture is taken from the flip side. My personal feeling is this wheel, if indeed it is one wheel, was not carved with geometric precision, why I can't say.  It might have been deliberate for the reasons I indicated above, or it might have been a temporary error later amended, which in any case would not have been detected from the ground. Anyway, this is as far as I want to go with this topic. It was enjoyable discussing this with you.  Fowler&amp;fowler  «Talk»  23:27, 30 August 2022 (UTC)

An alternative interpretation
In light of our recently obtained photograph, I have a slightly different interpretation. The wheel remains in the Sarnath Museum seem to belong to two different wheels: they are different in size, shape and surface texture. You were right when you pointed out that the dimensions of the remains did not match. But the Mauryans did not build wobbly wheels or wobbly circles.... they built quite perfect ones, as can be infered from their other creations: the 24-spoked wheels on the abacus, the smooth and cylindrical shape of the Pillars of Ashoka, the ringstones, the rounded ceilings of the Barabar caves are all dimensionally perfect, and the craftmanship of the Lion Capital of Sarnath is otherwise outstanding. Commenting on Mauryan sculpture, John Marshall once wrote about the "extraordinary precision and accuracy which characterizes all Mauryan works, and which has never, we venture to say, been surpassed even by the finest workmanship on Athenian buildings". So here too I tend to think that, rather than a single wheel with wildly varying dimensions, we instead have the remains of two well-engineered, very regular wheels, which properly accounts for the discrepancies in measurements. Hopefully more research will be provided by Secondary sources. Thank you for the discussion! पाटलिपुत्र Pataliputra  (talk) 04:30, 31 August 2022 (UTC)
 * 1) The first, smaller, wheel can be reconstructed from the top two rims (4 fragments in all). Its surface is smooth and rounded (tapering towards the periphery). It is the wheel which was reconstructed by Agrawala with just these two rims (4 fragments) as seen in his photograph (with the possible addition of a very small fragment seen on top of the photograph, but not in the museum). The width of the rim was rather precisely measured by him: 3.65 inches, so that it can be used as a measuring stick for the other parts. By then measuring the curvature of the remains, we can determine that the external diameter was 32 inches (2ft 8ins), as affirmed by Agrawala . The inside diameter is 25 inches (2ft 1in), and a root diameter of 29 inches (2ft 5ins) can also be defined (mortise bottom to mortise bottom, see diagram). The spoke holes in the rim were of the order of 0.9 inches in diameter, separated by intercalary knobs. The spokes were 32 in number and inserted 2 inches deep inside the rim. They were likely rombhus-shaped as suggested by spoke remains, and shown in reconstructions.
 * 2) The second, larger, wheel can be reconstructed from the bottom rim fragment seen in the museum photograph. Its surface is rough, its section is squarish, the width of its rim is around 3 inches. The measure of the curvature gives an outside diameter of about 35 inches (2ft 11in, about 10% larger than the smaller wheel), and an inside diameter of 29 inches (2ft 5ins: this value might also correspond to the second internal diameter of 2ft 5ins given by Agrawala in his measurements, but in this case he would also logically have mentioned the larger external diameter of 2ft 11in, which he doesn't. Agrawala apparently did not integrate this rim fragment into in his reconstruction). This wheel could belong to a second, larger, capital mentioned by Asher, some remains of which were also discovered in Sarnath.


 * User:पाटलिपुत्र I'm sorry I can't engage in that speculation and don't see valid grounds for it. By this I mean, that I certainly do not agree with your interpretation about holes being described as giving rise to "root diameters"  and the vast (90%) hole-less rim being described as having the "tip diameter" without saying a word about the depth of the hole.  Simply not in the cards.
 * It doesn't mean that the Sarnath wheel was wobbly. It might have been a wheel like the late Gupta age one, thicker in the bottom half, and angled for greater stability. Or as I have previously stated, in order for it to appear like a perfect circle from the ground level, it might have been oblong in shape with the upper rim thicker. That way it would look like a circular rim of even thickness from the bottom (as a result of the foreshortening).
 * As for John Marshall, he said many things during his long career. Our influences section says, "In 1922, John Marshall was the first scholar to suggest that the Sarnath capital was the work of foreign artisans working in India. Comparing the capital to a male figure from Parkham, Marshall wrote, “While the Sārnāth capital is thus an exotic, alien to Indian ideas in expression and in execution, the statue of Pārkham falls naturally into line with other products of indigenous art and affords a valuable starting point for the study of its evolution." Unlike you I don't have any fixed notions about the perfection of the Mauryas or any other political dispensation that has ruled over parts of India or the world.  Fowler&amp;fowler  «Talk»  10:34, 31 August 2022 (UTC)  Fowler&amp;fowler  «Talk»  10:45, 31 August 2022 (UTC)
 * PS Also User:पाटलिपुत्र in any future exchange with me, please do not draw your own pictures in which you have smoothed out the objects. Only show your circles on the raw image, preferably with a thin linewidth. And please show the whole image. Your claim that the lowest fragment has external diameter 35 inches whereas the others have 32 is false. Please show me in the raw image how you have fit the circles to the various fragments in order to obtain those results.  Fowler&amp;fowler  «Talk»  15:14, 31 August 2022 (UTC)
 * Sarnath fragments external diameters.jpg've already provided you with the image of my attempt at external tracings above, drawn to the best of my ability, but certainly approximate. I think they're self-explanatory. Please tell me what different readings you might have, I'm interested. And please kindly tone down a bit, our exchanges are supposed to be civil and I'm not your housemaid nor your punching ball. पाटलिपुत्र  Pataliputra  (talk) 16:34, 31 August 2022 (UTC)
 * I would appreciate if you did strike the "Unlike you..." sentence above, as well as the expression "in any future exchange with me". पाटलिपुत्र  Pataliputra  (talk) 16:46, 31 August 2022 (UTC)
 * Struck, apologies. Fowler&amp;fowler  «Talk»  02:30, 1 September 2022 (UTC)
 * You have exaggerated the red and reduced the black and blue. That alone creates this effect and if corrected renders false your notion that they have vastly different external diameters of ratio 32:29. I will show you in a minute I will not bother with the top right wheel as it is in two parts with a significant break such that their boundaries together do not form a circle. Fowler&amp;fowler  «Talk»  21:37, 31 August 2022 (UTC)
 * Sarnath fragments external diameters-better fit.jpg I have attached the picture. Have not bothered with the top right wheel.  The lime green is a better fit for the bottom circle and the pink better for the top left. Also as the camera position is in the top half of the circle somewhere near the spoke fragment, there is more significant elliptical distortion of the lower fragment than the upper two; in other words a better simulation will require an flattened ellipse for the lower fragment and a normal circle for the upper two. These fragments have different inner rim curvature, but essentially the same outer rim curvature.  This does not prove that they are necessarily from different wheels, but it doesn't rule out that possibility either. The idea that Agrawala was writing in the jargon of cogs and wheels is not credible, not even remotely.  Fowler&amp;fowler  «Talk»  22:21, 31 August 2022 (UTC)
 * I thank you user:Pat for finding this new higher focus image. The ultimate verification will depend on the availability of an image taken with a telephoto lens from across the room in the museum and with the camera held at the same height as the center of the wheel; in other words, so that perspective projection may be reduced to orthographic projection. Until then, I will be moving on to other topics which require my attention.  Fowler&amp;fowler  «Talk»  23:37, 31 August 2022 (UTC)
 * The problem, of course, user:पाटलिपुत्र, as you must be aware, is that sometimes it is hard to get away, the problem keeps churning up in the brain. Well I found this circular cropping tool and it seems to verify what I have been thinking that the problem is with the upper half of the top right fragment.  As you can see it has a significant fracture and has been fixed with some cement or some adhesive.  I don't know that it has been done correctly, or maybe it has deteriorated over the years.
 * Circular cropping applied to the Sarnath surmounting wheel.jpg

Number of spokes

 * It seems like it is veering inwards as you move up the wheel after the fracture, almost falling off the brackets that hold it. The other parts, the bottom and left especially fit pretty well. Looking at the circular crop, I am now wondering about the 32 spokes. I have a feeling there might be more.  Nothing checked as yet but just a feeling. Best,  Fowler&amp;fowler  «Talk»  01:22, 1 September 2022 (UTC)
 * Sarnath wheel does it have 36 spokes.jpg
 * user:पाटलिपुत्र 360/32 = 11.25  That means that if there was space for 6 spokes, they should take up 6x11.25 = 67.5 degrees.  But the 6 spaces below seem to span just 60 degrees, maybe 61 max. Three spaces should span 33.75 degrees, but they span only 31 degrees. Five spaces would subtend 56.25 degrees, but here they do only 51 degrees.  This seems to be much closer to 10 degrees for each spaces, i.e. 36 spokes. I will double check, of course, and you also please double check, but something is not adding up. Best,  Fowler&amp;fowler  «Talk»  02:27, 1 September 2022 (UTC)
 * Of course, one possibility is that in File:Buddhist Wheel of the Law surmounting the Lion Capital of Ashoka-- best fit of fragments in a circle at the Museum of Archaeology Sarnath-outer and internal diameters.jpg as the green and purple radii are smaller, their centers will be closer to the rim, which means the angles will be greater, whereas the red radius being larger will for sure have 36 spokes. It would definitively prove that the top two and the bottom are not parts of the same wheel. Will check this later. Fowler&amp;fowler  «Talk»  03:12, 1 September 2022 (UTC)
 * Checked but did not really help. Well, it did in the top right fragment, but that is broken and therefore more curved anyway, but did not in the other two. Time to go to bed. Fowler&amp;fowler  «Talk»  04:00, 1 September 2022 (UTC)

