Higher-dimensional "analogs" of Dehn invariant?

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

Higher-dimensional "analogs" of Dehn invariant?

Postby New Kid on the 4D analog of a Block » Sun Sep 08, 2019 8:43 pm

Polygons have one invariant: the area. Any two polygons can be cut-and-pasted into one another if they are of equal area. The cuts can be said to take the form of lines.

Polyhedra have two invariants: the volume, and the Dehn invariant (the sum, over all edges, of the tensor product of that edge's length and dihedral angle). Polyhedra can be cut-and-pasted into one another if both of their invariants match, and the cuts can be said to take the form of planes.

Would there be 3 invariants for polychora? Hypervolume, sum over all faces of (face area ⊗ dichoral angle), and something else? Or would the Dehn invariant be calculated the same way? Might the new third invariant have to do with quantities unchanged by "plane cuts" to the polychoron?
And in general, would there be N-1 invariants for N-dimensional polytopes?
New Kid on the 4D analog of a Block
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