dodecahedron wrote:Would someone like to hep me find nonconvex johnson solids.
wendy wrote:One of the interesting ones is the pentagon-pentagram tegum, which is made of 25 regular tetrahedra.
dodecahedron wrote:I also suppose by pentagon-pentagram you mean pentagonal crossed cupola
quickfur wrote:dodecahedron wrote:I also suppose by pentagon-pentagram you mean pentagonal crossed cupola
No, she's talking about tegums, which are, loosely speaking, "diamond-like" shapes. The pentagon-pentagram tegum is what you get by placing a pentagon in the XY plane, a pentagram in the ZW plane, and taking the convex hull.
sqrt[(1/sqrt(2))^2 + (1/sqrt(2))^2] = sqrt[1/2 + 1/2] = 1 (i.e. unity again)
sqrt[(sqrt[(5+sqrt(5))/10])^2 + (sqrt[(5-sqrt(5))/10])^2] = sqrt[(5+sqrt(5))/10 + (5-sqrt(5))/10] = sqrt[(5+sqrt(5)+5-sqrt(5))/10] = sqrt[10/10] = 1
sqrt([1/(2*sin(pi*d/n))]^2 + [1/(2*sin(pi*b/m))]^2)
Klitzing wrote:We already have two solutions of that problem: n=m=4, d=b=1 and n=m=5, d=1, b=2. - My conjecture would be that those are the only ones; but I have no proof.
Someone would like to come in, providing one?
Klitzing wrote: We already have two solutions of that problem: n=m=4, d=b=1 and n=m=5, d=1, b=2. - My conjecture would be that those are the only ones; but I have no proof.
Someone would like to come in, providing one?
dodecahedron wrote:ok, i think the pentagon-pentagram tegum is interesting.however look at the topic it's polyhedron.
Klitzing wrote:Have not given any thought so far, but how about this setup:
Not considering regular integral faces and non-convex solids, but the other way round. I.e.
- allowing for polygrammal faces,
- but still asking for local convexity at least.
Local convexity differs from global convexity. Global convexity does not allow for polygrammal faces. But local convexity does. In fact it is defined as a solid, with all of its vertex figures are globally convex - with respect to that figure of one dimension less. E.g. the sissid = {5/2, 5} is locally convex. But gad = {5, 5/2} OTOH is NOT locally convex.
Has anyone some ideas on how to estimate the potential magnitude of this set?
--- rk
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