Talk:Conway polyhedron notation

I added an example table, using the cube as the seed and all the octahedral symmetry polyhedra with images on wikipedia now that can be generated from it. I added some quick operational titles, not standardized, but whatever seemed most useful explanation in a single word. I'm happy if anyone else wants to improve on this. I just wanted to get some examples up there. Tom Ruen 02:52, 31 December 2006 (UTC)

Okay, here's an ugly example chart, at least to visually explain the operations by a cube example. It shows a graph of sequential operations in black, as well as alternative paths and short-names in red. I won't add it to the page since it isn't pretty enough (or clear enough!) Tom Ruen 12:37, 31 December 2006 (UTC)
 * [[Image:Conway polyhedron notation-examples.png|800px]]

Generalization to 4D?
Just wondering if Conway's operators can be generalized to 4D, and how many polychora can be generated using simple seeds? (Analogous to generating the Platonic solids from the prisms and antiprisms, and the Archimedian and Catalan solids from those.)&mdash;Tetracube (talk) 04:15, 3 August 2008 (UTC)


 * I've thought about that as well, but not looked in detail, nor heard of it being done. Certainly some operations apply directly to higher dimensions like dual, truncate, rectify, kis, expand, bevel. Well, I imagine some may diverge into two forms in higher dimensions (acting on faces vs acting on facets/cells for instance), and some new operators will be needed. I imagine the combination of operators will act differently between dimensions. I'd bet Conway has already thought of this and it was never published. We'd not have his polyhedron operators without George Hart's efforts to show them on his website and VRML generator. (The Symmetry of Things, 2008, which still doesn't fully express what's here either.) Tom Ruen (talk) 02:45, 5 August 2008 (UTC)


 * Anyone in correspondence with him, who can ask him about this? :-) One thing for sure, in 4D there is a new kind of truncation. Consider the progressive truncation of the cube into the octahedron: you go from cube -> truncated cube -> cuboctahedron (rectified cube/octahedron) -> truncated octahedron -> octahedron. So in 3D, the rectified polyhedron is sorta of "midpoint" between two duals. In 4D, however, the rectified polychora of two duals no longer coincide; the midpoint is now the bitruncate, e.g.: 4-cube -> truncated 4-cube -> rectified 4-cube -> bitruncated 4-cube (= bitruncated 16-cell) -> rectified 16-cell -> truncated 16-cell -> 16-cell. So I'd imagine there should be at least a bitruncation operator in 4D.&mdash;Tetracube (talk) 20:06, 5 August 2008 (UTC)


 * The thing generalises to every dimension. In essence, one treats the seed as regular, and each of wythoff's Mirror-edge and Mirror-margin operators correspond to a Conway-Hart flip and conway-hart flop operator.  To see how the operator works, one draws a "flag" (triangle with vertex at the vertex, edge-middle, face-middle et.)  The thing corresponds to v--e--f of the dynkin symbol, and that half of the operators are of the form xo---xo---xo (ie x/o) and the other half is mo---mo---mo.


 * When you apply a conway operator to a figure, ye generate a new seed, and that this figure can then have a new conway operator applied. The only concatinations is that a rectate or truncate (ie oxooo! or axooo!) applied to a n-rectate (..oxo...) gives ..oxaxo.., gives a n-cantellate ...xox... or n-cantetruncate ..oxxxo...


 * See eg http://www.os2fan2.com/gloss/pglossc.html#PGCONWAYOPER Conwy-Hart rule, cantelate. --Wendy.krieger (talk) 04:55, 18 October 2009 (UTC)


 * I understand it generalizes to higher dimensions. What's harder to see is what "new" operations exist, and how sets of operations work out as equivalents. It definitely would be fun to work out the operations in 4D. Someone should write a paper on it and we can include something on Wikipedia!
 * Here's the ones I can see, how they apply in general, or quick generalized descriptions:
 * d = dual - topologically swap vertices-facets, edges-ridges, faces-peaks, etc.
 * a = ambo - rectification - Create vertices on mid-edges, rectify facets, new facets at old vertices
 * t = truncate - create pairs of new vertices on edges, truncate facets, new facets at old vertices.
 * k = kis - create new vertices at facet center, replace facets by "central pyramids"
 * e = expand - Split vertices, one for each facet. Insert new facets at old vertices, at old edges, at old facess...
 * o = ortho - Divide facets at mid-edges, mid-faces, etc. Replace facets with ones from each old facet vertex/edge/face centers.
 * My FIRST interest would be to see what minimum set of operators can generate all of the uniform polychorons from the regular seeds. Tom Ruen (talk) 20:03, 20 October 2009 (UTC)

