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Uniform polyhedra can be divided between convex forms with convex regular polygon faces and star forms. Star forms have either regular star polygon faces or vertex figures or both. This list includes these: all 75 nonprismatic uniform polyhedra; a few representatives of the infinite sets of prisms and antiprisms;
The {7/2} and {7/3} star polygons also have occult significance, particularly in the Kabbalah and in Wicca. The {8/3} star polygon ( octagram ) is a frequent geometrical motif in Mughal Islamic art and architecture ; the first is on the emblem of Azerbaijan .
Some of the polyhedrons do have eight faces aside from being square bipyramids in the following: Hexagonal prism: Two faces are parallel regular hexagons; six squares link corresponding pairs of hexagon edges. Heptagonal pyramid: One face is a heptagon (usually regular), and the remaining seven faces are triangles (usually isosceles). All ...
The elements of a polytope can be considered according to either their own dimensionality or how many dimensions "down" they are from the body.
The stellated octahedron is the first iteration of the 3D analogue of a Koch snowflake. A compound of two spherical tetrahedra can be constructed, as illustrated. The two tetrahedra of the compound view of the stellated octahedron are "desmic", meaning that (when interpreted as a line in projective space ) each edge of one tetrahedron crosses ...
A flag is a connected set of elements of each dimension – for a polyhedron that is the body, a face, an edge of the face, a vertex of the edge, and the null polytope. An abstract polytope is said to be regular if its combinatorial symmetries are transitive on its flags – that is to say, that any flag can be mapped onto any other under a ...
The polytopes of rank 2 (2-polytopes) are called polygons.Regular polygons are equilateral and cyclic.A p-gonal regular polygon is represented by Schläfli symbol {p}.. Many sources only consider convex polygons, but star polygons, like the pentagram, when considered, can also be regular.
Each polyhedron can contain either star polygon faces, star polygon vertex figures, or both. The complete set of 57 nonprismatic uniform star polyhedra includes the 4 regular ones, called the Kepler–Poinsot polyhedra, 14 quasiregular ones, and 39 semiregular ones. There are also two infinite sets of uniform star prisms and uniform star ...