Talk:Bilinski dodecahedron

A fact from Bilinski dodecahedron appeared on Wikipedia's Main Page in the Did you know column on 5 June 2016 (check views). The text of the entry was as follows:

Did you know... that Stanko Bilinski's 1960 rediscovery of the Bilinski dodecahedron corrected a 75-year-old omission from the list of convex polyhedra with congruent rhombic faces?

A record of the entry may be seen at Wikipedia:Recent additions/2016/June. The nomination discussion and review may be seen at Template:Did you know nominations/Bilinski dodecahedron.

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«These zonohedra are projection envelopes of the hypercubes...»[edit]

@Steelpillow (talk · contribs):

In the Bilinski dodecahedron#Zonohedra with golden rhombic faces section, in the § starting with «These zonohedra are projection envelopes of the hypercubes, with n-dimensional projection basis, with golden ratio (φ)...»:

I don't know whether «These zonohedra» include the 2 golden rhombohedra (projection envelopes of the cube?) & the golden rhombus (projection envelope of the square?); do you, please?
I don't understand how a given point of an n-cube is actually projected onto the corresponding zonohedron with golden rhombic faces; do you, please?

In advance, thank you very much for your answers! —JavBol (talk) 00:02, 11 November 2022 (UTC)[reply]

I am not familiar with the details of the individual projections here. The "Projective n-cube image" at the bottom of the table shows how the various j-faces of the n-cube map onto and into the image zonohedron. One can think of the n-cube's higher dimensions as collapsing into the internal structure of the 3D polyhedron; the outer surface of the zonohedron is formed only by those 0, 1 and 2-faces of the n-cube which happen to project onto that position and are not hidden inside. An example transformation matrix, for n = 6, is offered above the table. I agree that it is all badly explained and the language is ambiguous.

Does the term "hypercube" include the point, line segment, square and cube? The article on the Hypercube claims that it does, but does not cite a source for this claim so it could be wrong; I have now added a citation tag there. (One reason I prefer Coxeter's "measure polytope" for the general class). This indeed makes it difficult to say which figures "these zonohedra" is intended to cover.

Also, I think this whole section does not belong here, it should be within the main Zonohedron article.

As it happens, I have recently updated my own web page on this polyhedral dissection (though in the context of quasicrystals not n-cubes), here. But it is not peer reviewed, so should not be cited here.

Hope this helps. — Cheers, Steelpillow (Talk) 05:40, 11 November 2022 (UTC)[reply]