Talk:Regular icosahedron

Regular icosahedron is currently a Mathematics and mathematicians good article nominee. Nominated by Dedhert.Jr (talk) at 10:39, 29 July 2025 (UTC)

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Short description: Convex polyhedron with 20 triangular faces

Mathematics Mid‑priority

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Polyhedra

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Stellations

The stellations of the icosahedron are described in University of Toronto Studies Number 6 - The Fifty-Nine Icosahedra - by HSM Coxeter, P Du Val, HT Flather, and JF Petrie - University of Toronto Press 1938 (Derek Locke)

volume vs dodecahedron

(Comment from Trevor: I'd LOVE to see a proof on this. Til then, I don't buy it.) —Preceding unsigned comment added by Btrevoryoung (talk • contribs) 12:38, 14 June 2006

If you refer to the table of volumes in Platonic solid it is relatively easy to calculate. R, the circumradius, corresponds to the radius of the sphere that the polyhedron is inscribed in. If you do some calculations you will find that (volume of dodecahedron with circumradius R)/(volume of sphere with radius R) is greater than (volume of icosahedron with circumradius R)/(volume of sphere with radius R). This is an alternative way of explaining what the article states. This may be counter-intuitive to some because it is unlike the similar situation in regards to circles and polygons. I will leave the math to you.

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:

Rhombicdodecahedron.jpg

Participate in the deletion discussion at the nomination page. —Community Tech bot (talk) 17:54, 18 May 2019 (UTC)[reply]

Missing proofs

The formulas for dimensions etc are given with no proof or reference. Andreas ^(T) 13:48, 12 October 2023 (UTC)[reply]

Confusing construction

The section Construction describes in its second paragraph a method for constructing the regular icosahedron from a cube.

The edge-length of the cube is never mentioned, so this section needs work.

But even if the cube were assumed to have a specific edge-length, this paragraph is still extremely confusing.

I hope someone familiar with this construction can make this paragraph much clearer than it is.

The construction in Cartesian coordinates is also misleading. When reading an expression like

\left(0,\pm 1,\pm \varphi \right)

it is not immediately obvious that this is defining more than 2 points. The implication is of course that the signs are independent, but it could also be equally well understood as a shorthand for

\pm \left(0,1,\varphi \right)

. The article for the cube and the regular dodecahedron have the same quirk, but for the dodecahedron there is a figure that clarifies what is meant. Since the edit introducing

\left(0,\pm 1,\mp \varphi \right)

was reverted the question is how and whether this ambiguity should be addressed. — Preceding unsigned comment added by 2A01:CB08:8EA5:5F00:A4CB:7AE2:2DB6:9EFA (talk) 13:11, 21 January 2025 (UTC)[reply]

The FIRST construction claims you "attach" »two« pentagonal pyramids to each (of the two) pentagonal faces of the pentagonal antiprism. I doubt this is correct. Assuming "attach" means to overlap the two pentagonal faces with the equal size pentagonal base of the pent. pyramids, then that means you attach »ONE« pyramid to each, not two. Assuming you loose 1 face on each pyramid, that leaves 5 faces each. the antiprism has 10 triangular faces, and assuming its two pentagonal faces have been lost (by "attachment") that gives 20 faces total. Using TWO for each pentagon would mean 30 faces but you also be left with a bunch of edges hanging in space. What a mess.98.19.179.27 (talk) 08:40, 18 May 2025 (UTC)[reply]

Whether

\pm

or

\mp

, they represent the same meaning. Speaking of the first construction, I am not sure how would you put up those ideas; the source mentions the same as this page does. Dedhert.Jr (talk) 02:15, 19 May 2025 (UTC)[reply]

the 3d view

the 3d view has no edge lines which makes it difficult to discern the shape and faces. Jeb12321 (talk) 16:13, 2 June 2025 (UTC)[reply]

That's the STL image. If you're using a mobile phone, you probably cannot rotate the image. On the other hand, using a laptop enables you to rotate any STL image. Perhaps your screen shows a full white image without edges. I'll notify this in WikiProject. Dedhert.Jr (talk) 01:24, 3 June 2025 (UTC)[reply]