Talk:Disdyakis triacontahedron

Remove calculation comment: Tom Ruen 08:40, 29 July 2006 (UTC)

• Side lengths for r=12:

s1=4.35411996 s2=6.55839670 s3=7.69022184 (Calculated by Garrith McLean)

Dihedral angle

This solid is not edge-transitive. It has 3 different kinds of edges. Shouldn't it therefore have 3 different dihedral angles?

Hexakis icosahedron?

Why's it also called a hexakis icosahedron? Professor M. Fiendish, Esq. 02:53, 2 September 2009 (UTC)

That's what Williams calls it, I guess each triangular face of the icosahedron is divided into 6 faces. Tom Ruen (talk) 03:03, 2 September 2009 (UTC)

Holden (Shapes, space, and symmetry, 1971) calls it that too. [1] Tom Ruen (talk) 03:08, 2 September 2009 (UTC)

That's also what it is in French and German

Sides?

Something is wrong with the edges. In the pictures, you clearly see that the sides are right-angled. In the infobox however; It says that the edges are 4, 6 and 10. Whith the pythagorean theorem we see:

$a^{2}+b^{2}=c^{2}={\sqrt {a^{2}+b^{2}}}=c={\sqrt {4^{2}+6^{2}}}\approx 7.21110255093$

That means that it is not 4, 6, 10. Does anyone know what it is? --Berntisso (talk) 19:39, 7 September 2011 (UTC)

First, they are not right-angled. They are only right-angled if you project them at a right angle. Secondly, the face configuration (4,6,10) is not referring to edge lengths; they are referring to how many faces surround each type of vertex.—Tetracube (talk) 20:09, 7 September 2011 (UTC)

Hmmm... the faces have right angles only as a spherical tiling with spherical triangle faces. There you can extract the angles from the V4.6.10 notation; internal angles are 360/4, 360/6, 360/10 or 90,60,36. The polyhedral face angles are a bit smaller - specifically 88d58'31", 58d14'17",32d46'12" (adding to 180 degrees) by Robert Williams. SockPuppetForTomruen (talk) 22:01, 7 September 2011 (UTC)

So, which are the proportions of the lenghts of the triangle? If the ratio is irrational, is there a good integer approximation? --RokerHRO (talk) 15:41, 10 February 2015 (UTC)

Twisty puzzles

Whether this figure can be a puzzle mechanism is currently the biggest unsolved problem in the area of mechanical puzzles. In the realm of twisty puzzles, it's known as "big chop". EdPeggJr (talk) 18:44, 24 August 2015 (UTC)

I moved the statement into a new Uses section. It would be good to get more sources for the claims. Tom Ruen (talk) 19:54, 24 August 2015 (UTC)

The outer shape for a twisty puzzle is very flexible. If a good mechanism was found, many variations of this would be made. It's not particularly the dodecahedral shape that's wanted, it's the big chop mechanism. I'd want the sphere version myself. But currently it's an unsolved problem. Tens of thousands of twisty puzzle mechanisms have been developed, but this unsolved one is the biggie. 98.212.151.160 (talk) 01:46, 25 August 2015 (UTC)

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Brilliant logo

Should I mention that the logo for the website Brilliant is a disdyakis triacontahedron? If so, would it go under uses? It seems to me like it ought to go in an “In popular culture” section, but there currently is none, and I don’t want to unnecessarily make one. QuarterNotes (talk) 18:41, 29 November 2019 (UTC)

Suggested new R for scaling

In working to add value to this (and other) articles, I have added a diagram of convex hulls that are used in the construction.

The sources I use are Mehmet Koca's work with quaternions and Weyl Orbit group theory and of course Eric W. Weisstein at MathWorld. Both of these are consistent in scaling factors. Yet, when I compare these to the R used in the Disdyakis_triacontahedron#Cartesian coordinates section, it is off by a small amount. This whole section was added (w/o source citation) by User:Quack5quack in May of 2022 and updated again with a slightly different (1.6%) R scale in June.

Interestingly, the original R is off the MTM and Koca specs by only .0007, so I am suggesting we change $R={\frac {5}{\phi +3}}\approx 1.08271$ in order to be exactly consistent with MTM and Koca sources cited.

Please see the diagram for my calculations showing the numbers and math used.

Any objections? Jgmoxness (talk) 20:03, 14 February 2023 (UTC)

It seems that whoever uploaded the old values for the angles in the Faces section got those expressions wrong. I had used those values as my input to calculate R, so those were wrong by a few percent as well. Now that you have updated the R values, I find that the angles are incorrect, so I'll update them to be consistent with yours. -- Quack5quack (talk) 13:37, 6 April 2023 (UTC)

Done. Pls cross-check. -- Quack5quack (talk) 14:04, 6 April 2023 (UTC)

