Problems 21, 66, and 116 in my Open Problem Collection concern polytopes with lattice coordinates—that is, polygons, polyhedra, or higher-dimensional analogs with vertices the square or triangular grids. (In higher dimensions, I’m most interested in the \(n\)-dimensional integer lattice and the \(n\)-simplex honeycomb).
This was largely inspired by one of my favorite mathematical facts: given a triangular grid with \(n\) points per side, you can find exactly \(\binom{n+2}{4}\) equilateral triangles with vertices on the grid. However, it turns out that there isn’t a similarly nice polynomial description of tetrahedra in a tetrahedron or of triangles in a tetrahedron. (Thanks to Anders Kaseorg for his Rust program that computed the number of triangles in all tetrahedra with 1000 or fewer points per side.)
The \(4\)-simplex (the \(4\)-dimensional analog of a triangle or tetrahedron) with \(n-1\) points per side, has a total of \(\binom{n+2}{4}\) points, so there is some correspondence between points in some \(4\)-dimensional polytope, and triangles in the triangular grid. This extends to other analogs of this problem: the number of squares in the square grid is the same as the number of points in a \(4\)-dimensional pyramid.
The \(\binom{n+2}{4}\) equilateral triangles
I put a Javascript applet on my webpage that illustrates a bijection between size-\(4\) subsets of \(n+2\) objects and triangles in the \(n\)-points-per-side grid. You can choose different subsets and see the resulting triangles. (The applet does not work on mobile.)
Polygons with vertices in \(\mathbb{Z}^n\)
This was also inspired by Mathologer video “What does this prove? Some of the most gorgeous visual ‘shrink’ proofs ever invented”, where Burkard Polster visually illustrates that the only regular polygons with vertices in \(\mathbb{Z}^n\) (and thus the \(n\)-simplex honeycomb) are equilateral triangles, squares, and regular hexagons.
Polyhedra with vertices in \(\mathbb{Z}^3\)
There are some surprising examples of polyhedra in the grid, including cubes with no faces parallel to the \(xy\)-, \(xz\)-, or \(yz\)-planes.
While there are lots of polytopes that can be written with vertices in \(\mathbb{Z}^3\), Alaska resident and friend RavenclawPrefect cleverly uses Legendre’s three-square theorem to prove that there’s no way to write the uniform triangular prism this way! However, he provides a cute embedding in \(\mathbb{Z}^5\): the convex hull of $$\scriptsize{\{(0,0,1,0,0),(0,1,0,0,0),(1,0,0,0,0),(0,0,1,1,1),(0,1,0,1,1),(1,0,0,1,1)}\}.$$
Polygons on a “centered \(n\)-gon”
I asked a question on Math Stack Exchange, “When is it possible to find a regular \(k\)-gon in a centered \(n\)-gon“—where “centered \(n\)-gon” refers to the diagram that you get when illustrating central polygonal numbers. These diagrams are one of many possible generalizations of the triangular, square, and centered hexagonal grids. (Although it’s worth noting that the centered triangular grid is different from the ordinary triangular grid.)
If you have any ideas about this, let me know on Twitter or post an answer to the Stack Exchange question above.
A catalog of polytopes and grids
On my OEIS wiki page, I’ve created some tables that show different kinds of polytopes in different kinds of grids. There are quite a number of combinations of polygons/polyhedra and grids that either don’t have an OEIS sequence or that I have been unable to find.
| Square | Rectangular | Centered Square | Triangular | Centered Hexagonal | |
|---|---|---|---|---|---|
| Equilateral Triangle | – | – | – | A000332 | A008893 |
| Square | A002415 | A130684 | A006324 | – | – |
| Regular Hexagon | – | – | – | A011779 | A000537 |
| Regular Polygon | A002415 | A130684 | A006324 | ? | A339483* |
| Triangle | A045996 | A334705 | ? | ? | A241223 |
| Rectangle | A085582 | A289832 | ? | – | – |
| Right Triangle | A077435 | ? | ? | ? | A241225 |
Sequences marked with “*” are ones that I’ve authored, cells marked with “—” have no polygons, and cells marked with “?” do not have a corresponding sequence that I know of.
| Cubic | Tetrahedral | Octahedral | |
|---|---|---|---|
| Equilateral Triangle | A102698 | A334581* | A342353* |
| Square | A334881* | A334891* | ? |
| Regular Hexagon | A338322* | ? | ? |
| Regular Polygon | A338323* | ? | ? |
| Triangle | ? | ? | ? |
| Rectangle | ? | ? | ? |
| Right Triangle | ? | ? | ? |
| Regular Tetrahedron | A103158 | A269747 | ? |
| Cube | A098928 | ? | ? |
| Octahedron | A178797 | ? | ? |
| Platonic Solid | A338791 | ? | ? |
Sequences marked with “*” are ones that I’ve authored, and cells marked with “?” do not have a corresponding sequence that I know of.
If you’re interested in working on filling in some of the gaps in this table, I’d love it if you let me now! And if you’d like to collaborate or could use help getting started, send me a message on Twitter!
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