Thank you for the nice attempt at calculating the number of spokes. I hadn't realized that the quality of the picture now allowed us to make angular calculations. But then you have to take into account that your three internal circles matched to internal rim edges are non-confocal. We have to use each circle's center to measure the actual angle. With your internal circles, the top two fragments indicate a wheel of 32 spokes, while the bottom fragment corresponds to a wheel of 38 spokes. I suspect this is another reason why Agrawala only seemed to use the top fragments for his reconstruction. Picture attached. पाटलिपुत्र Pataliputra  (talk) 05:24, 1 September 2022 (UTC)


 * What are you not reading? I already mentioned that above. And tried it. But the rims are at different heights. You have to fit the circles to where the knobs are.
 * If you do that, the bottom is still 36, top left is 35.5, and the top right is about 30. The only thing that is sure is that the bottom fragment has 36 spokes. The top left is much closer to the bottom than it is to the top right. Fowler&amp;fowler  «Talk»  10:43, 1 September 2022 (UTC)


 * Hi "What are you not reading?"... did you get up on the wrong side of the bed this morning? Actually I'm afraid your comment is an impossibility from a geometrical standpoint: your circles along the inner edges of the rim (which I adopted for this simulation) have to be concentric with the circles along the knobs just below. Otherwise it would mean that there are wild variations in the thickness of the rim, which is contrary to observation and contrary to the statement by Agrawala (regular rim thickness of 2.62 inches). And depending on the knobs for the purpose of drawing a circle is highly unreliable since the knobs have become slightly irregular with time (heights, thicknesses) and are semi-hidden in the shadows of the photograph. And the variation of elevation of the rims between fragments, if there is any, is necessarily tiny, and anyway would not affect our calculations since we are perpendicular to the wheel. Since rim edge circles and their knob circles are necessarily concentric, then the angular readings are necessarily the same, so we are still at 32 spokes for the two top rims and 38 spokes for the bottom one. Unless I am missing something in your reasonning? पाटलिपुत्र  Pataliputra  (talk) 11:42, 1 September 2022 (UTC)
 * What I meant by what is it you don't understand is that I already stated last night that I did that experiment. I stated the circles will have different radii. So what is it you did not understand that you are pretending this is all your new idea. And also please don't go off again about Agrawala only including two pieces. His is the same as ours.  If you examine the new picture's bottom fragment, only a little portion is the sandstone. It has the same shape as in Agrawala. The rest is cement.  Fowler&amp;fowler  «Talk»  11:49, 1 September 2022 (UTC)
 * Sarnath wheel 36 and 24 spokes.jpg In this picture if you examine the bottom fragment only a small portion in the middle has the characteristic grai of the other two fragments. It also has the tell-tale inverted S shape on one side.  The same shows in Agrawala.  Fowler&amp;fowler  «Talk»  11:42, 1 September 2022 (UTC)
 * I mean characteristic grain Fowler&amp;fowler  «Talk»  11:50, 1 September 2022 (UTC)
 * So the bottom line is that what they have at the Sarnath Museum is the same as what Agrawala has in his book. We can't on our own pretend the pieces belong to different wheels.  If we then choose the center the ASI(?) has chosen for the reconstruction, in File:Sarnath wheel does it have 36 spokes.jpg, the wheel clearly has 36 spokes.
 * Fowler&amp;fowler «Talk»  12:01, 1 September 2022 (UTC)

Another proposal
Thank you for identifying the bottom fragment in Agrawala's picture! This shows that, in the museum photograph, we are seeing it tilted at a rather great angle, which explains why it is odd-shaped, the rim seems narrower (3 inches instead of 3.5 inches for the others), and the inside of the rim seems to be much wider. But when using the photograph from Agrawala, which is properly centered and properly positions the fragments in a cast, it becomes obvious that this fragment (which has the same characteristic marks as the bottom fragment in the museum as you noticed) has the same width and same shape characteristics as the two others. So now we have no reason to doubt that the three pieces belong to the same wheel, as in Agrawala's photograph. Thank you again. With the new perpective for the bottom fragment, we now can see that all fragments have approximately the same curvature and the same spacing for the knobs (see image). When measuring the angle from the geometrical center of each of the segments and counting the intercalary spaces, we can see that they all belong to a wheel with 24 32 spokes (my calculation above ). The only remaining issue is that, in the setup of the Sarnath Museum photograph the three pieces are non-confocal: they are set a little bit too far away, and their centers do not meet. When pushing them a little closer to make their centers coincide, it becomes apparent that they share the same circumferences, and also that the knobs conform perfectly to the geometric shape of a 32-spoked wheel, as in Agrawala's photograph (see image below, and Agrawala ). The resulting wheel is 2ft 8 inches in external diameter and 2ft 5 inches in internal diameter, and properly has 32 spokes as illustrated by Agrawala. All other measures conform to his description, so our source was right after all: I think this resolves the issue. Please do not hesitate to discuss should you have remaining issues with this interpretation. Thank you again for this discussion. पाटलिपुत्र Pataliputra  (talk) 15:03, 1 September 2022 (UTC)


 * user:पाटलिपुत्र I am afraid your argument is not quite valid. We already know that the inner rim of bottom fragment has lower curvature than the inner rim of the upper two fragments. Their outer rims however have roughly the same diameter.  That means that the the circles that best fit the three inner rims, i.e. the osculating circles will have different centers but also that the osculating circles of the three outer rims are the same one circle.  But you cannot "push in" fragments to make them concentric, for their radius will change&mdash;both inner rim radius and outer rim radius&mdash;as will their curvature = 1/radius of osculating circle.
 * You might make their centers fit, but they won't have the same outer radius, and therefore will not be part of one wheel. The only thing you can say is that:
 * "Conclusion The wheel that best fits the fragments has been constructed at the Sarnath museum. In that configuration, the inner rims of the old fragments point to a wheel of 36 spokes." Fowler&amp;fowler  «Talk»  16:08, 1 September 2022 (UTC)
 * PS It maybe that the anomaly has resulted from the distortion in the image from the perspective projection (foreshortening). That can best be decided by the availability of an image taken from a good distance away and aligned to the same axis as the camera.  A telephoto lens will help as the details can then be seen.  Fowler&amp;fowler  «Talk»  16:19, 1 September 2022 (UTC)
 * PS 2 Agrawala's reconstruction is poor. Some sectors (angular wedges) of the wheel have a spoke rate of 36; others have 24.  Fowler&amp;fowler  «Talk»  16:30, 1 September 2022 (UTC)
 * PS3 I guess what worries me now is: who "completed"/changed the bottom fragment from its original state (marked 6673 and showing the grain) to the version we see with sharp edges? Was it done by Agrawala when preparing the "cast?" If so, maybe it is plaster of Paris and not cement, but regardless, is it acceptable archaeological practice: to damage an archeological artefact for making a nationalistic point? Fowler&amp;fowler  «Talk»  16:34, 1 September 2022 (UTC)


