Blue and red lines of reflection are drawn
An isogonal truncated cuboctahedron, seen as cube with its edges beveled and its vertices truncated.
In geometry, a polytope (a polygon, polyhedron or tiling, for example) is isogonal or vertex-transitive if, loosely speaking, all its vertices are the same. That implies that each vertex is surrounded by the same kinds of face in the same or reverse order, and with the same angles between corresponding faces.
Technically, we say that for any two vertices there exists a symmetry of the polytope mapping the first isometrically onto the second. Other ways of saying this are that the group of automorphisms of the polytope is transitive on its vertices, or that the vertices lie within a single symmetry orbit.
The pseudorhombicuboctahedron — which is not isogonal — demonstrates that simply asserting that "all vertices look the same" is not as restrictive as the definition used here, which involves the group of isometries preserving the polyhedron or tiling.
2 dimensions: Isogonal polygons
Some even-sided polygons which alternate two edge lengths, for example rectangle, are isogonal.
All such 2n-gons have dihedral symmetry (Dn, n=2,3,...) with reflection lines across the mid-edge points.
3 dimensions: Isogonal polyhedra
Isogonal polyhedra may be classified:
- Regular if it is also isohedral (face-transitive) and isotoxal (edge-transitive); this implies that every face is the same kind of regular polygon.
- Quasi-regular if it is also isotoxal (edge-transitive) but not isohedral (face-transitive).
- Semi-regular if every face is a regular polygon but it is not isohedral (face-transitive) or isotoxal (edge-transitive). (Definition varies among authors; e.g. some exclude solids with dihedral symmetry, or nonconvex solids.)
- Uniform if every face is a regular polygon, i.e. it is regular, quasiregular or semi-regular.
- Noble if it is also isohedral (face-transitive).
An isogonal polyhedron has a single kind of vertex figure. If the faces are regular (and the polyhedron is thus uniform) it can be represented by a vertex configuration notation sequencing the faces around each vertex.
N dimensions: Isogonal polytopes and tessellations
These definitions can be extended to higher-dimensional polytopes and tessellations. Most generally, all uniform polytopes are isogonal, for example, the uniform 4-polytopes and convex uniform honeycombs.
A polytope or tiling may be called k-isogonal if its vertices form k transitivity classes.
This truncated rhombic dodecahedron is 2-isogonal because it contains two transitivity classes of vertices. This polyhedron is made of squares and flattened hexagons.
This demiregular tiling is also 2-isogonal. This tiling is made of equilateral triangle, square and regular hexagonal faces.
2-isogonal 9/4 enneagram
- Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 0-521-55432-2, p. 369 Transitivity
- Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman and Company. ISBN 0-7167-1193-1. (p. 33 k-isogonal tiling, p. 65 k-uniform tilings)
- Weisstein, Eric W., "Vertex-transitive graph", MathWorld.
- Olshevsky, George, Transitivity at Glossary for Hyperspace.
- Olshevsky, George, Isogonal at Glossary for Hyperspace.
- Isogonal Kaleidoscopical Polyhedra Vladimir L. Bulatov, Physics Department, Oregon State University, Corvallis, Presented at Mosaic2000, Millennial Open Symposium on the Arts and Interdisciplinary Computing, 21–24 August 2000, Seattle, WA
- Steven Dutch uses the term k-uniform for enumerating k-isogonal tilings
- List of n-uniform tilings
- Weisstein, Eric W., "Demiregular tessellations", MathWorld. (Also uses term k-uniform for k-isogonal)