## Polytopes with Lattice Coordinates

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.)

## 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.

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!

## Parity Bitmaps from the OEIS

My friend Alec Jones and I wrote a Python script that takes a two-dimensional sequence in the On-Line Encyclopedia of Integer Sequences and uses it to create a one-bit-per-pixel (1BPP) “parity bitmaps“. The program is simple: it colors a given pixel is black or white depending on whether the corresponding value is even or odd.

### An Unexpected Fractal

We’ve now run the script on over a thousand sequences, but we still both agree on our favorite: the fractal generated by OEIS sequence A279212.

Fill an array by antidiagonals upwards; in the top left cell enter $$a(0)=1$$; thereafter, in the $$n$$-th cell, enter the sum of the entries of those earlier cells that can be “seen” from that cell.

Notice that in the images below, increasing the rows and columns by a factor of $$2^n$$ seems to increase the “resolution”, because the parity bitmap is self similar at 2x the scale. We still don’t have a good explanation for why we’d expect these images are fractals. If you know, please answer our question about it on Math Stack Exchange. (Alec and I have generated these images up to 16384 × 32768 resolution, roughly 536 megapixels.)

#### The Construction of the Sequence

The sequence is built up by “antidiagonals”, as shown in the GIF below. In the definition, “seen” means every direction a chess queen can move that already has numbers written down (i.e. north, west, northwest, or southwest). That is, look at all of the positions you can move to, add them all up, write that number in your square, move to the next square, and repeat. (The number in cell $$C$$ also counts the number of paths a queen can make from $$C$$ to the northwest corner using only N, NW, W, SW moves.)

(Interestingly, but only tangentially related: Code Golf Stack Exchange User flawr noticed that the number of north/west rook walks is related to the number of ways of partitioning a $$1 \times n$$ grid into triangles.)

## Parity Bitmaps for Other Sequences

It’s worth noting that many sequences are all black, consist of simple repeating patterns, or look like static. However, chess-type constructions, as illustrated by the GIF above, the one above yield images that look like the Sierpiński triangle. (See A132439 and A334017 below, and look at A334016 and A334745 via their OEIS entries.) Look below for a couple other sequences with interesting images too.

I ordered a poster-sized print of the A279212 fractal for Alec, and he framed it in his office.

Some ideas for further exploration: