These symbols then represent types of edges of vertex-intransitive isotoxal polyhedra. Since all edges of a given polyhedron must be of the same type, the types can form the basis of a compact identification tag for each polyhedron, augmented only by a sequential number. Thus for example, "I53b_1" refers to the first polyhedron with edge type I53b.
I use the notation "[m/d]" (or simply "[m]" when d = 1) to denote a polygon having m edges and rotation number (or density) d. (The Grünbaum/Shephard notation, oddly, halves the number of vertices.) This is similar to the "{m/d}" notation commonly used for regular polygons, with the difference that when the greatest common factor (m, d) = h is greater than 1 this refers to an m-gon rather than to a compound of h (m/h)-gons. A face whose border is such a polygon is a surface that can be understood as made up of triangular patches, each defined by an edge and the center of the face, stitched together along their other sides. Star-polygon faces have density greater than 1, with a branch point at the center of the face.
When m = 4, one obtains a rhombus or square. I refer to the latter as a "regular [4]" for consistency. Here is a table of the types and subtypes for m = 6, 8, 10. An aligned face has sets of 3 or 4 non-consecutive collinear vertices. The overlapped [10/3] is a special case in which some edges partially overlap. To clarify its structure, the table includes a representation of it with its edges shifted slightly. Each inner vertex is incident with two edges and also lies on two other edges that pass through the same location in space but do not intersect. In the surface that constitutes a face they are as distant from the vertex as are two layers of a Riemann surface.
| Central face | a face that passes through the center of symmetry |
| Covering face | a face that contributes to covering the circumsphere, that is, any face that is not central |
| Hexagrammic | having [6/2]s as faces |
| Decagrammic | having [10/d]s as faces where d>1 |
| 3-Decagrammic | specifically having [10/3]s as faces |
| 4-Decagrammic | specifically having [10/4]s as faces |
| Cubo- | having 6 covering faces with 4-fold symmetry |
| Octa- | having 8 covering faces with 3-fold symmetry |
| -hemi- | having central (also known as equatorial or hemispherical) face planes |
| -hemiicositetra- | having 12 central face planes that are parallel to those of a 24-hedron |
| Hemihexeconta- | having 30 central face planes that are parallel to those of a 60-hedron |
| Dihemidodeca- | having 6 central face planes that are parallel to those of a Dodecahedron, with 2 faces per plane |
| Dihemiicosa- | having 10 central face planes that are parallel to those of an Icosahedron, with 2 faces per plane |
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