Isotoxal figure

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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.

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

Isotoxal polygons

An isotoxal polygon is an even-sided equilateral polygon, but not all equilateral polygons are isotoxal. The duals of isotoxal polygons are isogonal polygons. 4n-gonal are centrally symmetric so are also zonogons.

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.

An isotoxal 2n-gon can be labeled as {nα} with outer most internal angle α. The second internal angle β may be greater or less than 180 degrees, making convex or concave polygons. Star polygons can also be isotoxal, labeled as {(n/q)α}, with q<n-1 and gcd(n,q)=1, with q as the turning number or density. [1] Concave inner vertices can be defined for q<n/2. If there is a largest common divisor, like a, {(na/qa)α} can be reduced as a compound a{(n/q)α}, with a copied rotated.

A set of uniform tilings can be defined with isotoxal polygons as a lower type of regular faces.

Example irregular isotoxal polygons and compounds
Sides (2n) 4 6 8 10 12 14 16
{nα}
Convex β<180
Concave β>180
Isotoxal rhombus.svg
{2α}
Isotoxal hexagon.svg Concave isotoxal hexagon.svg
{3α}
Isotoxal octagon.svg Concave isotoxal octagon.svg
{4α}
Isotoxal decagon.svg Concave isotoxal decagon.svg
{5α}
Isotoxal dodecagon.svg Isotoxal hexagram.svg
{6α}
Isotoxal tetradecagon.svg Concave isotoxal tetradecagon.svg
{7α}
Isotoxal hexadecagon.svg Concave isotoxal hexadecagon.svg
{8α}
2-turn
{(n/2)α}
-- Intersecting isotoxal hexagon.svg
{(3/2)α}
Isotoxal rhombus compound2.svg
2{2α}
Intersecting isotoxal decagon2.svg Intersecting isotoxal decagon2b.svg
{(5/2)α}
Isotoxal hexagon compound2.svg Concave isotoxal hexagon compound2.svg
2{3α}
Intersecting isotoxal tetradecagon.svg Isotoxal heptagram.svg
{(7/2)α}
Isotoxal octagon compound2.svg Concave isotoxal octagon compound2.svg
2{4α}
3-turn
{(n/3)α}
---- Intersecting isotoxal octagon.svg
{(4/3)α}
Isotoxal decagram.svg
{(5/3)α}
Isotoxal rhombus compound3.svg
3{2α}
Intersecting isotoxal tetradecagon3.svg Intersecting isotoxal tetradecagon3b.svg
{(7/3)α}
Intersecting isotoxal hexadecagon3.svg Concave intersecting isotoxal hexadecagon3.svg
{(8/3)α}
4-turn
{(n/4)α}
------ Intersecting isotoxal decagon.svg
{(5/4)α}
Intersecting isotoxal hexagon compound2.svg
2{(3/2)α}
Intersecting isotoxal tetradecagon4.svg
{(7/4)α}
Isotoxal rhombus compound4.svg
4{2α}
5-turn
{(n/5)α}
-------- Intersecting isotoxal dodecagon.svg
{(6/5)α}
Intersecting isotoxal tetradecagon5.svg
{(7/5)α}
Intersecting isotoxal hexadecagon5.svg
{(8/5)α}
6-turn
{(n/6)α}
---------- Intersecting isotoxal tetradecagon6.svg
{(7/6)α}
Intersecting isotoxal octagon compound2.svg
2{(4/3)α}
7-turn
{(n/7)α}
------------ Intersecting isotoxal hexadecagon7.svg
{(8/7)α}

Isotoxal polyhedra and tilings

Regular polyhedra are isohedral (face-transitive), isogonal (vertex-transitive), and isotoxal (edge-transitive).

Quasiregular polyhedra, like the cuboctahedron and the icosidodecahedron, are isogonal and isotoxal, but not isohedral. Their duals, including the rhombic dodecahedron and the rhombic triacontahedron, 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.svg
A cuboctahedron is an isogonal and isotoxal polyhedron
Rhombicdodecahedron.jpg
A rhombic dodecahedron is an isohedral and isotoxal polyhedron
Great icosidodecahedron.png
A great icosidodecahedron is an isogonal and isotoxal star polyhedron
DU54 great rhombic triacontahedron.png
A great rhombic triacontahedron is an isohedral and isotoxal star polyhedron
Tiling Semiregular 3-6-3-6 Trihexagonal.svg
The trihexagonal tiling is an isogonal and isotoxal tiling
Star rhombic lattice.png
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) is not isotoxal, as it has two edge types: 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.

