Packing of polytopes

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

Packing of polytopes

Postby snayperx » Tue Dec 11, 2012 8:11 pm

I've searching for a way how to check if a given polytope is able for packing in it its smaller copies in such a way that these copies have one common vertex with a "big" polytope and meet with other small polytopes in their vertices. For the regular polytopes it is trivial, however I tried to find a way to do it for other (uniform) polytopes. Please guide me in which direction I should search. Maybe the Petrie polygons could help?
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Re: Packing of polytopes

Postby Klitzing » Tue Dec 11, 2012 9:04 pm

Any non-convex uniform polytope scaled down in size with respect to to one of its vertices (as center of scaling), would peak out of the un-scaled form. Therefore "able for packing in it" would not be possible, whatever you mean with "packing". (Probably no overlaps. But would gaps be allowed?)

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Re: Packing of polytopes

Postby snayperx » Tue Dec 11, 2012 9:38 pm

It is obviously that for non-convex polytopes such a packing procedure is not possible. It could be determined e.g. basing on non-convex polyhedra. However, if the initial object is a convex uniform polytope, say convex uniform polychoron, I suppose the packing could be constructed of some of them. The question is how to examine this?
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Re: Packing of polytopes

Postby wendy » Wed Dec 12, 2012 11:04 am

A group of three squares in an L shape, copied four times, will make a larger shape of three-squares. So i can't see how the convex rule applies.

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     o----o----o
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     o    o----o
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     o----o    o----o----o
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     o    o----o----o    o
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     o----o----o----o----o



There are plenty of non-convex polytopes in any dimension, which tile all-space, purely without rotation. For example, the Z, N, V, L and W pentimos tile without rotation, but form a larger cell only by way of rotation, if a perfect larger copy is sought.

There are then the cases where one can make shapes out of the equals of the penrose tiles, which lead to larger copies of themselves.
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Re: Packing of polytopes

Postby Klitzing » Wed Dec 12, 2012 1:23 pm

First of all, Wendy, your L-tiling is not uniform. Moreover those Ls are placed within in a different orientation, they aren't scaled copies with respect to the respective vertex!

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Re: Packing of polytopes

Postby wendy » Thu Dec 13, 2012 8:33 am

The question asked does not seem very clear as to whether all-space is to be filled, or if just 'v' polytopes be packed inside the figure, or whatever.

If for example, it means that, for example, you take a truncated tetrahedron as the P, and get small truncated tetrahedra p, then you could place a small p at the vertices of P, and then enlargen p until they touch and share vertices. The necessary conditions here is that it's always possible to put p at every vertex of P, but the increase in size has to be compared at every kind of edge (ringed vertex). For example, the truncated tetrahedron would be x3x3o, so there are two different kinds of edge to measure, and it may, or may not meet at both at the same time. My suspicion is that it will always work for a single crossed node, and generally fail (with exceptions) for multiple cross-nodes. Obviously, x3o3x4o works, since it's o3x4o3o.

Snubs have in general, three degeees of freedom (sC, sD), but the s24ch has only two degrees of freedom, It may work with this one, though.

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Re: Packing of polytopes

Postby snayperx » Fri Dec 14, 2012 5:34 am

Thank you for the answers. In 3D, the truncated tetrahedron P is able to pack in it its small copies p in such a way, that small copies meet in common vertices and vertices of P. For this purpose small copies should be scaled three times with respect to P. My question is how to perform such a procedure for 4D convex polytopes?
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Re: Packing of polytopes

Postby Klitzing » Fri Dec 14, 2012 10:24 am

snayperx wrote:Thank you for the answers. In 3D, the truncated tetrahedron P is able to pack in it its small copies p in such a way, that small copies meet in common vertices and vertices of P. For this purpose small copies should be scaled three times with respect to P. My question is how to perform such a procedure for 4D convex polytopes?


Exactly the same as you do in 3D. :)

In fact, you consider some vertex of your convex uniform polytope. (Thus any would serve.) Now for any incident edge you would elongate that much beyond the other vertex, and consider a perpendicular hyperplane getting from infinity closer and closer to the polytope. (Kind of a sliding caliper.) When it hits the polytope you'll stop. Then you read off from that caliper the respective extend.

As the larger polytope was uniform, the larger edges would be all of the same size. Therefore, your question just comes down to whether all read off measures would be the same for any edge type.

Clearly, whenever the original polytope was quasiregular, i.e. has exactly one node ringed within its Dynkin symbol, then all edges are of the same type. Thus your quest would be fulfilled for free. In any other case your quest comes down to do those measures and to compare those. Most generally those would be different then.

Well, there is no real 4D caliper gauge. Thus you would have to do that by calculation for sure. This can be done as follows. First you consider the right triangle vertex - edge-center - body-center. This shows, if the polytope would be scaled according to unit edges, and further more you would align the polytope such that the edge under consideration would point along the x-axis (while the polytope itself is centered at the origin), that the fixed vertex would have x-coordinate -1/2. Further you would have to calculate the coordinates (at least the x-coord.s) of all other vertices. Then take the maximum thereof. Accordingly the searched for gauge measure for that edge direction would be the maximum value plus 1/2.

Then apply the same for the other edge directions as well, compare those measures, and you are done.

The calculation of the coordinates for all polytopes of consideration, and moreover for any relevant orientation, clearly would be the hardest job of your quest.

BTW., the scaling factor from the large to the small polytope versions would be, if all those direction values coincide, twice that calculated maximum value plus 1.

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