There are eighteen operators in four dimensions, of which only two are constructable from others, eg rectify.rectify = cantellate and truncated.rectified = truncicantellate. You can use dual + Edge to get the corresponding Margin. --Wendy.krieger (talk) 11:01, 31 December 2009 (UTC)

Conway operators for 4-polytopes?
Perhaps consider seeds: S=(4-simplex), C=(4-cube)Tesseract, O=(4-orthoplex)16-cell, D=dodecaplex, T=Tetraplex

Wendy, can you fill-out or expand this table for 4-polytopes?


 * There is a corresponding Conway operator, for each of the Wythoff mirror-edge and mirror-margin constructions, along with a variety of snub operators. Moreover, since both call down to pennants (simplexes with numbered vertices), the wythoff construction can be used for these figures too.


 * The wythoff mirror-margin operators (j, k, o, m, g) produce figures with single faces and several kinds of vertices, while the wythoff mirror-edge operators, (a, t, e, b, s) produce figures with single vertices and several kinds of face.


 * The 3d operators correspond to the WMM j=omo, k=omm, o=mom, m=mmm, and g=ggg, while WME are a=oxo, t=xxo, e=xox, b=xxx, and s=sss. In four dimensions, there are four elements, eg d = ooox or ooom.


 * The main difference is that the numbers between the nodes change from flag to flag. So while xPx still gives a 2P, P changes from node to node.--Wendy.krieger (talk) 10:03, 31 December 2009 (UTC)

It's not possible to add the number of elements in every case, since an edge in the seed might give rise to a prism of q or 2q, where there are q cells on that edge. Different edges can give rise to different q, and hence different prisms.

If you know the total number of flags (f) (eg for hexagonal prism = 12 + 12 + 6*8 = 72), then you can evaluate edges and vertices where only one flag-wall is used. These are the cases where there are no instances of adjacent 'o' nodes. The rule follows the one set out by Coxeter on the dynkin graph (you get something like f, f/2 or f/4), enumerated once for the vertex, and each x removed constitute an edge. You can always test this on a real polyhedron, which gives a simple-to-calculate result.

--Wendy.krieger (talk) 10:46, 31 December 2009 (UTC)

Kis operator
I notice that the kis operator is listed among the other "main" operators, and then described again as a shorter alternate notation (knX = dtndX). Is this a mistake? I made a graph of how various polyhedra are related to each other via these operators, and I can confirm that kX = dtdX at least for the Archimedean/Catalan polyhedra.&mdash;Tetracube (talk) 20:10, 5 August 2008 (UTC)


 * The n refers to a limited kis, or limited truncation. Like k3 means "kis only triangles", and t3 means "truncate order 3 vertices". Tom Ruen (talk) 05:27, 7 August 2008 (UTC)

Operations table
Aren't many of the v/e/f entries incorrect? In particular, ambo should have v+f faces unless I'm mistaken. Bmreiniger (talk) 01:57, 8 April 2014 (UTC)
 * Errors may exist, but possibly you're missing a spherical Euler characteristic substitution, with f+v=e+2, although this would be incorrect for using it for Euclidean tilings. Tom Ruen (talk) 03:13, 8 April 2014 (UTC)
 * OK, I see. In the back of my mind I'm thinking about arbitrary surfaces (when the operations make sense), which this page is not about.  Thanks.  Bmreiniger (talk) 17:15, 8 April 2014 (UTC)
 * The operations could apply to toroidal_polyhedron, although you have really define the surface. Hart's javaScript only works for spherical polyhedra seeds. This alternate script also works from spherical seeds, but has trouble with nonplanar faces, and doesn't conform points onto a sphere. Tom Ruen (talk) 22:27, 8 April 2014 (UTC)
 * Is there any reason not to list v+f faces for ambo? I mean, it makes it clearer what's going on; using the Euler formula substitution obscures things by one step.  v+f seems much more natural. I'd recommend the change.  And in any case, the entries for chamfer are wrong: chamfer has e+f faces and 2e+v vertices. Anyone reading this who agrees should go ahead and make the edit to the page itself, or I will if I come back in a while and there's no disagreement. — Preceding unsigned comment added by 98.116.5.101 (talk) 06:07, 12 May 2014 (UTC)

What is "cab ne see"?
What means cab ne see in sentence

The zero index cab ne see to represent vertices...