The dual of a convex polyhedron is also a convex polyhedron. [2]

The dual of a non-convex polyhedron is also a non-convex polyhedron. [2] (By contraposition.)

The dual of an isotoxal polyhedron is also an isotoxal polyhedron. (See the Dual polyhedron article.)

There are nine convex isotoxal polyhedra: the five (regular) Platonic solids, the two (quasiregular) common cores of dual Platonic solids, and their two duals.

There are fourteen non-convex isotoxal polyhedra: the four (regular) Kepler–Poinsot polyhedra, the two (quasiregular) common cores of dual Kepler–Poinsot polyhedra, and their two duals, plus the three quasiregular ditrigonal (3 | pq) star polyhedra, and their three duals.

There are at least five isotoxal polyhedral compounds: the five regular polyhedral compounds; their five duals are also the five regular polyhedral compounds (or one chiral twin).

There are at least five isotoxal polygonal tilings of the Euclidean plane, 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

Related Research Articles

Bipyramid Polyhedron formed by joining a pyramid and its mirror image base-to-base

A (symmetric) n-gonal bipyramid or dipyramid is a polyhedron formed by joining an n-gonal pyramid and its mirror image base-to-base. An n-gonal bipyramid has 2n triangle faces, 3n edges, and 2 + n vertices.

Cuboctahedron

A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is the only radially equilateral convex polyhedron.

In geometry, a dodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.

Dual polyhedron

In geometry, any polyhedron is associated with a second dual figure, where the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all are also geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron.

Polyhedron Three-dimensional shape with flat polygonal faces, straight edges and sharp corners

In geometry, a polyhedron is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from the Classical Greek πολύεδρον, as poly- + -hedron.

A polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram.

Star polygon

In geometry, a star polygon is a type of non-convex polygon. Only the regular star polygons have been studied in any depth; star polygons in general appear not to have been formally defined, however certain notable ones can arise through truncation operations on regular simple and star polygons.

A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are congruent regular polygons which are assembled in the same way around each vertex.

Schläfli symbol Notation that defines regular polytopes and tessellations

In geometry, the Schläfli symbol is a notation of the form {p,q,r,...} that defines regular polytopes and tessellations.

In geometry, a polytope 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.

Uniform polyhedron Polyhedron which has regular polygons as faces and is vertex-transitive

A uniform polyhedron has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent.

Chamfered dodecahedron

The chamfered dodecahedron is a convex polyhedron with 80 vertices, 120 edges, and 42 faces: 30 hexagons and 12 pentagons. It is constructed as a chamfer (geometry) (edge-truncation) of a regular dodecahedron. The pentagons are reduced in size and new hexagonal faces are added in place of all the original edges. Its dual is the pentakis icosidodecahedron.

Honeycomb (geometry) Tiling of 3-or-more dimensional euclidian or hyperbolic space

In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions. Its dimension can be clarified as n-honeycomb for a honeycomb of n-dimensional space.

Isohedral figure

In geometry, a polytope of dimension 3 or higher is isohedral or face-transitive when all its faces are the same. More specifically, all faces must be not merely congruent but must be transitive, i.e. must lie within the same symmetry orbit. In other words, for any faces A and B, there must be a symmetry of the entire solid by rotations and reflections that maps A onto B. For this reason, convex isohedral polyhedra are the shapes that will make fair dice.

In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the semiregular, which are merely vertex-transitive.

In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive.

Density (polytope)

In geometry, the density of a star polyhedron is a generalization of the concept of winding number from two dimensions to higher dimensions, representing the number of windings of the polyhedron around the center of symmetry of the polyhedron. It can be determined by passing a ray from the center to infinity, passing only through the facets of the polytope and not through any lower dimensional features, and counting how many facets it passes through. For polyhedra for which this count does not depend on the choice of the ray, and for which the central point is not itself on any facet, the density is given by this count of crossed facets.

Chamfer (geometry)

In geometry, chamfering or edge-truncation is a topological operator that modifies one polyhedron into another. It is similar to expansion, moving faces apart and outward, but also maintains the original vertices. For polyhedra, this operation adds a new hexagonal face in place of each original edge.

References

  1. Tilings and Patterns Branko Gruenbaum, G.C. Shephard, 1987. 2.5 Tilings using star polygons, pp.82-85.
  2. 1 2 "duality". maths.ac-noumea.nc. Retrieved 2020-09-30.