Talk:Pentagonal hexecontahedron

Geometry and coordinates

This addition was reverted as from a nonreliable source, while I have no reason to believe it incorrect. If anyone wants to try to defend or cite it better, I'll leave it here for reference. Tom Ruen (talk) 02:26, 20 August 2017 (UTC)

The faces are irregular pentagons with two long edges and three short edges. The long edges adjoin each other.

• Define phi = (1+sqrt(5))/2.
• Define x = cbrt((phi+sqrt(phi−5/27))/2)+cbrt((phi−sqrt(phi−5/27))/2).
• Then the short edges are 1/x in length. The long edges are (x*(7*phi+2)+(5*phi-3)+2*(8-3*phi)/x)/31 in length, as calculated by D. Mccooey at [[1]]
• The dihedral angle between all faces is acos(−(2*(x+(2/x))*(15*phi+1) +(16*phi+15))/209).
• The Cartesian coordinates are given at [2]

Edge length ratio

The edge length ratio should be 1:1.7498525667362 instead of 1:1.7489525667362. (The digits 98 instead of 89)

Rationale (from German page):

Let

$t={\frac {1}{12}}\left({\sqrt[{3}]{44+12\Phi \,(9+{\sqrt {81\Phi -15}})}}+{\sqrt[{3}]{44+12\Phi \,(9-{\sqrt {81\Phi -15}})}}-4\right)$

then the ratio between the two side lengths a and b is:

$2a\left(1-2t^{2}\right)=b\left(1+2t\right)$

Setting b=1 we find

$a={\frac {1+2t}{2(1-2t^{2})}}$

which has a value of 1.7498525667362 which agrees with the value given except the digits 8 and 9 are flipped.

Additionally, Mathematica gives the (largest real) root of the polynomial:

$1-11x+51x^{2}-128x^{3}+177x^{4}-122x^{5}+31x^{6}$

which is again 1.74985256673620. (Corrected in article page).

Examples

Coral lamp by David Trubridge Coral lamp

Weyl Orbit Isn't Defined

Twice the article refers to icosahedral symmetry of the Weyl orbits. Unfortunately, the term "Weyl orbits" isn't defined anywhere, and it doesn't have a Wikipedia entry. The entry Weyl group uses the word orbits once, and doesn't define it either.

Since none of the other Catalan solid articles use Weyl orbits (possibly because it doesn't apply; I don't know) I am tempted to just remove the passages. But it would help if this can be fixed with a parenthetical explanation, or a link to a longer section in the Weyl group article. John Gamble (talk) 08:13, 14 April 2024 (UTC)

I modified the wording to be more precise based on the cited article. Jgmoxness (talk) 01:18, 24 August 2024 (UTC)