Wikipedia:Reference desk/Archives/Mathematics/2023 January 30

= January 30 =

Where is tau?!
Thinking about yet another reasons why τ is perhaps more succinct than π in many cases, I decided to take a look at the Wikipedia article on the constant...which does not appear to exist! How can this be so? Surely "THE" circle constant deserves its own page. Just how many times are we going to have to write "2π" before we finally realize that tau is very much a thing of its own? Ok, all rants aside, I am seriously considering creating the article myself. **EDIT** Ok, I see now, it's currently listed as a subsection of the article Turn(angle). I still say tau needs to have it's own article. Comments? Earl of Arundel (talk) 15:57, 30 January 2023 (UTC)


 * Probably not, at least not yet. Even the link you point to says only that one educational website accepts it, a few programming languages implement it, and it's been used in one research article. --jpgordon&#x1d122;&#x1d106;&#x1D110;&#x1d107; 16:25, 30 January 2023 (UTC)
 * It's not clear to me that turn (angle) is the best host for the section (I think I would put it as a section of pi), but the content of that section seems pretty good. –jacobolus (t) 17:17, 30 January 2023 (UTC)
 * There is no lack of redirects:
 * Tau (2π)
 * Tau (circle constant)
 * Tau (2pi)
 * Tau (twice pi)
 * Tau (proposed mathematical constant)
 * Tau (number)
 * Tau (mathematical constant)
 * Tau (constant)
 * Tau constant
 * Tau circle constant
 * --Lambiam 17:57, 30 January 2023 (UTC)
 * That's a lot of redirects! There's a little more aboot tau under Pi and Pi Day. NadVolum (talk) 18:23, 30 January 2023 (UTC)
 * It's also the first entry at Tau (disambiguation) and has a direct link in the hatnote on Tau. It's hard to miss for anyone looking up tau. PrimeHunter (talk) 18:29, 30 January 2023 (UTC)
 * I have seen the argument that if we should demonstrate intelligence with radio signals to possible aliens, or look for intelligence, then binary digits of 2&pi; might be more universally identifiable than &pi;. (I made the same post in 2008 if it sounds vaguely familiar to any oldies here) PrimeHunter (talk) 18:44, 30 January 2023 (UTC)
 * Like broadcasting 1100100100001111110110101... instead of 1100100100001111110110101...? --Lambiam 18:56, 30 January 2023 (UTC)
 * Ha ha ;-) NadVolum (talk) 19:26, 30 January 2023 (UTC)