 * You write "to damage an archeological artefact for making a nationalistic point?": Better not to make personal attack on past scholars too... Let's focus on the facts.
 * 1) We know that the apparent curvature of a wheel will change with the view angle. Here, in addition to a level of foreshortening, the fragment is visibly tilted toward the front (20-30 degrees I would say), so this would explain the apparent wider curvature mechanically. Here are a few pictures from far away that also tend to confirm the forward tilting of that portion: ... bad museum setup I'm afraid. This one is fairly good and balanced though ... before the forward slide?
 * 2) My reconstruction obtains the convergence of centers using the inner cercles you have established for the top rim fragments, and the inner cercle of the bottom rim fragment from Agrawala's photograph, and this convergence results in a near-perfect match of the knobs to a 32-spoke geometrical wheel above, so the probability is very high that this is indeed the proper configuration (geometrically here, getting a match basically means it is the right configuration).
 * 3) Besides your recent idea that there were 36 spokes, I have never seen this claim a single time in literature (wheareas many references describe the 32 spokes -listed above ). There is a high risk that this could be the result of taking a falty museum setup too much at face value (and it also implies having a wobbly wheel made of three non-concentric fragments, if we are to follow the exact museum setup, which is highly unlikely).
 * 4) I think that in all probability the bottom fragment is original Chunar stone, because had they been willing to use cement they would have done it for the whole wheel.... why stop with one piece? (especially since it does not bring anything new to the reconstruction, for example the 32 spokes are already easy to demonstrate with the top fragments only). Your original drawing Fragments of big wheel Lion Capital Sarnath found by FO Oertel 1905 mentions "Fragments of big wheel Lion Capital Sarnath found by FO Oertel 1905 (based on ASI reports and pictures)". Do you know the dates of the reports/pictures in question (that is, are they anterior to Agrawala)? Wouldn't they help establish the antiquity of the fragment in their specific shape? Are there any pre-Agrawala graphical sources describing these fragments? Best पाटलिपुत्र  Pataliputra  (talk) 17:22, 1 September 2022 (UTC)
 * user:पाटलिपुत्र
 * First let me congratulate you on finding picture 162; it is a brilliant find. It is brilliant not just because it is taken from a distance (that is a definite plus), but that it is the first image thus far in which you can see the knobs of the left upper fragment clearly.  So, now let us consider the problem. First, we will transform the problem to:
 * New Problem: how many knobs are there in the wheel? (For the number of knobs = number of holes = number of spokes). That is the problem we will solve, and I will work with the golden image in 162.
 * A: Convention: For simplification, I will refer to the bottom fragment and its associated structures and constructions as "red," (because that was the color of the initial osculating circles); the top left fragment as "purple," and the top right as "green."
 * B: In order to solve it, we will fit osculating circles to the (horizontal) bases of the knobs. As they are also situated vertically lower in the wheel, there will be less perspective effect.
 * C: I have just carefully fitted three circles to the three fragments' knobs. The good news is that they are almost concentric.  The red and purple have almost identical centers and the green is not far to the south-south-east across the museum's center.
 * D: The outer rims fit in one circle (within milimeters).
 * E: So we don't really need to move any center, or make any deforming transformations. They are close enough.
 * F: I measured the red angle from the leftmost knob to the fifth over to the right (i.e. a total of 6 knobs; I did not bother with the crumbled one). The total angle between these six is a clear 50 degrees, giving 10 degrees for each space.
 * G: I measured the purple angle. There are five knobs clearly visible. I measured that angle and it was a clear 40 degrees, and again giving 10 degrees for each space.
 * H: Now the green. This is the problematic one. There are two issues:
 * first: now that you can see the purple clearly, it is obvious that the red and purple have clear ridge-like inner edges. The green does not.  It starts out as a ridge but after the fourth knob the edge begins to round out progressively and by the time it is at the top, the ridge has disappeared and you see the shading effect. This has nothing to do with the lighting overhead. For the purple also receives the same overhead lighting and the ridge is visible until the top end (clear as day)
 * second: the green has a fracture, a significant one which begins as a single one on the inner rim and splits into two midway to the outer. How this has been repaired is not clear, but it has resulted in the rim being bent a little inward (toward the center) upward of the crack. That is why if you fit a perfect circle to the green outer rim (below the crack) it will miss the rim above the crack; vice versa, if you fit a perfect circle to the green outer rim above the crack it will miss the rim below the crack, not by much but the difference is clear. Note: for the outer rim, there are no edge or shading issues.
 * first conclusion the inner green rim has slightly higher curvature than the inner red or purple because of the fracture. Also, the green inner edge is hard to figure out where the rim itself becomes rounded, so:
 * second assumption: to avoid these complications, we will measure the green angles from the first knob to the fourth just above the crack.
 * I: There are four green knobs and the angle between them is a clear 30 degrees.
 * K Final conclusion: For all three color of knobs, red, purple and green, the angle between the knobs is a clear 10 degrees. Therefore there are 36 knobs and 36 spaces between them. QED
 * Some considerations:
 * L: What is happening to the green rim above the crack. It seems that it is becoming flatter, a little wider and rounding out on the inside so the edge is no longer clearly distinguishable. I don't know the reason for this: I could conjecture a few reasons but I would rather not.
 * M: What about the green knobs above the crack? I did not bother with them because if you fit a circle to them, the center moves away from the previous three centers, the red, the purple, and the green (which were close enough), creating additional issues.
 * N: To the objection why no one has noticed this before, all I can say is I don't know. I could hazard semi-serious but semi-true guesses, but they would detract from my main argument.
 * O Please read my argument above carefully, discarding all previous assumptions. I have made the calculations on image 162 very carefully.
 * Best, Fowler&amp;fowler  «Talk»  21:16, 1 September 2022 (UTC)
 * Your argument (2) above is false. That much I am entirely sure about.  You are welcome to take it to a mathematics board.  Fowler&amp;fowler  «Talk»  21:59, 1 September 2022 (UTC)
 * But please tell me when you do post there. Fowler&amp;fowler  «Talk»  22:11, 1 September 2022 (UTC)
 * [[File:Best argument for 36 spokes in surmounting wheel of Lion capital of Ashoka in Sarnath.jpg|thumb|right|The final proof that there might be 36 spokes in the surmounting wheel of the Lion capital of Sarnath]]
 * user:पाटलिपुत्र and (as you are here, user:Johnbod) To the right is the image we already have on WP. It is the picture of the extant fragments of the surmounting wheel of the lion capital at the site museum in Sarnath. The following is the final evidence.
 * Theorem There are 36 spokes in the surmounting wheel Lion capital of Sarnath.
 * Proof As above, we will count the spokes by counting the little knobs (they are actually half disks) that stick out of the inner rim of the wheel fragments. The spokes would have met the rim between the knobs.  I have drawn two red circles, one fitted to the upper edge of the inner rim of the bottom fragment and the second (inside it) fitted to the base of the knobs.  Note that the first circle rides too high for the top fragments, i.e. it missed their inner edges. The second circle, though does not do too badly on the knobs of the top left fragment  and the bottom half (below the crack) on the top right fragment.  (I could make it a little smaller and fix this, but it is way past my bed time here, so you will excuse me.  The rest can be seen in the picture.  You can bring out your protractor and measure the angles.  The angle between the knobs is a clear 10 degrees, making 36 knobs in all.  But here is the:
 * Clinching evidence: The Sarnath Museum has its own display on the wall behind the wheel. In tan and white they show the 32 spokes, knobs and the spaces between. Please examine the bottom fragment in the picture and start counting from the left. Do you notice how  their numbering is running ahead? i.e. by the end of the fragment they are almost a full space ahead, i.e. by the time we are ready to count our seventh, they would have only counted six.  You do this once every 90 degrees, and lo and behold, a 36-spoked wheel looks only 32-spoked. QED
 * I suspect that this is why Oertel wrote "apparently 32 spokes" He most likely determined the total number of spokes from the limited evidence to be considerably larger than the 24 on the abacus's rim.  He and/or other then reasoned on grounds of tradition that if it is more than 24, then it must be the next multiple of 8, i.e. 32, discarding the evidence for 36 to be outliers.
 * Fowler&amp;fowler «Talk»  06:24, 2 September 2022 (UTC)
 * As for your question about the picture I received, Pat. It is not older than Agrawala's.  It is probably 20 years old, max.  It did not have the display on the wall behind, so its older than the current set of Flickr pictures.  Fowler&amp;fowler  «Talk»  06:30, 2 September 2022 (UTC)
 * Unfortunately, I don't have the original image file any more. I was requested to delete it after my use. I now regret this.  At that time I had less interest in the spokes question.  Fowler&amp;fowler  «Talk»  06:58, 2 September 2022 (UTC)

The issue of non concentric fitted circles
Thanks for the detailed answer Fowler&fowler! 1) First, to understand the extent of the problem, let's remember you are talking about 36 spokes, and the general claim is that there were 32 spokes : I'd like to point out that this is a difference of 4 additional spokes on the whole wheel, one additional spoke per quadrant (+12.5%). What I mean is that a slight error in assumptions or measurements will also create this kind of difference. 2) Let me also point out that for the first time in a photograph we are seeing a very close correspondence between the knobs and the painted 32-spoked background. We can see the intention of the museum is clear, but the knobs are still spread a little bit too close to fit perfectly that "ideal" wheel. 3) I think your measurements are probably right when assuming the Sarnath Museum layout of the fragments is absolutely exact (nice to see your appreciation of ASI's work at this point!). The problem is that the three fragments in the Sarnath Museum as they are laid out are not concentric (left drawing ), just one internal circle tells us that: they are positionned a bit too far appart. So your 36 spokes are inseparable from the fact that the wheel would be a "wobbly wheel" with at least three different centers (in effect a sort of daisy, where the three round segments are positionned too far appart from their concentric circle). So, in a word, if we assume that the Sarnarth wheel was a "wobbly wheel", then, yes, there were 36 spokes (drawing attached, slightly exagerated for the sake of illustration). 4) Let's imagine for a moment, just for the sake of argument, that the Sarnath wheel was originally a proper, regular, concentric wheel. In that case we have to move the fragments a little bit closer to the center, so that each of their centers match (the width of one or two knobs towards the inside as seen from figure 162 ). Once we have this regular, slightly smaller circle, we can immediately see that all the knobs match perfectly with the painted 32-spoked background. So, in a word, if we assume that the Sarnath wheel was a regular wheel, then the number of spokes was 32. 5) Now maybe we'll never know... the Sarnath wheel may have been wobbly, or it may have been regular. It is after all a matter of conjecture, as, technically, the Mauryans could have designed any shapes they wanted. It might be that there was indeed a "Great Wobbly Ashokan wheel", but maybe historian tend to choose the simpler solution of a regular wheel (how foolish and boring!). It is a bit like Copernicus against the theory of the Celestial spheres: the Celestial spheres worked, more or less, although with many convolutions and strange arrangements. But in the end, everything was actually much simpler: just a big (Ashokan?) sun at the center, with (roughly) concentric planetary orbits around it. पाटलिपुत्र  Pataliputra  (talk) 06:50, 2 September 2022 (UTC)


