The Shape Hidden Inside a Stack of Cubes

Added: 2026-06-01

[00:00:02]All right, welcome back. Today we're going to look at a stratified cubicle seed triple and its dimensional tower.

[00:00:20]So for the setup, let X be a smooth open judically convex N manifold whose closure has the combinotaurics of a regular N cube.

[00:00:29]Keep in mind the following um where v is a finite subset of the boundary of x. These are vertices uh of the cubical boundary and define the corresponding closed block by x union partial x.

[00:00:53]So a folational completion of the block B is obtained by attaching or selecting code dimension one foliations whose leaves accumulate at specified vertices of the boundary. For a pair of vertices let f subi j code dimension one foliation whose leaves accumulate at the two boundary vertices v subi v subj.

[00:01:22]So for a leaf L in F, we require the closure of the leaf intersecting with the boundary of X to be these two vertices. This this vertex pair on the boundary. So each leaf has two distinguished ideal end points, namely the vertices V sub I, V subj.

[00:01:50]Vertex distances in the n cube. In the n cubical case, the possible distances between vertices are 1 square root of two up to square of n. The largest possible vertex distance uh is the long diagonal distance. The supreum of the distances over the vertex pairs equals square of n. So we restrict attention to the antipal pairs of vertices of the n cube.

[00:02:23]Since the n cub has 2 to the n vertices, it has exactly 2 to the n minus one antipital vertex pairs. Let a subn denote the set of antipital vertex pairs.

[00:02:48]For each antipital pair A is an element of A subn, consider a co-dimension one foliation F sub A whose leaves are mutually difforphic to the model class L subn.

[00:03:08]So each leaf is topologically an open cylinder of dimension n minus one with its two ends accumulating at a pair of antipital vertices of the cube.

[00:03:20]So among the leaves of each foliation f sub a select a distinguished leaf which is volume maximizing inside the cubicle block and satisfies the appropriate constant positive curvature condition.

[00:03:33]Denote this rigidified leaf by sigma sub a after completion at its two ideal points. Sigma may be viewed as a topological n minus one sphere with two distinguished cone points. For n= 3, these are spindle or football surfaces.

[00:03:53]Since there are 2 to the n minus one antipital vertex pairs, this produces exactly 2 to the n minus one distinguished rigidified sheets. Define the n dimensional seed object by I subn defined to be the union of these sigma surfaces.

[00:04:20]So the interaction locus the singular or interaction locus is the subset of i subn where two or more sheets meet.

[00:04:31]We'll define it as the pair wise intersections. The union of the pair wise intersections where beta is not equal to gamma.

[00:04:42]It's a subset of i subn. This is just where the the surfaces intersect. That's intersection locus.

[00:04:52]So gamma subn carries a natural stratification by sheet multiplicity.

[00:04:57]For each k greater than equal to two, one may define the k-fold interaction stratum by the point in I subn where p lies on exactly k distinct sheets when n= 3. The sheets sigma sub beta are two-dimensional. So their pair-wise intersections are generically onedimensional. In that case, gamma sub 3 is a seam graph. For n greater than three, however, the pair-wise intersections are generically n minus 2dimensional.

[00:05:38]So gamma subn should be regarded as a seam complex or interaction complex rather than merely a graph.

[00:05:50]The symmetry group pi subn finally denote pi subn the finite group of stratum preserving symmetries acting on the pair. The full cubical symmetry group is the hyper ocahedral group and this group acts naturally on the vertices of the n cube and hence on the set of antipital pairs a subn.

[00:06:12]The actual symmetry group of the rigid rigidified stratified object may be a proper subgroup of B subn depending on if we use global or pie-wise isometries uh which will have preserved the selected sheets and their incidents.

[00:06:32]So one may define is subn defined to be the automorphism group of the strata stratum preserving um of the pair is suben gamma subn.

[00:06:46]If one wants to retain only symmetries induced from the cubicle block, we define py like that.

[00:06:58]And the stratified cubicle seed triple in dimension n. The construction produces a stratified cubicle seed triple where I subn is the union of 2 to the n minus one rigidified antipital sheets.

[00:07:14]Gamma is the higher dimensional interaction complex and pi is a finite symmetry group of stratum preserving symmetries.

[00:07:27]We'll look at the case n= 3. Now for n= 3 this recovers the concrete seed where i sub3 is the union of four maximally inflated constant curvature football surfaces.

[00:07:41]G uh gamma sub3 is their seam graph and pi sub3 is the corresponding finite symmetry group.

[00:07:52]So now let's pose a question.

[00:07:56]The preceding construction produces for each finite dimension n greater than equal to 3 a cubical seed triple where i subn is assembled from the 2 to the n minus one and typical rigidified sheets of the n cube. Gamma is the corresponding sheet interaction complex and pi is the finite symmetry group of stratum preserving symmetries.

[00:08:24]One might naturally ask is there a limiting object is there a natural limiting object in which we take n to infinity of the triple obtained from the tower of finite dimensional cubicle seeds.

[00:08:41]So we get this uh postulate postulated or hypothesized tower of objects for each n dimension.

[00:08:55]And can the finite seed triples be organized into a compatible system of stratified spaces with symmetry?

[00:09:03]So that one may define through inverse limit or direct limit or some other construction.

[00:09:30]If such a limit exists, what geometric, cominatorial, and spectral information is retained by the infinite seed triple?

[00:09:39]Does gamma sub infinity become a universal interaction complex encoding the stable incident, salon, symmetry, and spectral data of all finite dimensional cubicle seeds?

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