## Regular compounds in 4D

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

### Re: Regular compounds in 4D

There is at least one more. It is a compound of 72 ocas.
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### Re: Regular compounds in 4D

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.
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### Re: Regular compounds in 4D

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?
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### Re: Regular compounds in 4D

I made a mistake. There are actually 2 regular compounds of 1920 enes, not 1.
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?

The compounds of 2^(n-1)/(n+1) have the same vertex arrangements as demihypercubes, and the compounds of 2^n/(n+1) have the same vertex arrangements as hypercubes.
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### Re: Regular compounds in 4D

These would exist for all n = 2^k-1, wouldn't they? I see no reason for n to be a Mersenne prime specifically.
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### Re: Regular compounds in 4D

I said Mersenne number, not Mersenne prime.
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### Re: Regular compounds in 4D

Sorry, wasn't aware of that terminology.
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### Re: Regular compounds in 4D

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.

Mersenne numbers are of the form n=2k-1.

The latter one for k=2 (n=3) happens to be just "so" = {4,3}[2{3,3}]{3,4}.
The fromer one here is degenerate: it is just tet itself.

For the next k=3 (n=7) we get for the latter {4,3,3,3,3,3}[16{3,3,3,3,3,3}]{3,3,3,3,3,4}. - OBSA?
and the former then is its hemiation, the 8-oca compound within the hesa as its hull. - OBSA?

The latter seems to be always a fully regular Coxeter compound, the former then its hemiation.

--- rk
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### Re: Regular compounds in 4D

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?
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### Re: Regular compounds in 4D

I realized there is actually a third regular compound of 1920 enes.

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.
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### Re: Regular compounds in 4D

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.

Yep, just realized. But this still leaves questions. Can this be generalized to all dimensions 2^k – 1? Can this not be generalized to other dimensions?

I've asked this question on StackExchange: https://math.stackexchange.com/questions/3645941/when-can-a-regular-simplex-be-inscribed-in-a-regular-hypercube. Hopefully, someone will be able to answer.
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### Re: Regular compounds in 4D

So, there are simplexes in hypercubes for every dimension of the form 4k – 3 up to 663. Essentially, we can take a Hadamard matrix of size n with its first column identical, remove this column, and read rows as points. I don’t know if this construction is unique.

The question now is: are the compounds generated by constructing all such simplexes in a hypercube regular?
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### Re: Regular compounds in 4D

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 this is relatively straight-forward. Proof by example:

The vertex-rings of an n-simplex is the row n of pascal's triangles. Row 3 is 1,3,3,1, and holding a cube by opposite vertices reveals these as layers. The rings of a cross-polytope is 1, (2n-2), 1, so we see for the octahedron, 1,4,1. The cube does not hold an octahedron, because when set to equal diameters, the octahedron's vertices are on the equator and the cubes are above and below, by 19.4712 degrees.

The vertex-rings of a 4-simplex are 1,4,6,4,1. The four orthotope lies then in 1,6,1. In this case, there is a compound, since the 6 is further derived as a polytope being o3x3o, with rings 1,4,1.

We skip all odd dimensions, since the cube has no equatorial section.

In six dimensions, the section is 1,6,15,20,15,6,1 or row 6 of the pascal triangle. The 20 is derived by passing through 1,3,3,1 twice. That is, having 1 way to the first 1, there is only 1 way to the 20. Likewise, a path to the 3 is made in 3 ways, and from that 3, there are 3 paths to the 20. The middle number of an even row of a pascal triangle, ie row 2n, is made from the sum of squares of rown n, so 20 = 1, 3², 3², 1 = 1, 9, 9, 1. But this has no middle layer, so there is no compounds over the shared square between the 6-cube and 6-orthotope.

In eight dimensions, row 8 is 1,8,28,56,70,56,28,8,1, the middle row of 70 is o3o3o3x3o3o3o, gives 1,4²,6²,4²,1 or 1,16,36,16,1. Since the 36 derives from the square of a row of 4, we note there may be compounds there. The shape of the middle section, in any dimension, is a lace tower, comprising of layers of the rectates and co-rectates, ie x3o3o × o3o3x, o3x3o × o3x3o, and o3o3x × x3o3o. In 6D, we get x3o × o3x, then o3x × x3o.

Since the middle layer is itself a product of medial rectates, it suffices to note that one must have a succession of middle layers, to exhaustion, and hence you find cross-polytope only where the dimension is a power of 2.

