New & old LEGO seqs; production cuts

[Jonathan Post]

Dear Søren,

Consider how two cube-cells in a hypercube are adjacent, analogously to how
two squares are adjacent in a cube.

Now stretch the hypercube along 1, 2, or 3 axes to make a 4-D Lego block. an
axbxcxd can overlap and stick to an exbxcxd along 1 dimension, or a exfxcxd
along 2 directions, or an exfxgxd along 3 directions.

Remember also a 4-D block rotates, not around any of 3 orthogonal axes (and
combinations of them, noncommutatively) as in 3-space, but around a plane,
in particular about any of 6 orthogonal planes: xy, xz, zt, yz, yt, zt (and
combinations of them).

I think that we exclude the transformation which turns a block inside out,
and thus are looking at a subset of the automorphisms of a hypercube, the
same as for its dual, the 16-cell.

Hypercubes are also called tesseracts. A
tesseract<http://mathworld.wolfram.com/Tesseract.html>has 16 polytope
vertices <http://mathworld.wolfram.com/PolytopeVertex.html>, 32 polytope
edges <http://mathworld.wolfram.com/PolytopeEdge.html>, 24
squares<http://mathworld.wolfram.com/Square.html>,
and eight cubes <http://mathworld.wolfram.com/Cube.html>.

So how many cubes in common are two stuck-together hypercubes?  So how many
cubes in common are two stuck-together rectangular stretched hypercubes?

There are people in seqfans with better 4-D visualization powers than I.


On 11/30/06, Søren Eilers <eilers at math.ku.dk> wrote:
>
>  On 02/11/06 21:54, "Jonathan Post" <jvospost3 at gmail.com> wrote:
>
> He and I agree, however, that it's a shame that the manufacturer doesn't
> make 4-D Lego blocks, or, more properly, the ones that they do make are of
> extension along the t-axis only in the trivial way, and cannot be freely
> rotated in Minkowski space.
>
> So, I wonder, how well does your approach to counting under symmetry work
> if extended to an additional spacial dimension?
>
> I suppose that we can consider an AxBxC 4-D Lego to be actually an AxBxCx1
> Lego in Z^4. I think that adjacency is well-defined, all rotations are about
> planes, and there is a relationship to 4-D analogues of polyominoes, i.e.
> to polyhypercubes.
>
>
> Dear Jonathan
>
> These are very intriguing questions, especially because my main goal is to
> understand how large an increase of flexibility there is when going from
> building with (3D) legos in a restricted way (like making them into towers
> of maximal length)  to working freely in space. Such questions could be
> asked in higher dimensions as well.
>
> What kind of attachments would you allow? With 3D legos, of course, we
> allow the block with a corner in (x,y,z) to be put on one with a corner in
> (x',y',z') if |z-z'|=1 and the projections of the blocks to the XY plane
> overlap, so there is an obvious lack of symmetry between the axes which I do
> not see off hand how best to resolve in 4D.
>
> Best,
> Søren
>
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