So it's just me then. Tau has been relegated a mere side-note to some other article and no one seems to care. We might as well move pi to the Turn (angle) article then, under sub-heading "The Half Turn"... Earl of Arundel (talk) 20:34, 30 January 2023 (UTC)
 * I don't think that's a Wikipedia thing, though. In the wider math world, while it's not quite true to say no one cares, it's probably accurate to say not a lot of people care very much.  I actually think it would be reasonable to cover the tau movement, as a movement, in its own article, but the goals of the movement are effectively hopeless; if we were redoing math from scratch there would be some decent arguments for using tau, but as a reform it's doomed. --Trovatore (talk) 20:42, 30 January 2023 (UTC)
 * It's mentioned at Pi with a link on $τ = 2π$. Maybe it should be mentioned elsewhere but I don't see a need for its own article. Some things are so closely related that splitting them seems unhelpful and will just cause repetition. For example, even number and odd number redirect to the same article parity (mathematics). PrimeHunter (talk) 20:48, 30 January 2023 (UTC)
 * If over a dozen redirects to a subsection of an article isn't reason enough for a dedicated article, I don't know what is. Earl of Arundel (talk) 21:25, 30 January 2023 (UTC)
 * Tau (2π) was an article but was merged to Pi in 2012 at Talk:Tau (2π)/Archive 3. PrimeHunter (talk) 22:03, 30 January 2023 (UTC)
 * So I see. Well that makes more sense now. I kept thinking, "am I really the only person who has thought of having a dedicated page for this?". Clearly not. Earl of Arundel (talk) 13:37, 31 January 2023 (UTC)
 * As far as I can see, the listing made by above contains less than a dozen, not over a dozen redirects. Possibly there are more, I didn't bother to check, but it's not immediately obvious there actually are more than 12 of them. BTW, number of redirects reflects just a number of plausible search terms, which results from many possible ways of expressing tau itself, pi itself, theis descriptions and their combinations. It has nothing to do with notability, which is the primary criterion for the article to exist. And the notability has not been proven so far.... --CiaPan (talk) 22:31, 30 January 2023 (UTC)
 * Actually, insofar as notability goes, tau seems to satisfy that criteria just fine. A cursory search yields numerous articles, books, even programming languages (eg. Python, Julia) where it has been implemented. Earl of Arundel (talk) 13:52, 31 January 2023 (UTC)
 * @Earl of Arundel If you go to the trouble of gathering a bunch of citations to mentions/uses in reliable sources, you could probably make enough notability case to keep an article alive at tau (circle constant) or similar, mostly focusing on the social movement aspect. For better or worse, more than a decade after the “tau manifesto” it’s not clear whether it will have much lasting impact. In practice few people seem to be rushing to replace their pis with taus. –jacobolus (t) 01:23, 2 February 2023 (UTC)
 * It would definitely be worth the effort, I think. And who knows? It may even go on to inspire future mathematicians. Earl of Arundel (talk) 20:48, 5 February 2023 (UTC)
 * Almost all uses of one of these redirects in mainspace are in the context of a discussion of the proposal to use $τ$ as an alternative for $\pi$ (such as in Pi Day: "Alternative dates for the holiday include ... June 28 (6.28, an approximation of 2π or tau)." Other uses are in disambiguation pages for "Tau", or in lists of mathematical constants. The only genuine use I saw was in . --Lambiam 23:22, 30 January 2023 (UTC)
 * To be clear, I certainly do not think tau should have a separate article as a number. I think it would be justifiable to have an article on the movement, which appears to me to satisfy the GNG. --Trovatore (talk) 20:52, 30 January 2023 (UTC)
 * It's not really about "redoing math from scratch" though. It's just recognizing a very important mathematical constant for what it is. Otherwise we may never clean up all of that "2π" clutter we see sprinkled throughout SO MANY equations. It's atrocious. It's also not very helpful. Knowing that a 90 degree rotation is π/2 radians doesn't really illuminate the relationship properly. Whereas telling the student that it is simply 1/4τ radians, that makes all the sense in the world. Here's another. We say that the area of a circle is πr^2 and the surface area of a sphere 4πr^2. It almost seems arbitrary. If instead you were to explain that the former is the area of a r*r triangle scaled by τ (thus 1/2τr^2) and the latter a tetrahedron formed by assembling of 4 of those triangular sheets (having the area of a circle "drawn above" said sphere), that makes way more sense. It's intuitive. Which is my point. As to an article dedicated to "the movement", now THAT truly would be an article that no one really would care much about! Earl of Arundel (talk) 21:18, 30 January 2023 (UTC)
 * It's not going to get cleaned up. Live with it.  Just my advice; obviously you can do what you want. --Trovatore (talk) 21:32, 30 January 2023 (UTC)
 * "What it is" is a matter of convention, personal preference, and convenience in particular contexts. For example, in the context of Schwarz–Christoffel mapping in complex analysis, we construct conformal maps from the half plane or disk to an arbitrary polygon using maps of the form

