A polyhedron is a polyisohedron if its faces can be grouped into sets that are equivalent. Definitions:
As an example, consider a Johnson solid with an entertaining name, the gyrobifastigium. It has 4 equilateral triangle faces and 4 square faces. You can pair up each triangle with a square so you have 4 polyfaces. Here’s one way to arrange them into congruent polyfaces, creating an order 2 polyisohedron with 4 polyfaces: Here's a second way to create congruent polyfaces: With this next way of pairing up the triangles and squares, the 4 polysides aren’t congruent, hence it isn't polyisohedral: Latlong polyisohedronA latlong polyisohedron uses a generalization of latitude and longitude on a sphere. To make one, start with a series of connected line segments. Create multiple equally spaced copies around the axis that connects the start and end point of the segments. The following example shows an order 4 latlong polyishedron with 5 polyfaces. If there are m line segments and n copies, the result will be an order m polyisohedron with n polysides. Each polyside will consist of two isosceles triangles and m2 trapezoids. The latlong polyisohedron construction demonstrates that polyisohedra of any order 2 and greater exists with any number of sides 3 and greater. In contrast, order 1 polyisohedra, the isohedra, can't have an odd number of sides. They can have any number of even sides 4 and greater. Other examples

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