3

Disclaimer: i am bioinformatician and programmer, please excuse if my wording and definitions are far from elegant and occasionally imprecise.

Intro: I am interested in space tessellations of n dimensional spaces with following properties:

1 a tessellation must be made with only one type of element, with identical angles and edge lengths
e.g. cubic honeycomb is okay, pythagorean tiling is not (one type but two sub-types of different size) and https://en.wikipedia.org/wiki/Truncated_trihexagonal_tiling is also not (three types of elements)

2 for n dimensional space, each n dimensional element is built out of one type of n-1 dimensional elements,
e.g. cubic honeycomb is okay, hexagonal prismatic honeycomb is not

3 elements must connect only through n-1 dimensional elements
e.g. hexagons on a 2D plane are okay, squares are not okay for 2D plane (they connect to 8 others, 4 through edges, 4 through vertices)

Question: For 2D, hexagons are an example which satisfy these conditions. Are there any examples for N > 2?
Sub-question: Does https://en.wikipedia.org/wiki/Rhombic_dodecahedral_honeycomb satisfy these conditions? (i cannot come up with an idea on how to verify this)
Note 1: I am mainly interested in N = 3 and N = 4, but general solution would be best.
Note 2: If they exist, pointers what to look for (keywords, books, articles, etc) would be greatly appreciated

EDIT: A related question: Are there higher-dimensional tessellations touching only nearest neighbours?

Maciej
  • 161
  • Rhombic dodecahedral honeycomb fails the 3rd condition. Maybe you are asking for too much. – Ivan Neretin Sep 03 '18 at 16:30
  • @IvanNeretin Could you demonstrate how does it fail the 3rd condition? It's not obvious to me from the pictures I've found, and in general a method to check this properly would be nice. Also: i do not think that 'too much' is a thing in math. Either something exists, or not. If it only happens in 2D and not higher dimensions, it would be interesting to see why. – Maciej Sep 04 '18 at 07:50
  • 1
    Read the Wikipedia link till the words: "...The honeycomb is thus cell-transitive, face-transitive and edge-transitive; but it is not vertex-transitive, as it has two kinds of vertex. The vertices with the obtuse rhombic face angles have 4 cells. The vertices with the acute rhombic face angles have 6 cells." Now if there are 6 cells meeting at a vertex, then each cell must have 5 neighbors, but it only has 4 faces meeting at that vertex, hence there are four face-neighbors and one single-point-neighbor. – Ivan Neretin Sep 04 '18 at 08:12

1 Answers1

0

Here is an example that satisfies your criteria, I think:

The (equilateral, most symmetric) diplo-simplex with $2N+2$ vertices tiles space in 2D, 3D, and 7D.

2D diplo-simplex is hexagon, vertices of tesselation correspond to non-lattice hexagonal packing.

3D diplo-simplex is cube, vertices correspond to trivial cubic lattice. Doesn't the cubic lattice in arbitrary dimension satisfy your criteria?

7D diplo-simplex tesselation's vertices correspond to E7* lattice, i.e. the sole Delaunay polytope of the E7* lattice is the 7D diplo-simplex.

311411
  • 3,576
Dan Haxton
  • 1
  • Cubic lattices do not satisfy point 3 - they touch each others vertices. Thanks for the effort though! – Maciej Jan 18 '22 at 08:25
  • Oh I see; sorry I was reading too quickly. That requirement number 3 does not seem entirely natural imho because it excludes the triangular aka hexagonal lattice "A2" in 2D. The packing of triangles as opposed to hexagons is a very symmetric tiling of 2D space with one element, arguably superior to the hexagonal packing you describe (which is 2/3 of the A2 lattice corresponding to the packing of triangles). Imho in 2D, you want to exclude a rectangular packing but include the square packing to define an interesting category. 7D diplo-simplex is actually excluded by (1), not equilateral – Dan Haxton Jan 19 '22 at 04:21
  • In other words it seems that you are imposing a maximum convexity requirement with (3) and I do not think this can be satisfied.. I think you are requiring that the coordination number of a vertex in N dimensions must be N+1, i.e. vertex figure and voronoi polytope is a simplex, and while the hexagonal packing satisfies your criteria because its voronoi tesselation is the A2 lattice, the A2 lattice is the only simplex packing, so I think the hexagonal packing is provably the only example that satisfies your criteria – Dan Haxton Jan 19 '22 at 04:43
  • I think that the Schlafli 2_21 polytope is equilateral, having edges of the same length, and I think that this is the best you can do. But it does not have only one kind of 5-face, having both orthoplex and simplex 5-faces. I am not sure what high-n n-dimensional shape besides cube, orthoplex, simplex has only one n-1 face. – Dan Haxton Jan 19 '22 at 04:48
  • The thing is, i dont want this for any kind of 'beauty' or 'mathematically interesting' reasons. The requirements are dictated by the project i am thinking about. Perhaps i can loosen the third requirement a bit, I would have to think. The fun thing is, you've mentioned voronoi diagrams - I am aware of them, and they were in fact my first shot for the project, my question kinda stemmed from the fact i couldnt find any alternative. It would be -very curious- if this was my only choice. – Maciej Jan 19 '22 at 20:19