 * Final argument for 36 spokes-2.jpg Hello user: पाटलिपुत्र ! Great to hear from you.
 * With respect, may I request that you not draw your own interpretative pictures when I ask a question specific to one picture?
 * In the new picture on the right, I have drawn three osculating circles fitted to the bases of the knobs on the three fragments. Two have near identical centers (these are for the bottom and left fragments). The one on the right (circle in yellow) has a different center, but when you measure the angles subtended by the three rims, their sum remains the same.  For the bottom and left fragments the angles starting with the left fragment and proceeding clockwise are 40 + 50 + 30.  For the new center of the yellow circle, they are 39 + 50 + 31.  The sums of the angles are the same, and will remain the same in a transformation that deforms both circles into one.  That's all that matters to us.  If for 12 (=5 + 3 + 4) we have a total angle of 120 degrees, for 360 degrees we will have 36 spokes.  On the other hand if we had 32 spokes, the angle between them would be 360/32=11.25 degrees.
 * Therefore for 12 spaces between spokes, the angle would be 12x11.25 = 135
 * So again, please do not draw your own pictures instead please tell me what transformation will you be applying to the two dimensional plane R2 of the picture on the right so that the sum of total angle subtended by the fragments will change from 120 degrees to 135 degrees? The change 120 to 135 is not insignificant. May I request you again, to answer the question directly, not be drawing interpretative pictures. So again what is that transformation?  Can you specify it?
 * Best regards, Fowler&amp;fowler  «Talk»  08:39, 2 September 2022 (UTC)


 * Hi As previously pointed out, your new internal circles drawn from the bases of the knobs are contradictory with your previous internal circles which were built from the internal edges of the rims. Mathematically, these stucturally closely connected circles have to be concentric with one another . It seems also quite obvious that your initial circles were much more precise, being built from a comparitively clear and large topological feature (circular rim edge). A circle based on the bases of the knobs is necessarily affected by the fact that the knobs are more or less worn, more or less clogged at their basis, have different individual shadows and are in the most shadowy parts of the photograph, and may not be individually positionned as precisely as the inner rim edge is carved, since the edge at least is defined by a clear and wide geometrical curve. So I'm afraid we cannot consider these new circles as reliable, and I suggest you should revert to your original rim circles for analysis and angular calculations (and these calcutions showed 32 spokes , except for the lower edge which we now know is foreshortened and most probably tilted forward in the photograph, which gives a wider curvature and will mathematically increase spoke count).
 * Please try to understand my schemetic drawings above, because I trust that they explain clearly and geometrically why you were counting 36 spokes, when our sources count 32. Best पाटलिपुत्र  Pataliputra  (talk) 10:48, 2 September 2022 (UTC)
 * user:पाटलिपुत्र Thank you for your response. With great respect, I have made the calculations very precisely.  I have made them even more precisely on the "Golden" copyrighted image. The results are the same.  The inter-knob angles for the rim fragments under consideration are 10 degrees. Considering that I was the one who proposed the idea of wider base curvature and fitting circles, that the circles had different centers, that they might account for the fragments belonging to different wheels, and so forth, ... and eventually that that was false as the bottom fragment was the same as Agrawala's top, please give me some credit and some trust that  I would not be making those kinds of careless errors that would change 135 degrees to 120 degrees. I acknowledged your fabulous pictures and gave you full credit. So again, I request in earnest to listen to my argument, not mix it in with extraneous factors.
 * Yes, of course, they contradict the rim fitting data. They point to the fact that when the camera is held close to the picture, and neither vertically, nor horizontally centered, you will get foreshortening in the rim data that you will not in the knob data, (as the knobs are flat against the wall or nearly flat) and the foreshortening is asymmetric. That is what has happened in the picture.  Also the odd-man-out among the rims as I've said many times before is the "green" on the right, which is not only seriously fractured and misaligned, but is also much flatter than the other two. Its inner rim moreover is rounded above the crack and the rounding is intrinsic, independent of the lighting above. I am suggesting this is a serious issue, in the rim-based fittings but which disappear partially in the knob based ones.
 * And what do you make of the Sarnath museum counting on the wall behind, which a full 1 knobspace and 1/3 behind the actual knobs, over the same 120 degrees under consideration For 4/3 times 3 = 4. That's not an minor error.
 * I don't know when this spoke counting display was introduced. I don't remember it in 2015 or 2016 when I visited. I suspect it was brought out from the storage rooms and made a part of the display sometime after, probably the same time they allowed photography. Please keep looking and employ your excellent image finding abilities to find a copyright free image taken with a telephoto lens from across the room, and centered to the axis of the display.  That would be the ideal data. Maybe that image will prove my hypothesis false, but as yet I am suggesting the data and the museum's miscounting seems to suggest a serious case for 36.   Best regards,  Fowler&amp;fowler  «Talk»  14:22, 2 September 2022 (UTC)
 * PS You had mentioned User:पाटलिपुत्र, "Since rim edge circles and their knob circles are necessarily concentric, then the angular readings are necessarily the same,"
 * I don't believe this is true. When an image is deformed because of camera angle, parts of the image (in this case the rim) will be deformed convexly so as to become more curved and parts concavely so as to become less curved. The knobs however are positioned lower down, i.e. farther away from the camera. They experience less deformation. When you fit circles to the rim, you are measuring angles between the knobs by measuring their angles projected on the rim.  But when you straighten the camera, the knobs will experience less corrected deformation than the rim.  The old marking on the rim that we used to measure the angle between the knobs will now no longer show the angle between the knobs. i.e. they will be a little off kilter, and I am suggesting this is what explains the difference in angle measure from 135 or 130 to 120.  Fowler&amp;fowler  «Talk»  15:24, 2 September 2022 (UTC)

Coloured knobs
Hi You know my issues about the knob technique to make osculating circles, but I have no doubt about your sincere effort to find a solution. I checked your technique, and have come up with a slightly different approach: first, I contoured precisely and colored every single knob (image 1). Then, I compared this to you knob-based osculating circles (image 2): you will see that many do not match very well (especially several of the outlying ones, circled). Again, I'm not blaming anyone (it's simply difficult to find the right approach), and just trying to find the best solutions and move forward. Then I drew my own circles along these knobs (image 3): you will see that two of my circles come out much smaller (but the red one larger), and, curiously they are quite concentric with your previous rim edge-based osculating circles, which, mathematically, is good news (since our photographer was quite centered in respect to the wheel, I believe any foreshorting would have to be fairly uniform on short distances around the rim edge). I'm not drawing any conclusions yet, nor making any calculations (because I'm really not sure that the knob technique is good anyway), but could you check this "coloured-knob approach" on your side and see if it can help? (PS: I have no claim to perfection, so please double-check the contouring of the individual knobs too). Best पाटलिपुत्र  Pataliputra  (talk) 18:54, 2 September 2022 (UTC)


 * Again, with great respect, I have told you again and again, please do not create your own pictures, that means please do not add your own version of what the knobs look like. Just give me the raw data that you have fitted.  Again, please do not simply with your simulations, user:पाटलिपुत्र. So please show my your circles without anything else.  Fowler&amp;fowler  «Talk»  20:57, 2 September 2022 (UTC)
 * And you are aware that I have fitted only the bottom four on the right, because together neither the knobs nor the rim form a circle. I have checked the same on the "golden image.'  The right hand rim is fractured.  In the top left I have fitted the top four.  The fifth is not visible clearly.  I added that later.  Actually wait a minute, user:पाटलिपुत्र I will fit them again, and you can tell me how your results are different. So hold on.  Fowler&amp;fowler  «Talk»  21:15, 2 September 2022 (UTC)
 * [[File:Sum total of angles subtended from 7 sample points to 14 knobbed spaces of the fragments of the lion capital of Sarnath.jpg|thumb|right|The total angle subtended to 14 knobbed spaces (a knobbed space is the space between two consecutive knobs) from seven sample points within the inner hub.]] user:पाटलिपुत्र user:Johnbod.  Here is a completely different way of considering the matter. We won't bother with fitting circles.
 * Instead we will consider the angles made at a sample point in the hub by the three fragments (or rather the spread of the knobs in each fragment). i.e. geometrically we will consider the total angle subtended to 14 knobbed spaces (a knobbed space is the space between two consecutive knobs) from seven sample points within the inner hub in the Sarnath museum display. If there were 32 spokes this total angle would be 157 degrees.  If there were 36 spokes this angle would be 140 degrees. The average of 140 and 157 is 148.5  So again we can now forget about osculating circles. The center of the mahachakra has to be somewhere. It better be inside that inner hub; otherwise we are in serious trouble. So we take seven sample points as shown by the stars in the diagram. The average of 140 and 157 again is 148.5.  Here are the seven totals: White (144); Green (145.5); Orange (147); Red (147); Purple (142); Turquoise (143); Black (146.5); Average of 7 = 144.9 < 148.5 The values never venture into the 150s; If you consider only the first four knobs from the bottom in the right hand fragment, the average drops by two points to 142.9 were it extrapolated to 14 spaces; if you further consider only the first three knobs from the bottom in the right fragment, the average drops to 141.2 degrees if you extrapolated it to 14 spaces, and 141.2 is pretty close to 140.  The reason for doing this last is that the upper half (above the fracture) has been poorly joined and hangs/wedges into the big circle; in other words if you follow it, you will move along a spiral and eventually reach the center. For none of the sample points does the average get anywhere close to the mean let along cross it.
 * The mahachakra has to have a center, where are you going to choose it; for wherever you do in the hub the sum total of the angles the center makes with the fragments is what a 36-spoked wheel would make, not a 32. Added to this if you consider Sarnath museum's undercounting, described above, there is no doubt that the  number of knobs = number of spokes is 36, not 32.
 * If you have objections to sampling. I can do a proper mathematical argument by choosing polar coordinates for the hub and show that nowhere in the inner hub will the sum total of the angles be more than 144 where the mean between 140 (36 spokes) and 157 ( 32 spokes) is 148.5. We won't have to measure the angles as the polar coordinates function will compute it. I'll have to dust off my college math books, but I know how to do this. I still not 100% certain, but more confident, that the fragments in the Sarnath museum do not give evidence for a 32-spoked mahachakra, but a 36-spoked one. I will likely write to someone in India and ask them if they can find someone who will take a proper picture.  Fowler&amp;fowler  «Talk»  00:59, 3 September 2022 (UTC)
 * PS I don't mean that the original mahachakra had 36 spokes, only that the fragments recovered do not give evidence for 32, but rather 36. They might not be fragments from the same wheel, for example. There are other possibilities. Fowler&amp;fowler  «Talk»  01:11, 3 September 2022 (UTC)