There is definitely something wrong with the Cartesian coordinates as currently entered by User:Jgmoxness. You can verify that one of the triangles has vertices (5ϕ / (3 + ϕ), 0, 0), (ϕ, ϕ-1, 0), and (ϕ, 0, 1). You can also verify with elementary vector algebra that the angles of that triangle are 32.527 degrees, 59.081 degrees, and 88.391 degrees. But the angles ought to be 32.770, 58.238, and 88.992 degrees, so the given coordinates are not correct. You can also cross-check the dihedral angles for the given coordinates as being three different values: 155.773, 165.096, and 172.564 degrees, but all of them ought to be the single value 164.888 degrees. Pls debug the coordinates -- you're probably off by some radius factor on the 20 dodecahedral vertices. It will take me a while to correct it and if I do it, it will be numerical instead of symbolic. -- Quack5quack (talk) 00:18, 23 July 2023 (UTC)

I found the radii numerically. Start with the face centers of the truncated icosidodecahedron (the Archimedean solid). Those face centers will give the vertices of the disdyakis triacontahedron with scaling. Leaving the 10-valent vertices where they are, the 6-valent vertices need to be pushed in to R6 = 0.879774 of their former value, and the 4-valent vertices need to be pushed in to 0.848259 of their former value. At that point, all face angles are correct to 3 decimal places and all dihedrals are equal to 2 or 3 decimal places. If you can figure out the expressions behind those numbers, add them to the article. Otherwise I'll add the numbers in a few days. -- Quack5quack (talk) 14:36, 23 July 2023 (UTC)

A bit of clarification is needed for the statement "...with the Cartesian coordinates as currently entered by User:Jgmoxness". I only modified the factor R. The coordinate specifications were not entered or modified by me. As for the issue related to dihedral angles, you give no citations or sources for the statement "But the angles ought to be 32.770, 58.238, and 88.992 degrees." Please provide more detail for your calculations and/or citations.

The suggestion that "...you're probably off by some radius factor on the 20 dodecahedral vertices" is simply wrong since the coordinates for that are given and have Norm=${\sqrt {3}}$, which is what my data shows.

As I did above, I again verified that R must be as modified in order to be consistent with Mathworld and M. Koca's quaternion Weyl Orbit paper. That improved update is available in the form of a blog post on my website here, with links to a Mathematica Notebook, a PDF, and SVG image of the detail.

BTW - contrary to your method of "finding the R numerically" in order to construct the icosidodecahedron vertices, Koca (and thus my work) use the Coxeter-Dynkin H3 Weyl orbits to produce the icosidodecahedron directly from the quaternions, such that R is merely derived (symbolically) from those results.

I will try to document the dihedral angles you are concerned about using Koca's quaternions with the normalized scalar products $\oplus$. But at this point - if the coordinates in the article, Koca, and Mathworld only match using my new R, your dihedral angle calculations are likely the problem.

Jgmoxness (talk) 18:58, 26 July 2023 (UTC)

Ok, I created a graphic of the dihedral calculation, along with the angles for AB,BC,AC. The dihedral angle checks, but the face triangle angles are shown to be different. The full update is in the blog post reference above, but I've also loaded it into WP to the right.

Jgmoxness (talk) 14:50, 27 July 2023 (UTC)

Thanks for checking. The angles are calculated using the standard relation for Catalan solids (in fact any face transitive solid) as described at Catalan_solid#geometry. There are four angles to calculate: the three angles of the triangular face and the dihedral angle. There are three equations linking each plane angle to the dihedral:

$\sin(\theta /2)=\cos(\pi /p)/\cos(\alpha _{p}/2)$

$\sin(\theta /2)=\cos(\pi /q)/\cos(\alpha _{q}/2)$

$\sin(\theta /2)=\cos(\pi /r)/\cos(\alpha _{r}/2)$

Where p, q, r are 4, 6, and 10 in some order, theta is the dihedral, and the fourth equation is that the sum of the three alphas should be 180 degrees. This works out eventually to the expressions given in the article itself, and whose numerical values I gave above on the talk page.

Are you getting those four angles with your coordinate values? The dihedral needs to be equal for all three edges of the triangle. I'm getting a discrepancy as described above. I could get all the angles to agree by tweaking the radii of two of the vertex subsets also as described above. -- Quack5quack (talk) 00:07, 28 July 2023 (UTC)

They are not mine, actually you (and Mennotk) added them, as well as all the content you refer to in Catalan_solid#geometry and in your last comment. If you change the radii, you change the norms and thus mismatch Mathworld and Koca and even your own coordinates. Your face triangle angles don't even add to 180?? I am not going to try to recreate those numbers using your and Mennotk's suggested math, since my numbers seem to work just fine WRT Mathworld and Koca, as well as the current coordinates in the article. I have shown all of my work and given you peer reviewed citations, so I suggest you and Mennotk do the same for your numbers (not simply verbal descriptions of the process) in order for us to figure out what the correct numbers are. Jgmoxness (talk) 02:45, 28 July 2023 (UTC)

BTW - I've just added more detail to the various ABC triangle edges and angles, with comparison between the comments in the Talk section above Talk:Disdyakis_triacontahedron#Sides?. It turns out even those numbers do not add to 180. They are ABC={32.77, 58.2381, 88.9753}$^{\circ }$ from DMS for a total of 179.983$^{\circ }$.

