Talk:Triakis icosahedron

A triakis icosahedron is a catalan solid, the dual polyhedron to the truncated icosahedron. ... This is the first stellation of the icosahedron ...

It can't be both. A Catalan, being the dual of a convex solid, is convex; a stellation is not. Each consists of 60 obtuse isosceles triangles, but in the Catalan they lie on 60 distinct planes (being dual to 60 distinct points), while in the stellation they are all on the twenty planes of the icosahedron (that's the nature of a stellation). —Anton Sherwood 06:53, 12 January 2006 (UTC)

You're right, there is a difference!

They're the same topology, but different geometric coordinates. However the name triakis icosahedron would still seem to apply to both, since both have an icosahedron with faces replaced by short tetrahedron pyramids.

A tetrahedron pyramid is a pentachoron, I think you mean triangular pyramid. 4 T C 14:46, 29 January 2010 (UTC)

I verified that Wenninger calls it by this name in his book, Polyhedron Models.

Obviously this difference needs to be noted in the article, but both sources should remain. Tom Ruen 07:27, 12 January 2006 (UTC)

Wenninger also uses the name Triakisicosahedron (one word) for the Catalan in Dual Models!

How about this: "The name triakis icosahedron has been applied to two different polyhedra whose surfaces each consist of 60 obtuse isosceles triangles. Its more usual meaning is a Catalan solid, bla bla bla. The same name has also been applied (e.g. by Magnus Wenninger) to the first stellation of the regular icosahedron ..."

—Anton Sherwood 07:52, 12 January 2006 (UTC)

I'm content with that - when in doubt, state the facts as known!

Interestingly, I just noticed my (quick&dirty) program generates dual polyhedron by connecting dual vertices at face centers and this can cause concave faces on nonregular polyhedra, and in this case "my" face-centered dual of the truncated dodecahedron looks like the First stellation of icosahedron! SO it is due to different (possible) definitions of dual geometry, even if the formal one is defined to make the duals convex.

Worse for me, my definition can cause nonplanar faces if not triangles!

With a bit more (unavailable) time I could confirm my guess that the vertices of the First stellation of icosahedron match the face centers of a truncated dodecahedron!

Tom Ruen 08:05, 12 January 2006 (UTC)

Okay probably different vertices anyway.

ALSO interesting. A Triakis octahedron is topologically identical to the Stellated octahedron BUT a polyhedron generated with verices as the face-centers of a regular Truncated cube are a bit too short to match vertices of the Stellated octahedron.

Tom Ruen 08:16, 12 January 2006 (UTC)

... A BIT CRAZY, but I added 3 related stellations with images. The "interpretation" is a little messy. Both Stellations and nonconvex uniform polyhedra have different interpretations of their surfaces. The "pure" approach introduces no new vertices on the intersections, while a "physical" model" approach computes the actual intersections and only shows the external surfaces. So the "external intersected surfaces" are topologically related.

This difference is similar to star polygons. For example, a "pure" pentagram has 5 vertices and is self-intersecting, while its "surface" can be considered a concave decagon polygon by computing vertices at the intersections and removing the interior edges.

A great dodecahedron is NOT a Triakis icosahedron, but its intersected surface IS! Tom Ruen 09:57, 12 January 2006 (UTC)

Yes. (sigh) A small stellated dodecahedron is a pentakis dodecahedron with very tall pyramids, a great dodecahedron is a triakis icosahedron with inverted pyramids, a great stellated dodecahedron is a triakis icosahedron with very tall pyramids and a [[great icosahedron] is a pentakis dodecahedron with inverted pyramids and pentagrammic pyramids inserted into the dimples. Am I right? Professor M. Fiendish, Esq. 01:24, 2 September 2009 (UTC)

I'm not sure. Pentagrammic pyramid or pentagonal pyramid? Tom Ruen (talk) 01:41, 2 September 2009 (UTC)

You know something, if you try making a triakis icosahedron using equilateral triangles you get something that looks more like a great stellated dodecahedron! (You want proof? I built one!) It's getting even crazier now! 4 T C 14:45, 29 January 2010 (UTC)

I think I know why - the effect is like stellating a great dodecahedron. 4 T C 14:46, 29 January 2010 (UTC)

File:First stellation of icosahedron.png Nominated for Deletion

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Hypnohedron?