Since a cross-polytope is the antiprism of a simplex of n-1, and that a cube is a line-prism of a cube n-1, this is a bi-directional relation (either makes the other), and hence you have simplexes in cubes, where n+1 is a power of 2.
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### Re: Regular compounds in 4D

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.

But regular simplexes can be inscribed in hypercubes of dimension other than 2^(k – 1). Here's a regular 11-simplex, inscribed in the 11-hypercube with ±1 coordinates:

(–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)
(–1,1,–1,–1,–1,1,1,1,–1,1,1)

In fact, the dimensions n in which an n-simplex can be inscribed in an n-hypercube are precisely those that are one less than the size of some Hadamard matrix. It's conjectured that the latter numbers are precisely the multiples of 4. Theorem 5.3 in https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Markov.pdf proves this.
Last edited by URL on Mon Apr 27, 2020 5:21 pm, edited 2 times in total.
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### Re: Regular compounds in 4D

[quote=Markov]Theorem 4.5.A regular simplex of dimensionn can be inscribed in a hypercube of dimensionn if and only if n= 2m−1for somem.[/quote]

While you can doggy up something, it does not equal a simplex. The figure in the table is not a simplex.
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### Re: Regular compounds in 4D

The text quite sneakily states on the next page:

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].

They then go on to prove, correctly,

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.

What theorem 4.3 actually proves is that for n ≠ 2^k + 1, not all of the simplexes' symmetries are symmetries of the hypercube. So, for these cases, the compounds creating by taking all such simplexes on the hypercube aren't regular, though they'd be vertex uniform.

And by the way, the coordinates I gave do actually describe a simplex (though the first coordinates were redundant in my original post, I've since fixed that). Here's a short JS script you can run to verify it. It should log "Distance: sqrt(2*6)" once for every pair of vertices.

Code: Select all
`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.  }`
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### Re: Regular compounds in 4D

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
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 -0.707106781186547520.00000000000000000   -1.70710678118654752   1.00000000000000000   0.707106781186547520.00000000000000000   -0.70710678118654752   1.00000000000000000   -1.707106781186547520.00000000000000000   -0.70710678118654752   1.00000000000000000   1.707106781186547520.00000000000000000   0.70710678118654752   1.00000000000000000   -1.707106781186547520.00000000000000000   0.70710678118654752   1.00000000000000000   1.707106781186547520.00000000000000000   1.70710678118654752   1.00000000000000000   -0.707106781186547520.00000000000000000   1.70710678118654752   1.00000000000000000   0.70710678118654752#20.00000000000000000   -1.00000000000000000   -1.70710678118654752   -0.707106781186547520.00000000000000000   -1.00000000000000000   -1.70710678118654752   0.707106781186547520.00000000000000000   -1.00000000000000000   -0.70710678118654752   -1.707106781186547520.00000000000000000   -1.00000000000000000   -0.70710678118654752   1.707106781186547520.00000000000000000  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How did you rotate it?
Mecejide
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### Re: Regular compounds in 4D

Mecejide wrote:How did you rotate it?

The same way you would rotate the ico to create the 2-ico compound.
ndl
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Posts: 77
Joined: Tue Nov 27, 2018 2:13 pm
Location: Louisville, KY

### Re: Regular compounds in 4D

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.

Peter Cromwell in his book "Polyhedra" uses a definition for regular compounds in which the vertices of the hull or faces of the kernel all lie on symmetry axes of the same kind, which means the hulls can be regular or quasi-regular (includes the rectifieds) or kernels must be duals of those. According to that definition the 2-pen, etc. compounds are surely included.
ndl
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Posts: 77
Joined: Tue Nov 27, 2018 2:13 pm
Location: Louisville, KY

### Re: Regular compounds in 4D

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
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How did you "rotate" the compound? Is this something you can do with Stella4D or externally? If the latter, how do you actually build a polychoron from that?
galoomba
Mononian

Posts: 6
Joined: Mon Oct 19, 2020 10:01 am

### Re: Regular compounds in 4D

ndl wrote:A small discovery:

The dual of Odip (8,8 duoprism) can be facetted to create 2-hex and 4-hex uniform compounds.

Hex is the 4-4 duotegum, so any (4n/n)-(4n/n) duotegum is a uniform compound of hexes.
galoomba
Mononian

Posts: 6
Joined: Mon Oct 19, 2020 10:01 am

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