Hi The problem with your method is that the way you count spokes, or calculate angles for that matter, is completely dependent on how the rim fragments are spread apart (ie. their respective distances), whatever the "sample point in the hub" you take. If you put the rim fragments farther apart, then you will have smaller angles and a bigger total spoke count. If you pull the rim fragments closer together, then you will get wider angles and a smaller total spoke count. Please see the two examples attached: if the fragments are set far apart... you get a Great 60-Spoked Ashokan Wheel (which your calculation method will also confirm as a 60-spoked wheel); if the fragments are set close to one another... you get a not-so Great 20-Spoked Ashokan Wheel (which your calculation method again will confirm). Between these two extremes, there is one point where the curvatures of the three rim fragments coincide with one another, and they form a unique concentric circle. This is the point where the original proportions of the wheel are reproduced, and where we can reliably count the actual number of spokes. Geometrically, the basic issue with your analysis is your assumption that "the center of the mahachakra has to be somewhere", referring to the setup in the Sarnath Museum. Actually, since the fragments are not precisely disposed in a concentric way in the Sarnath Museum, there is not one center, but there are currently three (the three centers defined by the three curved segments: they do not coincide in the museum set up). The Sarnath setup is a bit too wide, the fragments are a bit too far apart. Before making any calculation, you need to pull the rim fragments a bit closer (about one or two inches each) so that their centers coincide and the inner and outer circumferences are aligned. Then you have a regular circle with one center, then you can calculate angles and count spokes. Then you will find 32 spokes (approximately). PS: I am making pictures to illustrate my points more clearly, that's all. Best पाटलिपुत्र  Pataliputra  (talk) 03:56, 3 September 2022 (UTC)


 * Thank you for your input, which is always very helpful.  In a few minutes I will post a new analysis of the Sarnath museum pictures and arrangements.  Please bear with me.  Best regards,  Fowler&amp;fowler  «Talk»  18:07, 3 September 2022 (UTC)

A new analysis by F&f
I have benefitted much from user:पाटलिपुत्र's valuable feedback. Like them, I was initially thinking about expanding or shrinking the various fragments. But I was always stumped by an anomaly of piece 4 in the pictures here. I could fit a circle to the outer boundaries of pieces 1, 2, and 3, but not to 4. In all the pictures, its top end was leaning into the circle, i.e. toward the centre. user:पाटलिपुत्र, very helpfully first found a high res picture which is the subject of our analysis, and then miraculously found a picture taken from across the museum's large hall&mdash;a picture in which the perspective effects would be minimal. I tried to fit a circle to that picture, fully expecting it to fit, but it had the same issue, the top end was leaning in, stumping me a little. Mulling over it more, this is what I realized: The problem was that pieces 3 and 4 were presented as one piece, but as you will see below, the original museum catalogue spoke of 4 pieces. This is when I realized that 3 and 4 had been joined with plaster casting, that the position they were in is not their correct position, that they did not need to be shrunk or expanded, and only piece 4 needed to be moved up the circle a bit and rotated. This is shown in Figure C below. Why the plaster casting has turned dark is partly explained by Emma Payne's paper, and I'd be grateful if user:Johnbod could give some input. I know this thread has taken up much space, and I apologize, but I think we might have reached some resolution.
 * In 1908, Oertel's report of the 1904/1905 excavation in Sarnath stated on page 69: "As the photograph shows, it is surmounted by four magnificent lions standing back to back and in their middle was a large stone wheel, the sacred dharma chakra symbol. A few fragments of the rim found near the column and the smaller wheels below the lions enable the wheel to be restored with some certainty. It apparently had 32 spokes, while the four smaller wheels below the lions have only 24 spokes."
 * In 1914, Sahni's, Catalogue of the Museum of Archaeology at Sarnath, stated on page 29: "Of the wheel itself, four small fragments were found. The ends of thirteen spokes remained on these pieces. Their total number was presumably thirty-two."
 * In 1946, Agrawala fit the four pieces in cast plaster, i.e. made a plaster casting. This can be seen in his 1964 book in Figure 6 (a)
 * The four fragments have been on display at the Museum of Archaeology at Sarnath. The top fragment in Agrawala's casting is now the bottom piece.  You can see the same inverted S shape in it if you magnify the figure the rest is cast plaster (as it does not have Chunar sandstone's tell-tale flecked grain).  So, is the bottom piece still in the cast made by Agrawala? I can't be sure, but I would not be surprised.
 * Similarly pieces 3 and 4 are joined with plaster, by a thin sliver, which also does not have the grain of sandstone.  Whether this was done by Agrawala or later is not clear.  This is the data available in the high-resolution picture.  Why does the plaster appear much darker than in Agrawala's picture?  According to Emma Payne in "The Conservation of Plaster Casts in the Nineteenth Century" does say, "The only casts known to have been routinely coated at the British Museum were the ‘store casts’. These were those specifically allocated for storage rather than display in order to keep a spare set from which further moulds and casts could be produced if necessary. The store casts now appear a dark brown colour because of the shellac coating applied to protect their surfaces."
 * In any event that is the configuration at the Sarnath museum. The problem is with Piece 4.  It does not need to be shrunk or expanded, only moved up (i.e. further counter-clockwise) and rotated a little to fit.  I can fit in the same circle. This will be seen in the next figure.
 * Piece 4 has to be moved up along the perimeter to fit; it does not need to be shrunk. In other words, the casting will need to be redone.
 * Note in the new alignment, point A where the crack between pieces 3 and 4 began has been moved up. It is now a perfect fit. That means if we were casting again, the portion of the rim between the two A's will be the new cast plaster.
 * Note also that Piece 3 and Piece 4 cannot be considered one piece as now there is properly the requisite cast plaster between them. So this is the new configuration.
 * In this configuration, piece 1 has four knob spaces; piece 2 has five; piece 3 has one, and piece 4 has three. Their total is 4 + 5 + 1 + 3 = 13, which is the same as Sahni's statement in the Museum's catalogue that four small pieces were found with the ends of 13 spokes on them.
 * The four pieces are now properly aligned in a circle with centre C. The total angle subtended by the knob spaces to C is = 41.5 + 48 + 10 + 33 = 132.5 degrees
 * This means each knob space subtends 132.5/13 = 10.192 degrees. If there were 36 spokes each knob space would have 360/36 = 10 degrees, and thus 13 would subtend the angle 130.  If there were 32 spokes, which is the same as 32 knob spaces, each angle would be 360/32=11.25.  Thus the total for 13 would be 146.25.  The mean value between 130 and 146.25 is 138.13.  Our value is 132.5, which is nowhere near the mean, let alone near 146.25. Also, 360/10.192 = 35.321.  So, we cannot even say, "The wheel had 35 spokes."  It had 36.    Fowler&amp;fowler  «Talk»  18:26, 3 September 2022 (UTC)  Fowler&amp;fowler  «Talk»  18:44, 3 September 2022 (UTC)


 * PS You'd be within your rights to ask why the sector (i.e. the wedge) with the new plaster makes 15 degrees, i.e. how many spokes can be fitted in 15 degrees? The answer is that this configuration was achieved only for the purpose of fitting the four pieces in the circle, but once they have been so fitted they can be freely rotated within to divy up the angles equitably  So piece 4 can be rotated counterclockwise by 5 degrees, making 20 degrees and then it could be two knob spaces, i.e. room for two spokes.  Fowler&amp;fowler  «Talk»  18:50, 3 September 2022 (UTC)

Thank you for the analysis. 1) First, an apparent contradiction (but it's comparatively a detail): I note that Sahni as early as 1914 explained that 13 spokes remained on the fragments, ie thirty fairly complete intercalary spaces remained with their spokes inserted page 29. Apparently, this necessarily means that Fragment 2 was whole as of 1914 don't you think? And it was whole well before Agrawala's time... so it would have to be a full block of Churnar stone and not plaster... I am missing something? 2) I am afraid your fragments are still far from being positionned in a concentric way. I have redrawn the internal circles rapidly (one for the rim edge and one for the bottom of the knobs of each fragment, removing Piece 3 now as it's too short), and it's fairly obvious that they are not centered on your center c. So this goes back to my previous post: if the fragments are positionned too far apart and not concentric, you will necessarily increase significantly the spoke count (in addition to having a wobbly wheel).... So my conclusion remains the same: we have to pull the fragments closer a bit (about 2 inches each) so that they become concentric (ie the centers of their inner circles roughly coincide) and we get a true circle, and only then can we start to measure angles and count spokes.... पाटलिपुत्र Pataliputra  (talk) 19:40, 3 September 2022 (UTC)


