This chart shows the number of subfacets of a given dimension in a hypercube of a given dimension. The x axis is the dimension of the hypercube, and the y axis is the dimension of the subfacet.

The right hand side is well-known and easy to read; a point has 1 vertex; a line segment has 2 vertices and 1 edge; a square has 4 vertices, 4 edges, and 1 face; and so on. It should be easy to see those values, and easy to check them if you have knowledge of how hypercubes work (each value is twice the one to its left, plus the one to its top-left).

The left hand side is the strange realm of a negative dimensions. A -1-hypercube has 1 -1face, -2 -2faces, 4 -3faces, -8 -4faces, and so on. Infinitely. Oscillating between positive and negative. Note that these are a mathematical consequence of extrapolating a hypercube backwards.

What does this mean? No idea. But it's pretty cool. Maybe you smart guys could think of some implications.

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