Jumpow (talk) 19:59, 22 October 2017 (UTC)

Original research
While this page is a great resource, I'm having a hard time tracking down references for some of the operators that don't appear on George Hart's list, and I kind of suspect that created some out of necessity. That would fall under WP:NOR. I'm hoping that's not the case and I'm just confused by the article lacking in-line citations. -Apocheir (talk) 21:55, 17 November 2017 (UTC)

Hi Apocheir. Yes, referencing sources here is tricky since the primary sources come from what actual program generators use, starting with Hart's. And you're right I took some liberties for defining extensions, like helping with the Goldberg polyhedra. The http://levskaya.github.io/polyhedronisme site is very useful since it can generate polyhedra within a URL address, but only implement a few operators beyond Hart's - chamfer and whirl which do help on the Goldberg forms. The strongest implementation of extended operators, working directly from Hart's algorithms is in http://www.antiprism.com, but its not had a recent release. I've not used it myself, but I have its documentation, copied here: User:Tomruen/Conway polyhedron notation/Antiprism notes. Tom Ruen (talk) 23:17, 18 November 2017 (UTC)


 * Neither Antiprism nor Polyhedronisme implement the parameterized uk, wj,k, and vj,k operators. I'm also concerned because Antiprism, Polyhedronisme, and this page have some definitions that conflict, for example n, x, and z. I'm not looking at Wikipedia policy for the sake of policy, it's just that this page is kind of turning into a zoo, and creating new operators ad-hoc is making it more crowded. There's a recent paper here that actually complains about the absence of a systematic organization for Conway operators (page 9 in particular), and it references this Wikipedia page as an example! -Apocheir (talk) 21:40, 27 November 2017 (UTC)
 * Agreed, its a problem, and there's no authorative source to set a standard. I have thought to rework the article with multiple operator columns for different generator implementations. I'm less worried about indexed operators like wj,k since it can be seen as a definition of a series, which has some instances derivable from other operations. Tom Ruen (talk) 21:55, 27 November 2017 (UTC)

I've been working on a major redesign of this page in User:Apocheir/sandbox to address both the unclear citations and the disorganization. Grouping the operators by duality has made it much more manageable. I'd like anyone who's interested to look it over, since statistically it's very likely that I omitted or typoed something in there somewhere. One of my main questions is with the example section: I don't get much out of most of the examples other than "hey, that's a cool looking polyhedra" and I wonder if it would be more appropriate for, say, a Commons gallery page. -Apocheir (talk) 21:27, 14 January 2018 (UTC)


 * It looks promising. The matrix representation looks interesting, but will take some getting used to. I agree the gallery section is more just examples of what can be done and could be done on a commons page. One of the open questions for me was to see what was possible, so that's why I tried making the derived tables with composite operators. Anyway, since no one had really written up a theory on what is possible or not possible, this article is necessarily exploratory, showing possibilities with the cubic seed starting point. The main theory is apparently that symmetry is preserved, except for chiral forms which can cut symmetry in half if original has reflective symmetry. Tom Ruen (talk) 21:54, 14 January 2018 (UTC)

Some comments as I push my changes:
 * I used Antiprism as the reference implementation. This is a little goofy because I believe Antiprism used this page as its reference for the extended operations, but I suppose that's water under the bridge now.
 * The only named operators that aren't being carried over in some form are gyro-3 and snub-3. Aside from WP:NOR questions, I couldn't verify the vertex/edge/face counts or the dual relation. (If x=xd, the equation for either vertexes or faces has to be in the form of either $$me$$ or $$v + ne + f$$.) I'm willing to readd them if they're attested somewhere outside of Wikipedia, e.g. if Antiprism implements a gyro/snub indexed operator.
 * I've removed the "derived operations" section entirely. This is another section where all I get out of it is "hey that's neat-looking", and the tables of operator counts by edge multiplier are dubious and uncited.
 * There are a bunch of images in the dx and dxd columns that might be reversed. This was true of the original page, it's just more obvious now. I'll attempt to go through the list if nobody gets to it before me.
 * Probably related: It looks like the image for v was actually for dv.
 * I tried to retain as much as I could, but the unclear reference style from before didn't help. If there's something I missed that's citeable, add it back in. I've only been able to get chunks of The Symmetries of Things from Google Books, so it's possible I missed something from that source.