This Spanish blog portraying mathematical absurdities shows what appears to be an animation of a puzzle resembling either a great dodecahedron (most likely an Alexander's Star) or a triakis icosahedron:

http://casarandom.blogspot.com/2012/01/comunicado-los-lectores-que-tienden-0.html?m=1

The animation's GIF file itself is called hypnoedron, but I think it should have been called hypnohedron instead. ​‑‑🌀⁠SilSinn^AL982100⁠💬 02:21, 4 February 2019 (UTC)

Orthogonal projections

has five special orthogonal projections, centered on a vertex, on two types of edges, and two types of faces: hexagonal and pentagonal. The last two correspond to the A₂ and H₂ Coxeter planes.
Copied from truncated dodecahedron. Can anyone fix the text? Turbojet (talk) 12:05, 7 April 2022 (UTC)

Cartesian coordinates

Re: User:David_Eppstein's comments on undoing a revision without any prior discussion:

unsourced

well, it is rather simple math, but one source I've used for verification of that is Koca - Catalan Solids Derived From 3D-Root Systems and Quaternions https://arxiv.org/abs/0908.3272

not very grammatical

This section wording was copied from other similar pages, so I am not sure of your point. Do you have specific changes as a WP editor to understand your specific issues?

too dogmatic (this is one possible realization of this shape, not the only realization)

Ok, I understand - so we could insert some weasel words to that affect.

For example:

---

One possible set of coordinates for the 32 vertices of the triakis icosahedron hulls fall in two sets:

• Twelve vertices of regular icosahedron.
• Twenty vertices of ${\frac {1}{11}}{\sqrt {75+6{\sqrt {5}}}}\approx 0.8548$ scaled regular dodecahedron.

These hulls are visualized in the figure below:

Jgmoxness (talk)

Re "not very grammatical": for instance, it should be "Twelve vertices of the regular icosahedron".

Re "this section wording was copied from other similar pages": many of our polyhedron pages are in extremely bad shape. In this article, I have made a recent attempt to put it into better shape, by using actual text rather than image-heavy demonstrations, using content specific to this shape rather than copying-and-pasting generic content for a wide class of polyhedra to all polyhedron articles in that class, and basing all content on reliable sources. Copying from the other bad polyhedron pages goes in the reverse direction, pushing the article back to being in as bad a state as most of the other polyhedron article.

Re using WP:WEASEL: No. One problem is that the coordinates you describe are only for the Catalan solid version of this polyhedron. If they belong anywhere, they need to be in that section, with proper sources for those coordinates. Another problem is that, even when the Catalan solid version is used, there is nothing requiring it to be centered at the origin, at that scale, and having the specific orientation used in that choice of coordinates. —David Eppstein (talk) 19:39, 31 January 2023 (UTC)

there is nothing requiring it to be centered at the origin

While I appreciate your attempts to clean up the page, this sounds like you're being pedantic (and I am being kind using that term) and don't see the pedagogical value in having an accurate and cited visualization of the structure. Images can be powerful tools and text-only misses that opportunity (IMO).

I don't have a problem moving it under the catalan solid section with your suggested changes.

Let's get another "expert" opinion before you remove it again.

