63 is a centered octahedral number. That means if you build an approximation of an octahedron with cubes, one size of octahedron will have 63 cubes.

In the late 1700’s René Just Haüy developed a theory about how crystals formed: successive layers of fundamental primitives in orderly arrangements. One of those arrangements was stacking cubes together to make an octahedron.

Start with one cube:

Add six more cubes around it, one on each face. Now we have seven:

Add another layer, adding a cube to touch each visible cube, making 25:

One more layer and we have a total of 63:

The remaining numbers in the sequence less than 10,000 are 129, 231, 377, 575, 833, 1159, 1561, 2047, 2625, 3303, 4089, 4991, 6017, 7175, 8473, 9919.

63 also shows up in the Delannoy numbers: the number of ways to traverse a grid from the lower left corner to upper right using only steps north, east, or northeast. Here are the 63 ways of moving on a 3×3 grid:

(Diagram from Wikipedia)

In fact, the number of cubes in a Haüy octahedron with N layers is the same as the number of Delannoy steps on a 3×N grid!

Since the two ideas are both geometric and fairly simple, I would love to find a geometric explanation for the correspondence. The octahedron is three-dimensional, and the Delannoy grids have that tantalizing 3 in them. It seems like there should be a way to convert Haüy coordinates to Delannoy coordinates to show how they relate. But I haven’t found one...

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Colophon: I made the octahedron diagrams by asking Claude to write a Python program to do it. It wasn’t a fast process because it took pushing and prodding to get the diagrams to come out the way I liked. But Claude was very competent, and I could think about the results rather than about projections or color spaces. I could dip into it for 10 minutes at a time over a number of days without having to somehow reconstruct a mental context.

This kind of casual hobby programming is perfect for AI assistance. I don’t need the code to be perfect or even good, I just want the diagrams to be nice. I don’t have the focus time to learn how to write the program, so I can leave it to an imperfect assistant.