Dual polychora?

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

Dual polychora?

Postby mood » Wed May 20, 2020 12:42 am

In 3D, each topologically distinct polyhedron has a "canonical form" where all of its edges are tangent to the same sphere. The corresponding dual polyhedron can then be constructed by cutting a vertex figure off the midpoints of the original polyhedron's edges, and then building pyramids on the original faces so that they lie on the same plane as the vertex figure.

However, I'm not sure if this should even apply to 4D, because edges invert to faces instead, but dual uniform polychora with constant dichoral angles must exist, so how are they defined?
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Re: Dual polychora?

Postby wendy » Wed May 20, 2020 7:45 am

The simplex is self-dual in every dimension, and the 24ch is also selfdual in 4d.

The n-edges are dual to n-margins, that is, the surtope of n dimensions, is against a (unrelated) surtope of s-n dimensions, where s is the dimension of the solid.

So the edges of {3,3,5} run through the centres of the margins (pentagons) of {5,3,3}. [Thanks, Klitzing]
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Re: Dual polychora?

Postby Klitzing » Wed May 20, 2020 8:32 am

wendy wrote:So the edges of {3,3,5} run through the centres of {5,3,3}.

Wendy missed to specify the type of centers in the above statement.

But you can go along the following lines.

The mere combinatorical topology of duals and their elements can be obtained as follows:
  • vertices of the dual polychoron are obtrained as the set of cell centers of the original
  • edge midpoints of the dual polychoron are obtrained as the face centers of the original
  • face centers of the dual polychoron are obtrained as the edge midpoints of the original
  • cell centers of the dual polychoron are obtrained as the vertices of the original

If you were after a metrical description instead, you probably would do best apply the construction of spherical reciprocation:
  • choose some sphere around the bulk center of your polychoron and use its radius for unity of measure
  • construct the hyperplanes perpendicular to the rays upto the originals vertices, displaced from the bulk center by the inverse of the former vertex distance (wrt. above chosen units!)
  • intersection of any pair of these hyperplanes will happen according to the rule that the according originals vertices are connected by an edge or not

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