 * I'll answer your second question later, but here is the answer to question 1: by a knob space I mean the space in the rim between two consecutive knobs; if pieces 3 and 4 were one piece, the ends of 14 spokes would be visible, not 13 (4 in piece 1, 5 in piece 2, and 5 in piece 3 and 4 together as shown in Figure A and B). Fowler&amp;fowler  «Talk»  19:59, 3 September 2022 (UTC)
 * Please read what I have written carefully. I have referred to Sahni twice. Fowler&amp;fowler  «Talk»  20:00, 3 September 2022 (UTC)
 * Now to answer your second question. Let us proceed in the manner you have indicated. The purple and the green circles are obviously smaller than the red one.  So, we'll find the centers of each.  The purple's will lie to the left; the green's to the right; and the red's somewhere in the middle. To make life easier, we'll assume that purple and green are of the same size and both are 80% of the radius of the red. In order to align them, we'll expand purple and green by 20% and superimpose the respective rims on the red aligning the centers. So now the three circles will have the same center and the same rim.  But as purple and green have been expanded by 20% their rim will increase in thickness by 20%, whereas red's will not. So the outer perimeter will not be a circle any more but a discontinuous piece of boundary.  Alternatively, you could shrink red by 20% and keep purple and green the same; then the thickness of red will decrease by 20% whereas purple and green will remain the same; so again there will be a discontinuity.   Fowler&amp;fowler  «Talk»  22:15, 3 September 2022 (UTC)
 * As to your question about how long the dharmacakra has been on display, Sahni has this to say in another book:
 * A few fragments only of this wheel were recovered by Mr Oertel and are exhibited in a show-case near the capital (see page 42 of Guide to the Buddhist ruins of Sarnath, one of ASI's most often reprinted volumes (according to Asher 2021 I think) So they've been on display in the museum from the get-go. Not sure why I did not see it.  Perhaps because I saw what was being shown to me, I didn't see it because it wasn't shown.  Maybe at that time, I thought it wasn't important,  Who knows? Whether Agrawala was the first to make a cast, and how he managed to swing this I can't say, given that he was in New Delhi and the cakra in Sarnath, I can't say either.  To what purpose I don't know either.  Fowler&amp;fowler  «Talk»  23:36, 3 September 2022 (UTC)
 * Btw, Pat, how do you draw these nice pictures? I had meant to ask earlier but the math got in the way. I mean like the colored knobs.  How are you able to trace their boundaries? Do you do it by hand or do you have an automated program?  Pretty nifty.  Fowler&amp;fowler  «Talk»  00:10, 4 September 2022 (UTC)


 * Yet another approach

We fit an ellipse to the reed-like fixtures in the wall to which the rims are affixed (where there are rims.) The ellipse it turns out is almost a circle. The semi-major axis = 19.5 cm (on my screen) and the semi-minor is 19 cm. So we shrink the image in the x direction by that factor. The result is not much different as shown in Figures H and I. Best, Fowler&amp;fowler  «Talk»  04:18, 4 September 2022 (UTC)

Solution by moving fragments toward the centre
Getting the curved fragments to become concentric, so as to form a proper wheel, is fairly easy stuff. There is no need to shrink, expand or make complicated analysis: just move them around (simple lateral translations) until the centers fit. Imagine you're an ASI staff in the Sarnath Museum (an enviable position...), trying to rectify the display so that the four fragments do align properly so as to form an exact circle/wheel (at the moment the individual centers do not match, neither do the inner circles, what a mess...): 1) You take Fragment 1 (top left fragment), and move it towards the center by 2 inches. Now the center of your fragment coincides with the center of the display... good start 2) You take Fragment 3 (middle right), move it to the center by 2 inches. Now the center of your fragment coincides with the center of the display... 3) You take Fragment 4 (top right), move it to the center by 2 inches. Now the center of your fragment coincides with the center of the display... After this strenuous activity, all the top fragments are now aligned properly and concentrically around one center, all their inner circles are matching like by magic, and you start to have a feeling of profound contentment (mathematical beauty...). 3) Now you see that Fragment 2 at the bottom has tilted forward by about 20 degrees. It used to be better positionned at the time of the Golden photograph, when it looked very similar to the other fragments (your predecessor at the Sarnath Museum had actually set up the fragment pretty well, but it did not hold in place very long apparently...). So you tilt it back up by 20 degrees or so, and, yes, you can see that it looks just like the other fragments, just as in reverred professor Agrawala's photograph (here shown by the black-and-white fragment delineated from Agrawala's photograph). And then, guess what, you move it up towards the center by 2 inches. To your great relief, everything is perfect now: all the fragments are properly aligned, their centers all match, their inner and outer circles all coincide!! Sadly, nobody ever notices this broken wheel in the museum, so overshadowed it is by the glaring beauty of the four lions capital next to it. But now, at least, you have an object of geometrical perfection: a pure circle/ regular wheel, 32 inches in diameter, with the marks of 32 spokes/ knobs perfectly matching the painting of the 32-spoked star behind it, and now also matching Agrawala's reconstruction ... Isn't life great at the museum? पाटलिपुत्र Pataliputra  (talk) 05:10, 4 September 2022 (UTC)
 * Thank you for your reply.
 * In your picture on the left. The radii of the purple and green circles is 83.333% of the radius of the red.  That means the red is 16.667% larger, ie at 32 inches, its diameter is 32x.1667 = 5.334 inches larger. You may align the centers, but how do you make the difference of 5.334 inches in the radii go away?  Tilting a rim forward by your own calculation will reduce the radius by two inches, i.e. the diameter by 4 inches. May I suggest again with great respect that you answer directly only with the raw data, not draw your own pictures, and not tell us tales? Best regards,  Fowler&amp;fowler  «Talk»  12:14, 4 September 2022 (UTC)
 * PS Alternatively, if tilting piece 2 toward the display, will reduce its radius by 2 inches, its inner radius before the tilting was 100/16.667 x 2 = 12 inches, and its internal diameter therefore 24 inches. After corrective re-tilting, the internal diameter you claim will become 10 inches, ie its diameter will be 20 inches.  The external diameter, however, is 2 ft 8 inches = 32 inches.  So the width of piece 2 will be 12 inches/2 = 6 inches. That is not credible.  Fowler&amp;fowler  «Talk»  13:15, 4 September 2022 (UTC)


 * Note: the sides of pieces 1 and 3 are also clearly visible in the picture on the left; only a bit less than piece 2's. In other words, when you bring piece 1 toward the center of the display so that its inner rim is no longer visible, its radius will decrease a little, increasing the difference between it and the red.  So, thereafter you will need to account for a difference of a full 6 inches.  How will tilting piece 2 accomplish that?  Fowler&amp;fowler  «Talk»  12:33, 4 September 2022 (UTC)




 * 1) Problematic viewing angle of Fragment 2 First of all, as we discussed before, it is clear that Fragment 2 is not photographed at the same angle as the other fragments, and appears tilted downward (when I say "tilt", I mean "rotated"): this is because a much wider portion on the inside surface of the rim is visible (about twice as large as the other fragments, all the more visible since it is over-exposed), and the rim surface appears to be receeding away from the camera. The Golden photograph gives a slightly better view of the fragment when seen more frontally (but still tilting downward...). This apparent downward tilt is probably the result of two factors: the simple effect of a photographer taking a picture from high up (a phenomenon similar to what you explained as "The elliptical distortion"), and probably a level of physical tilt downward, as photographs taken from afar still suggest Fragment 2 is somewhat dropping. Anyway, this combined "apparent tilt" roughly doubles the size of the visible part of the inside of the rim, which, given the section plan provided by Agrawala 7c, would correspond to as much as 20-30 degrees in apparent downward deviation compared to the plane of the other fragments.
 * 2) Drawing Fragment 2 in a plane This "apparent tilt" (deviation from the plane) is considerable, which implies that the apparent curvature of Fragment 2 as seen in the photograph cannot be taken at face value (nor the inner cercles derived from it), and the real curvature is significantly higher, and can only be seen when the bottom fragment is positionned in the same plane as the other fragments, under the same viewing angle. This is provided by Agrawala's photograph 6. The graphical grey fragment at the bottom of my reconstruction ("Concentric, regular wheel") was delineated precisely from Agrawala's picture, and therefore reflects the shape of Fragment 2 when seen in the same plane as the other fragments, and its curvature is identical, which means its inner circle will also coincide with the others.
 * But Fragment 2 is actually rather secondary, as the top fragments (left and right) are amply sufficient to define a perfect circular wheel in which the fragments are concentric and their inner circles coincide perfectly, thereby sufficing to define the proper Ashokan wheel. Actually even one single large fragment, such as Fragment 1 or Fragment 4, would be entirely sufficient to define the size of the wheel and the number of its spokes: we are only debating here because several fragments are available and they were set up awkwardly in the museum. Fragment 2 is just an added bonus, and its imperfect viewing angle doesn't affect the conclusion.... Best! पाटलिपुत्र  Pataliputra  (talk) 15:11, 4 September 2022 (UTC)
 * So are you suggesting that in the Museum of Archaeology at Sarnath the display board in which the wheel fits has an width of 36 inches, four inches wider than described in its own catalog and best-selling book?     Fowler&amp;fowler  «Talk»  21:58, 4 September 2022 (UTC)


