Isotoxal figure

In geometry, a polytope (for example, a polygon or a polyhedron), or a tiling, is isotoxal or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two edges, there is a translation, rotation and/or reflection that will move one edge to the other, while leaving the region occupied by the object unchanged.

The term isotoxal is derived from the Greek τοξον meaning arc.

Isotoxal polygons

An isotoxal polygon is an equilateral polygon, but not all equilateral polygons are isotoxal. The duals of isotoxal polygons are isogonal polygons.

In general, an isotoxal 2n-gon will have Dn (*nn) dihedral symmetry. A rhombus is an isotoxal polygon with D2 (*22) symmetry.

All regular polygons (equilateral triangle, square, etc.) are isotoxal, having double the minimum symmetry order: a regular n-gon has Dn (*nn) dihedral symmetry. A regular 2n-gon is an isotoxal polygon and can be marked with alternately colored vertices, removing the line of reflection through the mid-edges.

Example isotoxal polygons
D2 (*22) D3 (*33) D4 (*44) D5 (*55)
Rhombus Equilateral triangle Concave hexagon Self-intersecting hexagon Convex octagon Regular pentagon Self-intersecting (regular) pentagram Self-intersecting decagram
Lozenge - black simple Regular triangle Medial triambic icosahedron face Great triambic icosahedron face Regular polygon truncation 4 1 dual Isotoxal octagon Pentagon Pentagram green Isotoxal pentagram

Isotoxal polyhedra and tilings

Regular polyhedra are isohedral (face-transitive), isogonal (vertex-transitive) and isotoxal. Quasiregular polyhedra are isogonal and isotoxal, but not isohedral; their duals are isohedral and isotoxal, but not isogonal.

Examples
Quasiregular
polyhedron
Quasiregular dual
polyhedron
Quasiregular
star polyhedron
Quasiregular dual
star polyhedron
Quasiregular
tiling
Quasiregular dual
tiling
Uniform polyhedron-43-t1
A cuboctahedron is isogonal and isotoxal polyhedron
Rhombicdodecahedron
A rhombic dodecahedron is an isohedral and isotoxal polyhedron
Great icosidodecahedron
A great icosidodecahedron is isogonal and isotoxal star polyhedron
DU54 great rhombic triacontahedron
A great rhombic triacontahedron is an isohedral and isotoxal star polyhedron
Tiling Semiregular 3-6-3-6 Trihexagonal
The trihexagonal tiling is an isogonal and isotoxal tiling
Star rhombic lattice
The rhombille tiling is an isohedral and isotoxal tiling with p6m (*632) symmetry.

Not every polyhedron or 2-dimensional tessellation constructed from regular polygons is isotoxal. For instance, the truncated icosahedron (the familiar soccerball) has two types of edges: hexagon-hexagon and hexagon-pentagon, and it is not possible for a symmetry of the solid to move a hexagon-hexagon edge onto a hexagon-pentagon edge.

An isotoxal polyhedron has the same dihedral angle for all edges.

There are nine convex isotoxal polyhedra formed from the Platonic solids, 8 formed by the Kepler–Poinsot polyhedra, and six more as quasiregular (3 | p q) star polyhedra and their duals.

There are at least 5 polygonal tilings of the Euclidean plane that are isotoxal, and infinitely many isotoxal polygonal tilings of the hyperbolic plane, including the Wythoff constructions from the regular hyperbolic tilings {p,q}, and non-right (p q r) groups.

See also

References

  • Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 0-521-55432-2, p. 371 Transitivity
  • Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1. (6.4 Isotoxal tilings, 309-321)
  • Coxeter, Harold Scott MacDonald; Longuet-Higgins, M. S.; Miller, J. C. P. (1954), "Uniform polyhedra", Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences, 246: 401–450, doi:10.1098/rsta.1954.0003, ISSN 0080-4614, JSTOR 91532, MR 0062446
Isogonal figure

In geometry, a polytope (a polygon, polyhedron or tiling, for example) is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This 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.

All vertices of a finite n-dimensional isogonal figure exist on an (n-1)-sphere.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.

1–10 sides
11–20 sides
21–100 sides
(selected)
>100 sides
Star polygons
(5–12 sides)
Others
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