My calculations from a MTM defined polygon is ABC={32.5275, 59.0812, 88.3913}$^{\circ }$ for a total of 180$^{\circ }$. This gives a difference of {-0.242535, 0.84317, -0.583968}$^{\circ }$. As I had shown before, I included in the calculation for ABC projected to the XY plane, but as discussed above in Sides?, that has a 90$^{\circ }$ angle and is not precisely the same shape.

It seems there is a third set of angles from redcrab with angles of ABC={32.7703, 58.2381, 88.9917}, which add to 180$^{\circ }$. So I am looking into the consistency between the Great Rhombicosidodecahedron coordinates and the scaling of each orbit of the Disdyakis Triacontahedron. I will let you know what I come up with.

Jgmoxness (talk) 18:19, 28 July 2023 (UTC)

I see you updated the article with more of your Quack5quack and Mennotk material w/o citations. Can you please provide peer reviewed citation or reference (w/o using your own work on other pages) or at least explain why no citations? It's looking like a NOR problem here. It may be right, but at this point I just can't know w/o 3rd opinion or better documentation of your calculations.Jgmoxness (talk) 01:58, 29 July 2023 (UTC)

Please do verify my calculations. The only citations are those in the Catalan solid article. Nobody seems to have bothered to publish the disdyakis coordinates in a way that's comprehensible, so I had to do the tedious vector algebra myself. (This used to be the same problem with snub cube and snub dodecahedron in the past but over time those articles have improved.) There was a site called dmccooey.com that published a long laundry list of coords but nothing else. As you correctly point out, there are many sites that publish their own expressions, all of which are topologically equivalent to the disdyakis triacontahedron but the majority of them yield angles that do not agree, so they fall in the gap between topology and geometry. In fact, erecting pyramids on the rhombic triacontahedron fails for the exact reason that expanding an icosidodecahedron fails: the square faces end up as rectangles and that messes up the symmetry. I got started on this journey when trying to recreate what those articles said and finding nothing seemed to agree with each other exactly. It is a tough shape to get accurately, but the formulae in Catalan solid are a good start. -- Quack5quack (talk) 16:13, 29 July 2023 (UTC)

Addendum: the site I was referring to: http://dmccooey.com/polyhedra/DisdyakisTriacontahedron.html -- Quack5quack (talk) 16:15, 29 July 2023 (UTC)

I just added that to the expressions for the two radii R and S, and replaced the vector algebra with the citation. It at least checks out numerically and agrees with preexisting formulae from other articles. The ratio R is obtained by dividing the 6-vertex radius by the 10-vertex radius, both radii obtained from the cited site. The ratio S is obtained by dividing the 4-vertex radius by the 10-vertex radius. -- Quack5quack (talk) 18:25, 29 July 2023 (UTC)

Thanks for those updates - it would have saved me much time if I had had the D McCooey reference you were relying on. I have been able to figure out the issue. It seems for the Dodecahedron circumradius, in the previous hull graphic I used a factor related to the one that normalizes the unit icosahedron, specifically ${\sqrt {3}}/Norm({0,1,\phi })\approx .9106$, while D McCooey has a scaled Dodecahedron which is, in Koca parlance, based on the Norms of the quaternion definition of ${\sqrt {3}}*R={\frac {Norm(B)}{Norm(A)}}\approx .9378$. Your factor $S=1/Norm(A)\approx .92101$. As you determined, the Catalan angle equations are corrected using D McCooey's numbers. Thanks for fixing this. I have updated the articles hull graphic to be consistent with your changes.Jgmoxness (talk) 03:27, 30 July 2023 (UTC)

Thank you for updating the figure. Glad the article is stronger now all round. I hadn't referred to the dmccooey site in recent times, trying to derive the coordinates ab initio from the known angles instead, mainly to serve as a cross-check. The dmccooey numbers are accurate and self-consistent for the polyhedra I've cross-checked, but there was at least one Wikipedia article where that site was banned as "not a reliable source", which in retrospect might have just been someone's mathematical snobbery against empirically calculated values. All's well that ends well though. -- Quack5quack (talk) 14:25, 30 July 2023 (UTC)

You're welcome. I rechecked what I just wrote last night and needed to correct it. It turns out the Mathworld numbers DO match D McCooey's (and redcrab) numbers. It was in my attempt to correct your original R coordinates that prompted me to incorrectly change the dodecahedron circumradius to ${\sqrt {3}}/Norm({0,1,\phi })\approx .9106$. Anyway, I learned a few things and was able to confirm all your angle calculations, etc.

BTW - You made a reference above to "This used to be the same problem with snub cube and snub dodecahedron#Geometric relations in the past but over time those articles have improved...the square faces end up as rectangles and that messes up the symmetry". This is due to a really interesting chirality between the regular and irregular mirrored pairs of these, which I documented the coordinates for on that page.

Overlay of regular and semiregular truncated icosidodecahedra and snub dodecahedra

Cheers, Jgmoxness (talk) 15:42, 30 July 2023 (UTC)