Stand a line on end, you have a point.

Stand a 2-dimensional figure on end, align 2 pairs of points, you have a line.

Turn a four-dimensional object on end, align 4 pairs of points, you have 2-dimensional object.

The conjectures carry this out to any N. The third conjecture is clearly false; perform a mega-alignment of two four-dimensional objects (or one object separated in space over time), you have a 2-dimensional object. But it will be interesting to see at higher dimensions where the conjecture might hold true.

Credit to BLW for the relation P = 2^(P-1) implied here.

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