Jgmoxness (talk) 20:11, 31 January 2023 (UTC)

I have ignored your demand that I defer to someone else and that we based our edits here on expertise rather than on what reliable sources say. Your newer version is improved but still problematic:

• "One possible set of coordinates": but these are not coordinates. It's a geometric construction, but not described in terms of coordinates.
• "triakis icosahedron hulls": what do you mean by hulls?
• It is meaningless to talk about scaling a regular dodecahedron without any description of how big it was before you scaled it
• To construct the triakis icosahedron in this way, the icosahedron and dodecahedron need to be in a specific orientation with respect to each other.
• The figure has a lot of unexplained text that appears to be original research

—David Eppstein (talk) 21:17, 31 January 2023 (UTC)

You really should make an attempt to correct deficiencies rather than simply remove the content (this is WP after all). But I will play nice and correct the specific issues - but it would be nice for you be more supportive than antagonistic.

• "One possible set of coordinates": but these are not coordinates. It's a geometric construction, but not described in terms of coordinates.

Describing the coordinates by reference to well known geometry is much cleaner than redundantly copying them from the referenced pages (that is what other pages do). Would you accept my using the phrase "Cartesian coordinates using the vertices of the following Platonic solids..."

• "triakis icosahedron hulls": what do you mean by hulls?

There are two sets of radii (each generating a hull of points with the same norm). If you don't want to use the term hull, I can drop it (but it is used commonly elsewhere in WP Geometry pages). Do you have other word(s) you prefer to refer to this?

• It is meaningless to talk about scaling a regular dodecahedron without any description of how big it was before you scaled it

It is rather obvious the default would be unit radius (as in the icosahedron, but I can add the words "from unit radius".

• To construct the triakis icosahedron in this way, the icosahedron and dodecahedron need to be in a specific orientation with respect to each other.

Agreed, that is why showing the figure is pedagogical and the cited references re:quaternion Weyl orbits of A3, B3 and H3 groups describes that. It can be a bit tedious, but I will add the specific scaling and permutation descriptions that orient the orbits if you like. They are basically (before scaling to unity):

For the icosahedron $\pm (0,1,1/\phi )$

For the dodecahedron (being an icosahedron and a cube $\pm (0,1,1/\phi ^{2})$ and $\pm (1,1,1)/\phi$)

• The figure has a lot of unexplained text that appears to be original research

What specifically is in it that you think still needs explanation (given my answers above)?

This isn't OR - the published citation provides needed references to what is standard geometry and group theory.

Jgmoxness (talk) 22:40, 31 January 2023 (UTC)

I can link directly to the "Cartesian coordinate" section of the Icosahedron and Dodecahedron pages and add to description of the link that these are the vertex coordinates being scaled. Jgmoxness (talk) 23:22, 31 January 2023 (UTC)

You are still exhibiting the same sloppiness that mars most of our polyhedron articles. The geometric construction of this polyhedron is already described in the article; providing coordinates should mean providing formulas for the actual Cartesian (or polar) coordinates of the points, not just how one might go about calculating them. "Hull" in this context can only mean convex hull; it is not a word used for concentric co-circular systems of vertices. You talk about the radius of a regular dodecahedron and icosahedron as if there is only one possible radius, but in fact they have three, all important: the inradius, midradius, and circumradius. It is not obvious to me which one you mean. Quaternion Weyl orbits are far more WP:TECHNICAL than is needed here. —David Eppstein (talk) 00:02, 1 February 2023 (UTC)

Did you even read the CARTESIAN coordinate section of the icosahedron and dodecahedron pages referenced? They have all the information for vertex coordinates needed for full definition (with an added scaling factor for the dodecahedron). It seems those pages are the status quo, so why can't they be used for reference?

It is obvious when I clarified the hull was a reference to the two vertex coordinate norms that this by definition is the circumradius for each set of norms (aka. hulls, aka. circumradius, etc.). If you want me to use that word - no problem. But your pendantry is verging on idiocy.

I agree going into the detail of the Weyl orbits is not necessary for the casual reader (and that is why I did NOT include any words to that effect)! My additions are simply adding specific info on the precise coordinates for each vertex (some people like to see that w/o diving into assorted papers) and a visual to see how the scaled Platonic solids create the Triakis Icosahedron. This is very informative - no?

