Mercurial, the Spectre wrote:Bowers has a nice definition on these shapes as well as on the hi.gher.space wiki.
Powertopes are basically objects that exist only in perfect-power dimensions such as 4, 8, 9, etc. because it is made out of two shapes, one being the base shape A, and an exponent shape B.
The powertope is constructed by defining A as any geometric shape, and B as a specific shape with brick symmetry (for every coordinate (a,b,...,z), there are coordinates which represent all sign combinations, so for a rectangle with one coordinate (3,4), you have (3,-4), (-3,4), and (-3,-4) as coordinates.). Having brick symmetry entails that if B is n-dimensional, then it has a topological order of 2^n. It is denoted as A^B and written as the B of A.
Then you consider B 's vertices as a compound of congruent n-cuboids (ex. an octagon is represented as a compound of two congruent rectangles and a cuboctahedron is represented as a compound of 3 squares lying perpendicular to each other), then out of it, you take several cartesian products of shapes that are similar but not necessarily congruent to A representing the n-cuboids. (hence, for a triangle^cuboid, you can have a cartesian product of three similar triangles of different sizes that depend on the edge lengths of the cuboid). If you are taking a pentagon^octagon, you have two pentagon-pentagon duoprisms of sizes NxM and MxN lying perpendicular to each other; the convex hull is the powertope. The number of cartesian products is dependent on the number of n-cuboids present within B (usually number of vertices / 2^n).
In fact, B can be curved as long as it has brick symmetry (such as a sphere but not a cone), though it has to be noted that the powertope is also curved.
Focusing on isogonal 4D shapes, the n-n duoprisms are equal to n^square, where n is a regular n-gon. Cells are 2n n-gonal prisms with a symmetry of 8n^2. Vertex figures are generally tetragonal disphenoids; the 4-4 duoprism (tesseract) is instead a regular tetrahedron.
For the octagon as an exponent, consider it as having D4 symmetry (basically a truncated square). Then the powertope is non-trivial and is formed from what I described above on the pentagon^octagon powertope, now for n^octagon, you have two n-n duoprisms of sizes NxM and MxN lying perpendicular to each other. Cells are 2n n-gonal prisms connected by n^2 rectangular trapezoprisms (D2d symmetry like tetragonal disphenoids) with the same symmetry as an n-n duoprism. Vertex figures are triangular bipyramids (C2v symmetry) that may instead be realized as compounds of two tetrahedra (since they are generally not corealmic).
Due to that, the grand antiprism is closely related to the decagon^octagon as an alternation, except that two new tetrahedra fill each deleted vertex.
That is all I can say about powertopes and I would love to see curved ones.
Ilazhra! (Truth to everyone!)
Mercurial, the Spectre
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