Mecejide wrote:There is also a compound of 1920 enes plus two more infinite families: 2^(n-1)/(n+1) n-simplices and 2^n/(n+1) n-simplices when n is a mersenne number. I think that’s all of them, though.
URL wrote:Mecejide wrote:There is also a compound of 1920 enes plus two more infinite families: 2^(n-1)/(n+1) n-simplices and 2^n/(n+1) n-simplices when n is a mersenne number. I think that’s all of them, though.
Where do these infinite families come from? Can you give some sort of description of the arrangement of simplexes on these compounds?
Mercejide wrote:There is also a compound of […] 2^(n-1)/(n+1) n-simplices and 2^n/(n+1) n-simplices when n is a mersenne number.
URL wrote:Actually, Mecejide, I'm not convinced that the compounds you describe even exist. In 3 dimensions, a tetrahedron is trivially inscribible in a cube, but I can't see why something like this should be the case for higher dimensions. Even if it were, I'm not sure how one could build a regular compound out of these simplexes (unless there was a clearly unique one for each vertex).
Care to elaborate?
viewtopic.php?f=25&t=2468 provides explicit coordinates for the vertices of an oca with edge length 1 inscribed in a hept with edge length 1/2.
wendy wrote:The regular compounds in dimensions higher than four, are restricted to 2^n and 2^-1, the former being cross-polytopes, the latter being simplexes.
The proof of the “only if” part of the statement of the last theorem may have seemed a bit unclear to the reader, and that is because it is false. To see why, let us consider a construction suggested by the abstracts of [1] and [4].
Theorem 5.3. It is possible to inscribe a regular simplex of dimension n in a hypercube of dimension n if and only if a Hadamard matrix of order n + 1 exists.
Proof. Whenever a string simplex of dimension n exists, we can simply replace all 0’s with −1’s and add a constant column of 1’s or−1’s to get a Hadamard matrix of order n + 1. Whenever a Hadamard matrix of order n + 1 exists, we can use the previous lemma to get a constant column, and then reverse this process to get a string simplex of dimension n.
var v = [[-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1],
[1,-1,1,-1,-1,-1,1,1,1,-1,1],
[1,1,-1,1,-1,-1,-1,1,1,1,-1],
[-1,1,1,-1,1,-1,-1,-1,1,1,1],
[1,-1,1,1,-1,1,-1,-1,-1,1,1],
[1,1,-1,1,1,-1,1,-1,-1,-1,1],
[1,1,1,-1,1,1,-1,1,-1,-1,-1],
[-1,1,1,1,-1,1,1,-1,1,-1,-1],
[-1,-1,1,1,1,-1,1,1,-1,1,-1],
[-1,-1,-1,1,1,1,-1,1,1,-1,1],
[-1,1,-1,-1,-1,1,1,1,-1,1,1]];
for (var i = 0; i < 11; i++)
for (var j = i+1; j < 11; j++) {
var c = 0;
for (var k = 0; k < 11; k++)
if (v[i][k] != v[j][k])
c++;
console.log("Distance: sqrt(2*"+c+")"); //The distance between two points equals the square root of twice the amount of distinct coordinates.
}
ndl wrote:Mecejide wrote:ndl wrote:Mecejide wrote:That is the blend of 24 sidpiths, in category S12.
Yes, it appears I didn't read that earlier post by J.B. fully.Mecejide wrote:How did you make it?
Which are you referring to?
How did you make the compound of 6 sidpiths?
I took the compound of 3 sidpith with convex hull of x3w4o3o and rotated it along the icoic extended symmetry axis and put them together. As mentioned above this can be done with any polychoron with icoic symmetry.
Here's the list of vertices:
- Code: Select all
#1
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#2
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#3
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#4
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Mecejide wrote:How did you rotate it?
wendy wrote:In order to be regular, either the hull or the core must be regular.
In the case of two pen or two icos, the core is o3x3x3o and o3x4x3o.
ndl wrote:I took the compound of 3 sidpith with convex hull of x3w4o3o and rotated it along the icoic extended symmetry axis and put them together. As mentioned above this can be done with any polychoron with icoic symmetry.
Here's the list of vertices:
- Code: Select all
#1
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#3
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#4
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#5
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#6
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ndl wrote:A small discovery:
The dual of Odip (8,8 duoprism) can be facetted to create 2-hex and 4-hex uniform compounds.
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