 * Approximately yes. The Sarnath display is not built with geometrical precision: for example, if you analyse the painted wheel in the background, you will see that it's quite wobbly and that the spokes are completely off, as they do not converge at all, and tend to cross haphazardly at a point below the center of the template circle holding the wheel (image 1). It's not a modern, scientific, production. I suspect it was set up in the 1920s as you pointed out, and hasn't really been corrected since. Also, I've built an "Agrawala ruler", based on his precise mesurement of the width of the rim (3.65 inches): this ruler, applied to our photograph or the "Golden photograph" for example, gives 36.5 inches for the diameter of the Sarnath Museum display (image 2). So my impression is that the Sarnath Museum display is just an approximate reenactment, and that the work of actual academics (especially Agrawala or the archaeological reports, and essentially all the academics since then) should be given priority of authority, especially given their sometimes very precise analysis of the wheel. What do you think? Best  पाटलिपुत्र  Pataliputra  (talk) 05:52, 5 September 2022 (UTC)


 * thumb|right| A revised ruler for measuring the surmounting wheel of the lion capital of Ashoka at Sarnath.
 * Hello,
 * Steps:
 * I checked the "golden wheel." The width of the three pieces are the same there.  So, there is no issue of tilting of piece 2.  All differences in width and thickness follow from placement of the observer/camera.
 * using the trigonometry and surveying topics from high school, one can see that the ratios of the observed thickness and width si/wi give us the tangent of the angle the observer's/camera's axis makes with the horizontal. Based on that we can calculate the approximate position of the observer within the circle. (This is the basis of what I called foreshortening).
 * Now apply the same techniques to the ruler. In the front view (where the middle of the ruler is placed) there is no horizontal distortion, but as you move out from the middle, the successive pieces of "3.65"'s begin to appear shorter and shorter.  So the picture looks something like shown here.
 * Fowler&amp;fowler «Talk»  12:25, 5 September 2022 (UTC)
 * PS I did not understand Vincent Smith to say that the wheel had been on display with a background visual for counting the spokes since 1920. The current display might not be that old. In any case, it would be highly unusual for the oldest site museum of the ASI whose catalog was written by a future DG would make that kind of error.  Fowler&amp;fowler  «Talk»  12:37, 5 September 2022 (UTC)
 * PPS If theta_i is the angle a observer's eye/camera axis makes with the vertical, then theta_i = arctan(wi/si x 2.65/3.65). Fowler&amp;fowler  «Talk»  12:41, 5 September 2022 (UTC)

Thank you, but I would like to bow out
Thank you for your input. This discussion was a learning experience for me. Thank you also for finding some excellent images. From everything I have seen thus far, I remain unconvinced that there is definitive proof that the wheel fragments are best captured in a model of 32 spokes and not 36, which the geometric evidence thus far seems to point to me. But that ultimately is a minor issue. I remain unconvinced that Agrawala 1964b published by Agrawala's son is a reliable source. The book has been roundly criticized by John Irwin (on at least two occasions), Osmund Bopearachchi, and by Sudeshna Guha in modern reliable sources. The book has not been reviewed in any reliable journal that I am aware of. The text is wildly speculative and historically wrong. Assigning lions and tigers in the same category of cultural fauna on the one hand and horses and rhinoceros on the other, is about the most bizarre categorization I have read. So I will oppose the use of that book in this article. I will consider it to devalue if not degrade the value of this article and remove the edits that give it prominence. Thank you again for your engagement. I am very appreciative of your many images and of your very creative arguments. But I now have other things to do, including improving this article further. Best regards, Fowler&amp;fowler  «Talk»  16:52, 5 September 2022 (UTC)

PS By the way I did not appreciate the manner in which you posted at Neutral_point_of_view/Noticeboard, quickly interpreted a perfunctory reply there to be evidence of unanimity and in a few hours chose to force through an edit and then tell me about the discussion. I went to the same notice board and let the discussion proceed, and eventually, user:Aquillion described WP:FALSEBALANCE to me in greater nuance. And it was only thereafter that I reverted half of one edit on that principle. But I consider Agrawala 1964b to be a highly troubling and unreliable source and will continue to view its introduction as an instance of false balance. Best regards, Fowler&amp;fowler  «Talk»  16:52, 5 September 2022 (UTC)


 * Thank you too for the exchange!


 * 1) If we are to go into the details for calculations, you have to take into account the fact that the rim is closer to the camera than the painted wheel, by about 2.62 inches (per Agrawala, thickness of the rim). This means that the 3.65 inches of the rim actually coincide with an object with a width of about 4.00 inches on the plane of the painted wheel (my estimate, attached), which has to be taken as a reference so that everything can be computed with object-equivalents on the same plane. The consequence is that our "measuring unit" on your corrected "Agrawala ruler" should be taken as 4.00 inches, not 3.65. So your corrected 32 inches in diameter (after taking into account what you call "foreshortening") has to be increased back up, roughly by (4.00/3.65=9.6%) to account for the proximity of the rim, which again brings us to around 35 inches for the external diameter. We are thus back to a diameter significantly higher than 32, which is confirmed by the fact that the inner circles and the centers of the fragments in the Sarnath Museum do not match anyway.
 * Hopefully we have now exhausted all of the important variables, and I trust it seems rather obvious, both visually and from calculation, that the fragments are set too far away in the Sarnath Museum, which corrobates Agrawala's reconstruction. By the way, Agrawala is not alone, and the vast majority of sources over the last century have been mentioning 32 inches in diameter and 32 spokes for the wheel . In all our discussions, not a single source has been provided in favour of the new hypothesis that the wheel might have had 36 spokes.
 * 2) WP:FALSEBALANCE is a very good principle, and forces us to make sure our sources and claims are of comparable prominence and reliability. It should be determined through discussion in case editors disagree. The one very positive outcome from Neutral_point_of_view/Noticeboard in my opinion was that it dispelled the notion that in order to present two sources with competitive views, we necessarily have to have a tertiary source presenting these competing points of view. This is just not a principle supported by Wikipedia, except for the fact tertiary sources can help in establishing balance.
 * 3) There are a few minor things I did not appreciate as well, but overall I think we did great: we were able to exchange productively over a long period of time, in the most civil manner, despite disagrements. It was, in my opinion, very useful, and promissing for the future. Best regards पाटलिपुत्र  Pataliputra  (talk) 17:18, 5 September 2022 (UTC)
 * I have told you again and again Please do not draw simplistic pictures.  Work only with the data.  Are you still saying that the way the Sarnath wheel is arranged is flawed and that its four fragments need to be moved in by two inches radially to fit in a proper circle?
 * Are you still saying that if two people hold a tape measure across the display from the end of one fragment to the end of a point antipodal on another fragment, it will measure 36 inches? If you are, I will call the museum tonight and have them measure it, or find someone to do this for me. Are you saying that or not?  Fowler&amp;fowler  «Talk»  22:09, 5 September 2022 (UTC)
 * Agrawala ruler (Sarnath wheel) 3d.jpg You do this again and again.  You ask me a question.  I reply directly, as I did to your question about the scale. I drew how the scale will be visible in the camera view and what its distortion will look like.  What did you do in return? Made another simplified picture and changed the direct topic.  So, again: show me what your scale or ruler will look like.  We have a 3-dimensional situation that I have crudely drawn here.  Don't draw me another picture.  Don't try to improve my picture.  Draw your scale on this one. Otherwise, I will consider you to be a typical POV pusher. So please draw the scale on the accompanying picture.  Fowler&amp;fowler  «Talk»  04:25, 6 September 2022 (UTC)