-Apocheir (talk) 20:46, 20 January 2018 (UTC)

Special Tilings
Apocheir: I have since added a rigorous description to the operations of expand and ortho for uniform tilings which is entirely consistent with George Hart's Conway Polyhedron notation. Ambo connects consecutive midpoints of regular polygons to form new polygons, and expand is ambo applied twice, so I connected the midpoints again. It turns out that there is a compatibility between the ambo regular polygons and the vertex-type polygons as well (the midpoints of both types of polygons coincide and we can express this as a shared linear combination of original vertices). So this operation is undisputed according to George Hart's Conway Polyhedron notation, applied to uniform tilings. Moreover, ortho is compatible with dual superimposition over k-uniform tilings only because regular polygons are symmetric about any edge perpendicular bisector, and each regular polygon only shares up to a single edge with any other regular polygon (no half-edges). In George Hart's words, duality replaces faces with vertices and vertices with faces, and for uniform tilings, the resulting centroid-centroid edges are perpendicular bisectors because of the above symmetry. Thus, orthogonality.

I do feel like I am clear enough on this matter; I have since added the special tilings to the commons page and my sandbox as well. Thanks. Harry Princeton (talk) 15:23, 13 July 2019 (UTC)


 * I may have been unclear. I never had an issue with the expand operator. What I was questioning was the "elongation" operator you use. That is not cited, and I'm not sure what you mean by it. -Apocheir (talk) 23:10, 13 July 2019 (UTC)

Thanks for your reply, Apocheir. I'm sorry about this confusion - elongation was never a Conway operator, but I wanted to write e,o—S instead of e,o[Elongated Square Tiling] or e,o[3.3.3.4.4] for abbreviation (or it would be too unnecessarily 'cluttered' in your words, and would stick out in the table). If you could find any suitable cited abbreviation, please do use it. Harry Princeton (talk) 20:45, 15 July 2019 (UTC)


 * Do you have a citation for what elongation is defined as? And if elongation is not a Conway operator, why include it on this page? -Apocheir (talk) 20:50, 18 July 2019 (UTC)


 * Given that Harry Princeton has not responded to my concerns, and is currently banned due to adding OR to this and other pages, I'm removing it. -Apocheir (talk) 22:35, 4 December 2019 (UTC)

Can someone chime in on a disagreement between myself and Apochier regarding the meaning of xd and dx
We both seem to agree as to how matrix multiplication works (what $$\mathbf{M}_x$$$$\mathbf{M}_d$$ and $$\mathbf{M}_d$$$$\mathbf{M}_x$$ each are in matrix form) but disagree as to which one of xd and dx corresponds to which of those. Apochier believes that xd = $$\mathbf{M}_x$$$$\mathbf{M}_d$$ and dx = $$\mathbf{M}_d$$$$\mathbf{M}_x$$, while I believe that xd = $$\mathbf{M}_d$$$$\mathbf{M}_x$$ and dx = $$\mathbf{M}_x$$$$\mathbf{M}_d$$. Other places in the article seem to back up my assumption. Who is correct? Kevin Lamoreau (talk) 21:49, 17 October 2018 (UTC)
 * I now realize I was incorrect. What threw me off is that I hadn't realized that the operation itself (the operation of an operator on a polyhedron) has the operator matrix before the matrix representing the polyhedron being operated upon.  I've edited the Operators section to try and show this better, by using the transpose symbol and an actual 3x1 matrix representation of the column vectors. Kevin Lamoreau (talk) 22:56, 17 October 2018 (UTC)

Actualized illustrations?
I'm enjoying this article, though I don't have the training to understand even a fraction of the math and notation, or the patience to work it through and learn it by myself. What I'd really like is examples of what the resultant polyhedra would look like, rather than the formulaic blue-face/pink-face/dot diagrams, which require painstaking mental expansion to produce examples.

