Shadow Conjectures and Dimensional Projections

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Lyons-Weiler 2018. Conjecture 1: For every N, there exists at least one N+1 dimensional regularly shaped object that has a precise set of 2^N vertices that can simultaneously be aligned to produce the equivalent of an N-dimensional regularly shaped object.

Lyons-Weiler Conjecture 2 (Corollary): For every N, there exists at least one N-dimensional regularly shaped object that is equivalent to a N+1 dimensional regularly shaped object with precisely 2^N of its vertices aligned.

Lyons-Weiler Conjecture 3 (Hypothesis): For every N, there are no N-dimensional regularly shaped objects that are equivalent to any N+x dimensional regularly shaped objects with 2^N of its vertices aligned, where x > 1.
dims
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.

Original source: https://jameslyonsweiler.com/2018/07/24/shadow-conjectures-and-dimensional-projections/

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