You can't claim OR and then also claim the info is already in the article. You are all over the map here. I will repost with corrections you suggest and if you keep undoing I will call for arbitration (or it gets forced by the undo limits- either way, it's your call). Jgmoxness (talk) 00:35, 1 February 2023 (UTC)

What you think is obvious is not obvious to other people. Including me. I did not understand what you meant by "hull" and I did not understand what you meant by "radius". If I cannot understand these things then I think it is likely that other people will not understand these things.

Re "It seems those pages are the status quo, so why can't they be used for reference?": (1) Wikipedia articles cannot be used as references for other Wikipedia articles. (2) Did you even read my earlier comment about the bad shape of most of the polyhedron articles? Icosahedron and dodecahedron are examples of this.

Also, if you insist on escalating to higher Wikipedia authorities, it's your call, but you will be the one in violation of WP:BRD by pushing your edits through despite them being reverted. —David Eppstein (talk) 01:16, 1 February 2023 (UTC)

Au contraire my good friend... WP:BRD specifically states the you should "Revert an edit if it is not an improvement, AND only if you cannot immediately refine it. "

Every single one of your suggestions or questions was a trivial edit that you could have made yourself. I know your credentials, so it is a bit disingenuous for you to suggest you couldn't figure out what kind of radius I was referring to, etc. Actually, I suspect you are just being... pedantic and simply enjoy wrestling with pigs in the mud. I've been doing WP for quite a number of years and these math sites have a number of those personalities. Don't be one of them. Jgmoxness (talk) 01:39, 1 February 2023 (UTC)

Finding published sources (not MathWorld) that provide a clear description of actual Cartesian coordinates (not "go look at these other two Wikipedia articles, hope that the coordinates they describe are in the right orientation with respect to each other, and scale one of them) is not trivial. If it were trivial, I would have done it when I made other major cleanups to the article not long ago. Also, no, I really did not know whether you meant inradius, midradius, circumradius, or something else. There are too many possibilities and too many different ways for you to be making things up rather than relying on published sources for me to be confident in guessing which one you mean. —David Eppstein (talk) 01:50, 1 February 2023 (UTC)

But they ARE in the right orientation as referenced - and that article is not that bad - if it were you should fix it BEFORE you worry about this one. It's like your asking every (other) WP editor (other than yourself) to recreate redundant article content or fix the underlying linked ones. You are not the (only) arbiter of the extent to which the reference is good or bad. Arbitration is needed here IMO.

If you do the math you will see that the WP page references have good coordinates as needed (but you are not putting in any effort in order to even know that (e.g. pig in the mud). I will copy the coordinates and put in the unit norm factors from those pages so it will all be on one page - just for you, but I think it will be less clear and more distracting (IMO).

As for "I really did not know whether you meant inradius, midradius, circumradius, or something else.", the context of a cartesian vertex coordinate for the (convex) hull can't be anything other than a circumradius. Maybe you're not even reading my edits before undoing them. That would be simply rude (or worse).

Jgmoxness (talk) 02:06, 1 February 2023 (UTC)

Your latest version was still crammed full of errors. I am convinced that you still have no idea what the words "permutations" and "convex hull" mean. And your latest claim the context of a cartesian vertex coordinate for the (convex) hull can't be anything other than a circumradius is obvious and total bullshit. I would be interested in finding sources for the canonical polyhedron version of this polyhedron, where it is the midsphere radius of the icosahedron that controls the scale of the dodecahedron, for instance. But at least this iteration was close enough that I could finally copyedit it into something more accurate rather than just reverting. —David Eppstein (talk) 08:00, 1 February 2023 (UTC)

The article and our edits to it are much improved, thanks for all your help.

Jgmoxness (talk) 14:07, 1 February 2023 (UTC)