Edge lengths of x, f, o, etc.

Discussion of known convex regular-faced polytopes, including the Johnson solids in 3D, and higher dimensions; and the discovery of new ones.

Edge lengths of x, f, o, etc.

Postby student91 » Thu Jun 23, 2016 1:32 pm

Hello, I've just finished this year of university, and thus I've receded to the CRF-polytopes again. Now I've been thinking about our current writing system. There seems to be some ambiguity on what e.g. x5o3o means: does the letter 'x' mean a value of 1, or a value of 1/2?

When you look at x5o3o as a coordinate in a skew coordinate system ("position polytope" I thought wendy called it), it would mean the initial vertex is at a distance 1 from one plane, and incident to the other planes. When you then reflect this, it would result in a dodecahedron with edges of length 2, instead of 1.
When x, on the other hand means a value of 1/2, (And thus f=ϕ/2), things go horribly wrong during the EKF-process. (You'd have huge amounds of 1/2's piling up, making the reflections all go more and more inward.)

So does x mean a value of 1, and do all of our polytopes secretly have edge-lengths of 2, or is something else going on?
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Re: Edge lengths of x, f, o, etc.

Postby wendy » Fri Jun 24, 2016 8:01 am

The letters denote the short-chords of unit-edged polygons, with x and o standing in for {3} and {2} respectively.

The problem comes when you start dealing with the canonical forms of polytopes, which have a centre on the origion, and an edge of 2.

cube = (1,1,1,1...) ++ edge = 2
octa = (q,0,0,0...)++ edge = 2
icosa = (1,f,0)+ edge = 2
dodeca = (1,0,F)+ edge = 2

+ = all change of sign, even permutations
++ = all cange of sign, all permutations.

A 'unit sphere' is in the real world described as a '2-inch sphere'. This is because the rss(x,y,z) gives a diameter of 2.

The design of the matrix is meant to give the coordinate of the vector from the centre to the named point (ie 0,0,0 to 1,1,1) = sqrt 3, but the edge presented as a half-unit o----| which is reflected in a mirror (change of sign) as o----|----o

So when we write x5o3o, the edge goes from the vertex to the mirror. The x stands for 1 here but represents a double-edge.

But since x, f, o are held relative to an edge, we suppose that e=2 when mirror coordinates are taken to account.
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Re: Edge lengths of x, f, o, etc.

Postby quickfur » Wed Jun 29, 2016 6:27 pm

The way I think of it is that x = unit edge length, f = phi times edge length, ... etc.. While some people prefer working with edge length = 1 for obvious reasons, I find that in terms of manipulating Cartesian coordinates setting edge length = 2 produces nicer results, since it eliminates the 1/2 factor from many coordinates. That does mean x = 2, f = 2*phi, etc., for me, but that's OK because in terms of origin- or axis-centered coordinates it becomes ±1, ±phi, etc., which are simpler to write and work with. So I usually standardize on edge length = 2.
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