I realize/gather that this would require a lot of work and might well be WP:OR, so I'm not so much asking for it as wishing for it. Still, there's no harm in doing so here on the Talk page.

--Thnidu (talk) 16:27, 6 January 2019 (UTC)


 * Polyhedronisme is a good place to start. It's a web app where you can put in arbitrary strings of Conway notation and it generates a 3d model you can rotate and zoom on. It doesn't implement every operation that's listed on this page, but it has all of the original ones. Antiprism is a more complete implementation, but it's less user-friendly. -Apocheir (talk) 16:53, 6 January 2019 (UTC)
 * Θanx.--Thnidu (talk) 00:25, 7 January 2019 (UTC)

about Archimedean solids by Conway original notations
Conway's original set of operators can create all of the Archimedean solids and Catalan solids, using the Platonic solids as seeds.

I believe that the previous claims on this page are not correct. Because Truncated cuboctahedron as one of Archimedean solid can not be made by truncate Cuboctahedron (taC=bC). The new face that can be created is not a square, but a rectangle. The same is true for Truncated icosidodecahedron. Kepler also knew the above. Today, see 'Polyhedra' by Peter R. Cromwell.

--中川 宏 (talk) 06:30, 17 May 2019 (UTC)


 * I'm struggling somewhat to understand your comments, but I think you're distinguishing geometric equality and topological equality. Unfortunately the writings on this topic often muddles the two. Many implementations insert an implicit canonicalization step before returning the result. The above statement is true in the topological sense, and is true in the geometric sense after canonicalization, but may not be true when omitting the canonicalization step. This page could be clearer about that. -Apocheir (talk) 17:41, 22 May 2019 (UTC)

A Commons file used on this page has been nominated for deletion
The following Wikimedia Commons file used on this page has been nominated for deletion: Participate in the deletion discussion at the. —Community Tech bot (talk) 18:07, 18 May 2019 (UTC)
 * Rhombictriacontahedron.jpg

Illustration for Conway polyhedron notation
you were right about the jagged parts of the graphics that were there previously, commons:Category:Polygon_chambers_(red_and_white). I updated them to fix that, and to make the middle lines in the polygons a little thinner. We seem to be converging on the same design for this graphic, so let's cooperate and save ourselves some effort.

Here are my concerns about the graphic that you want to use.
 * It's distracting that each shape is a different size.
 * I do not agree that using the different colors improves the graphic. Having them be the same color emphasizes that it's the same fundamental domain on a different polygon. Doesn't matter what color, though it should be something high-contrast.
 * Putting labels in English within the graphic itself makes it more difficult to use the graphic on non-English-language Wikipedia projects. There are at least six other language Wikipedias that have an article on this topic. It's better to put the label in the page text, where it can be searched and automatically translated.
 * Relatedly, all the shapes are combined into one file in your graphic. If they're separate, they're easier to label.
 * Simple graphics like this are often preferred to be in scaleable vector graphics (SVG) format, so they can be displayed at any size without distortion. It can also produce smaller files. Your PNG graphic is 209 KB, while the SVGs all together only about 1% of that. Not that 209 KB is a huge file, but it can make a difference on poor-quality connections.

Like I said, I'm willing to make changes to the SVG files to create something everyone's happy with. In fact, I already have made some changes, as I mentioned above.

As a side note, since you seem to have an interest in creating graphics for math articles, you might want to look into a SVG drawing program like Inkscape. -Apocheir (talk) 23:52, 20 October 2019 (UTC)


 * To suggest on the SVGs, vertices at the center may be helpful :). Also, Asymptote tends to perform better on rounded line ends (for the vertices). Harry Princeton (talk) 14:38, 27 October 2019 (UTC)


 * I can't tell what your suggestion is? SVG can be displayed in any web browser without converting it to a raster file first, unlike Asymptote (vector graphics language), which must be converted to PNG first (which negates the benefits of using a vector graphic format). -Apocheir (talk) 17:48, 27 October 2019 (UTC)

Faulty image for the Cross operation
The image for dX, called Conway_dXC.png, is missing a green point on the top surface, in the lower left. (Each side of the cube should have 12 points.) 80.208.70.76 (talk) 17:37, 9 January 2024 (UTC)