Isogonal figure

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Isogonal octagon
(D4 symmetry)
Blue and red lines of reflection are drawn
Truncated rhombicuboctahedron nonuniform.png
An isogonal truncated cuboctahedron, seen as cube with its edges beveled and its vertices truncated.
(Oh symmetry)

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 term isogonal has long been used for polyhedra. Vertex-transitive is a synonym borrowed from modern ideas such as symmetry groups and graph theory.

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[edit]

All regular polygons and regular star polygons are isogonal.

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[edit]

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[edit]

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.

The dual of an isogonal polytope is called an isotope which is transitive on its facets.

k-isogonal figures[edit]

A polytope or tiling may be called k-isogonal if its vertices form k transitivity classes.

Truncated rhombic dodecahedron2.png
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.
Enneagram 9-4 icosahedral.svg
2-isogonal 9/4 enneagram

A more restrictive term, k-uniform figures is defined as an k-isogonal figure constructed only from regular polygons. They can be represented visually with colors by different uniform colorings.

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