f(z) = a + c\int^z { \prod_k{\left(\zeta - z_k\right)^{\!-\beta_k}} \mathrm{d}\zeta }, $$
 * where the exponents $\beta_k$ range from $$-1$$ to $$1,$$ so that $\pi\beta_k$  is the turning angle at each vertex of the polygon. So for example, a square root with $\beta_k = \tfrac12$  results in a turn of $\tfrac12\pi.$  In this context is is entirely reasonable to consider $$\pi$$ to be more natural/fundamental than $$2\pi$$.
 * Similarly, it is often more convenient to work with the diameter of a circle than its radius. (In my opinion, after comparing flat geometry to spherical and hyperbolic geometry, the diameter is actually a conceptually/formulaically clearer way to represent the circle’s size than the radius, and the latter mainly became dominant because we have a convention of drawing a circle using the apparatus of the pair of compasses. But there are several alternative (equivalent) geometrical definitions of a circle that lend themselves to thinking about the diameter instead of the radius. For example, if we think of the circle as the locus of points from which directions to the two ends of the diameter are perpendicular – which we might call the Thales's theorem description of the circle, also related to the idea of an orthoptic of an ellipse, when we take the limit as the ellipse becomes infinitely flat – then again it makes more sense to think about $$\pi,$$ the ratio of circumference to diameter.
 * Of course there are also contexts where $$2\pi$$ or $\tfrac12\pi$ are more natural constants to think of as fundamental. –jacobolus (t) 22:41, 30 January 2023 (UTC)
 * I am not advocating the abolishment of π here. Tau and pi both have their place. And yes, even τ/4 could be considered a constant in its own right. (The "orthogonality constant" perhaps?) Earl of Arundel (talk) 13:33, 31 January 2023 (UTC)

Rhombicosidodecahedron expansion/contraction question.
Consider the following four dimensional shape. at w=a it is a dodecahedron, at w = b it is a Rhombicosidodecahedron and at w = c it is an icosahedron. (sort of like https://upload.wikimedia.org/wikipedia/commons/5/5a/P4-A11-P5.gif without the rotation around the Z axis). I *think* it is possible to set the values of a, b and c so that all of the edges are the same size, right? This would basically be two different hypercupolae with the same "larger" side adjoined, right?

(Also, shouldn't there be an icoahedral hypercupola?)Naraht (talk) 20:18, 30 January 2023 (UTC)


 * It is possible to scale the regular convex dodecahedron, rhombicosidodecahedron and icosahedron such that all edges have unit length. Placing them disjointedly in $$n$$-dimensional space, $$n\ge 3,$$ results in a shape with 92 vertices and 152 edges, all the same length. Presumably, you want each to be embedded in one of three parallel hyperplanes of four-dimensional space and then to be connected somehow, but to make the question answerable you need to clarify which vertices in each of the three are connected to which in the other two. It is not clear what the relation is between the variable $$w$$ and this configuration; I suppose that what is relevant are the distances between the hyperplanes. --Lambiam 21:10, 30 January 2023 (UTC)
 * Yes, I'm looking for the values of w for the parallel hyperplanes. As for which vertices connect, I'd call it the "Natural" way. the vertices of the Icosahedron connect in pentagonal pyramids to the corners of the pentagons of the rhombicosidodecahedron and the vertices of the Dodecahedron connect in triangular pyramids to the triangles of the rhombicosidodecahedron.Naraht (talk) 21:36, 31 January 2023 (UTC)
 * If these are the only new edges, 60 between the icosahedron and the rhombicosidodecahedron and 60 between the dodecahedron and the rhombicosidodecahedron, just put the icosahedron and dodecahedron at distances $$\sqrt{(5-\sqrt{5})/10}$$ and $$\sqrt{2/3}$$ times the unit length away from the rhombicosidodecahedron. Adding these edges is not sufficient to view the combination as a polychoron; for that, you also have to define its cells.  --Lambiam 08:19, 1 February 2023 (UTC)
 * Defining the Cells is pretty easy, I think. On the "Icosahedron half", you have 12 pentagonal pyramids where just a point (at the top) is on the icosahedron vertex, 30 triangular prisms (with and edge between two squares) is on the icosahedron and 20 triangular prisms where a triangle is on the icosahedron. On the "Dodecahedron half", there are 20 triangular pyramids (where just a point is on the dodecahedron), 30 triangular prisms (with an edge between two squares of the prism on the dodecahedron) and 12 pentagonal prisms (where the pentagon is on the dodecahedron).
 * The problem with the icosahedral cupola is that it's like the heptagonal cupola: if you insist on unit-length edges, it won't fit in Euclidean space. It will work if you don't demand unit-length edges, or if you make it in hyperbolic space (then it appears as part of the expanded {3,5,3}). Double sharp (talk) 23:04, 31 January 2023 (UTC)
 * Ah. the opposite corners of the Icosahedron are "too far apart".Naraht (talk) 19:23, 1 February 2023 (UTC)