Sunday, September 20, 2020

[kcpeniqu] Planes through checkerboard

Color a 2D grid of squares in a checkerboard pattern.  One can draw a 45-degree diagonal line that goes through the centers of squares of just one color.

Color a 3D grid of cubes in a checkerboard pattern.  Consider a diagonal plane that goes through the centers of cubes of just one color.  A cube's intersection with that plane is the famously surprising regular hexagon cross section that shows up when you obliquely slice a cube in just the right way.  The plane also slices a bit off of the corners of cubes which are the opposite color.  The pattern of cube cross sections on the plane is the trihexagonal tiling. The pattern of cube centers on the plane is the equilateral triangle lattice.  The planar Voronoi diagram formed from those centers is the regular hexagon tiling.

What happens if we start with a 4D checkerboard?  Although 4D geometry is impossible to visualize, the diagonal hyperplane that goes through the centers hypercubes of one color is a 3D honeycomb, so it is accessible to us 3D beings.  Wild guess: it is the tetrahedral-octahedral honeycomb with cell centers at the octahedron centers, and the Voronoi diagram is the rhombic dodecahedral honeycomb.

A easy hyperplane through 4D points of only 1 color (the centers of tesseracts of only one color) is a hyperplane through points whose coordinates are integers summing to zero.  We then need to rotate this hyperplane so that one of the coordinates is always zero.  This is probably not too difficult, maybe Gram-Schmidt.

The diagonal line through 2D has lots of sqrt(2).  The trihexagonal tiling through 3D has lots of sqrt(3).  In 4D, things might work out really nicely with lots of sqrt(4)=2, i.e, things remaining integers.

What happens in higher dimensions?  The alternated hypercube honeycomb is likely relevant.

There are 2 diagonal lines through a point in 2D.  There are 4 of the above-described diagonal planes through a cube center in 3D.  I think there are 2^(d-1) diagonal hyperplanes in d dimensions, which is exponentially more than the d orthogonal hyperplanes.

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