 * Please understand that I am totally free to draw anything, especially if I believe that it can help explain a complex technical point... Yes, all the information we have so far points to the fact that the Sarnath fragments are set too far apart by a few inches in the museum display (our photographs, our measurements, and of course the fact that you, F&f, can count the space for 4 more spokes than the universally agreed 32...). I would be delighted if you could find someone to take the actual dimensions of the wheel display in the museum. It will be difficult since there is a display window, so there is massive potential for parallax: all individual measure points would have to be taken perfectly perpendicularly to the plane of the wheel. If that person has a camera, it would be fantastic too if she could take photographs of the wheel at each measuring point, so that we can understand the precise vertical position of the wheel againt the background drawing. And of course a photograph from afar, and a photograph of the bottom fragment up-close. This would just be great, and will help us confirm the exact situation.... (PS: I will draw the scale on top of your picture in a few hours if you wish, but if you read my point above carefully, you will understand that for calculation purposes you just have to replace the number 3.65 by 4.00 in your scale, to reflect the actual size of the reference rim fragment as projected from the point of the camera onto the plane of the 32-spoked drawing... if we have to go into camera optics, let's do it properly...). This will give you a total diameter of 35.1 inches, which is within the parameters that will confirm the properly restored 32-spoked 32 inches-wide concentric wheel of academics, after moving the museum fragments slightly towards the center). Best पाटलिपुत्र  Pataliputra  (talk) 04:59, 6 September 2022 (UTC)
 * So I had someone in India call the museum. The officer couldn't confirm what it looked like in 1920, but the display is a new one from when the museum was renovated in 2011-2012.  It is behind glass and its humidity is monitored. The bottom fragment is indeed in cast plaster and the top right fragment is hanging in as a result of a restoration of a break earlier. He did not know when he restoration of "crack" was done.  Upon being asked if it was at the same time as the cast, he said it could be.
 * (In other words, it could very much be laid at the doorstep of the museologist Agrawala's inglorious and vacuous nationalistic venture of 1946, per Guha.) The pieces are sunken into the display board. They are not uniform in cross-section. It is a pride of the museum and they take great care of it. They would not measure the outer diameter of the wheel, but said it is highly unlikely that the display would veer off the museum's own catalog by four inches no less, as the restoration was supervised by archaeologists.
 * So, you were telling me the bottom fragment did not have cast. It was all Chunar sandstone you said grandly.  A week ago, you did not know the word "foreshortening," or for that matter what a "Canny edge detector" or for that matter confocal ellipses. Yet you feel perfectly comfortable in lecturing me about their defects.  Are you doing this to further encyclopedic knowledge or to win brownie points in what you perceive to be a battle you need to win?
 * So, yes, you can give any answer you want, just has you can use a user name in a script that makes it difficult for other uses to communicate with you. They have to copy the name each time.  Just as you can copy and paste, images, articles, quotes, text, here there and everywhere, cut out Brahmi letters to write a POV-ridden article, until I came along and cleaned out the lead (Brahmi script) Just as you can perform reconstructive surgery on the Sanchi lions by pasting the Sarnath lions on them even after they told you at RS/N you can't.  So, I'm done with this discussion.
 * But let me be perfectly clear user:Pat, if you make a unilateral edit of assigning blame to Muslims for the destruction of Sarnath, I will remove it. If you add anything from Agrawala published by his son, I will removed it. If you add any more fringe sources which you have been for years to many articles, I will remove them from this article.  Fowler&amp;fowler  «Talk»  12:44, 6 September 2022 (UTC)
 * No one can say I did not engage your vanities. Fowler&amp;fowler  «Talk»  13:20, 6 September 2022 (UTC)
 * This is all pretty wild for someone who promissed to edit collaboratively and in a civil manner from now on .... You seem frustated simply because I have been challenging your new pet idea (of 4-5 days?) that the Sarnath wheel had 36 spokes, when no scholar in the world, in more than a century, has ever mentioned such a thing, and the wheel is universally thought to have had 32 spokes ? Why not just ask the museum about the number of spokes then (see the actual museum notice here 1:55, "32 spokes"). Sadly, your "36 spokes theory", although interesting, is obviously pure WP:OR and very unlikely, so I have been attempting with you to find the reason why your theory was probably wrong. The photographs, the measurements all point to the museum's set up being awkward... didn't yourself say that ASI's reconstructions are not to be trusted ? One day someone will actually measure this display and see that, oh surprise, it is a few inches too large: it is obvious just by looking at the photograph and the curvatures of the fragments that don't match and are not concentric, this is basic geometry. And if all reliable sources say 32 spokes, then it's probably 32 spokes... unless you have extraordinary proof... Well, never mind, I've only been devoting time and effort to answering to your questions here, I've done what I could to address your concerns. Do not lash out just because your latest theory is not vindicated... I'm glad to work with you, but please be more tolerant of the contributions of others, and stop making personal attacks everytime something is not going exactly your way. Best पाटलिपुत्र  Pataliputra  (talk) 13:30, 6 September 2022 (UTC)
 * I am angry not because of my theory not being established, but by your deceitful comments that indicate no knowledge only a desire to be contradictory. Did you not say there was no plaster in the bottom fragment? That the bottom fragment was turned up by 25 degrees? It all turned out to be nonsense. Did you not say there was no issue with segments 3 and 4, it was all a matter of lighting
 * Well, the top of fragment 4 is more rounded than the others. Now you are going on telling me about a simplified picture in Agrawala's 1946/64 and making that a basis of dubious calculations. I am not being abusive, you are being abusive of a process that requires honesty.  You are being deceitful.  Fowler&amp;fowler  «Talk»  13:40, 6 September 2022 (UTC)
 * I am concerned about the process of Wikpedia that you are abusing by turning it into a blog of your solipsism, whether on a talk page or on regular pages. Fowler&amp;fowler  «Talk»  13:45, 6 September 2022 (UTC)
 * Again, I am not concerned about 36/32/24. 32 is yours and Agrawala's obsession.  Fowler&amp;fowler  «Talk»  13:41, 6 September 2022 (UTC)
 * More personal attacks? Do you have any idea of what collaborative discussion is? We make hypotheses, some turn out right, some turn out wrong. That's how we move forward. We discuss and try to find the answers to our questions. It's a beautiful and interesting process, and I'm glad we followed it. I have been trying to respond to your questions to the best of my ability . I have been trying to point out the potential issues in a new theory of yours which flies in the face of common sense and established knowledge: the fact that the wheel had 32 spokes is not just "me" or Agrawala, it's basically everyone in academia who mentions this subject . The lonely obsession here is actually the one about "36 spokes" I'm afraid... We are all here in good faith spending a lot of our time trying to approach the truth, let's do this collaboratively and in a good spirit Fowler&fowler. पाटलिपुत्र  Pataliputra  (talk) 14:02, 6 September 2022 (UTC)


 * Pataliputra scale.jpgThe scale: So, if you want to fully take optics into account, you will have to include the foreshortening of the rim in the photograph (the rim is closer to the camera by 2.65 inches, my discussion above, with explanatory drawing)… what you see in a photograph is not the object itself, but is equivalent to its projection on the 2 dimensional surface at the back, with the camera at the center of the projection. If we are to take measures and compare them in this projection, we have to take into account the size of the components as projected. For example, the rim is 3.65 wide in real life, but its actual projection on the flat surface (as seen from the camera) is about 4.00 inches wide (explained above). If you don't take this foreshortening into account, it's like holding your thumb in front of your face, saying that it is 1 inch wide, and then deducing that the TV at the other end of the room has a real width of 10 inches... you have to take into account what the width of you thumb means in actual width when projected on the plane of the TV screen (in this specific case your thumb may have a "projectional value" of about 5 inches, see my drawing above). I've attached a scale which incorporates your trigonometrical correction when looking at a plane from a point (lateral correction), in addition to the foreshortening due to the proximity of the rim to the camera. Beyond that only an optical engineer will do... It's a rough estimate but it gives 35.1 inches for the diameter, which is coherent with the fact that the museum fragments are visibly a bit spread apart too much (their centers are not concentric), and need to be moved slightly inward to make a proper concentric wheel with 32 inches in diameter and 32 spokes, which are the values given by realiable sources (no author has ever written that there were 36 spokes, this is unheard of I believe, so maybe a bit more care would be needed when making such an original claim...). Of course it would be great to confirm the actual diameter of the wheel display in the museum! Best  पाटलिपुत्र  Pataliputra  (talk) 08:41, 6 September 2022 (UTC)
 * AA Sarnath wheel fragment ends 1.jpg Using a fringe source is disruptive and ultimately deceitful because we put out false knowledge. And what happened to your cog in wheel model? Still on board? Fowler&amp;fowler  «Talk»  15:09, 6 September 2022 (UTC)
 * AA Sarnath wheel fragment ends 2.jpg Fowler&amp;fowler  «Talk»  15:14, 6 September 2022 (UTC)
 * Did you not say the bottom fragment was all Chunar sandstone? Yes or No. Fowler&amp;fowler  «Talk»  15:33, 6 September 2022 (UTC)
 * [[File:AA Sarnath wheel fragment ends 3.jpg|thumb|right|
 * At what angle will 3.65 and 2.65 look like this? How will a 1 inch wide spoke be going in two inches into this one?]] Fowler&amp;fowler  «Talk»  15:26, 6 September 2022 (UTC)

Hi Please note that I do not intend to respond or continue this discussion (in which I am otherwise very interested from an archaeological standpoint), unless you tone down, and delete or strike your recent personal attacks above and below. Best पाटलिपुत्र  Pataliputra  (talk) 16:14, 7 September 2022 (UTC)


 * Didn't see this piece of chicanery. No worries there is no need for a response. I know how far your model is from the reality. The discussion is over.  Fowler&amp;fowler  «Talk»  06:30, 16 September 2022 (UTC)
 * As far as I am concerned, you have only been able to delve into WP:OR, trying to prove a new personal theory that the Sarnath wheel had 36 spokes, whether not a single academic source has never ever made such a claim. Of course, your theory is not supported, neither by sources nor by the direct facts as seen from the photographs of the object itself. Then, when your argument fails, you simply try to escape with WP:Personal attacks, as always.... so much for your claims to intellectual honesty . This is enlightening... By the way, you claimed that you were sending someone to the Sarnath Museum to obtain actual measurements of the wheel on display ... just empty talk again, or are we finally going to obtain some factual closure? Or you did obtain the measurements, but you are not sharing them because they finally confirmed that you were wrong all along?.... पाटलिपुत्र  Pataliputra  (talk) 08:07, 16 September 2022 (UTC)
 * Not biting. Like I said: no interest in chicanery.  Fowler&amp;fowler  «Talk»  11:26, 16 September 2022 (UTC)
 * Again, for the umpteenth time, please do not ping me. Please Do Not ping me.  Fowler&amp;fowler  «Talk»  11:47, 16 September 2022 (UTC)