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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, a polytope
Polytope

In geometry, polytope is a generic term that can refer to a two-dimensional polygon, a three-dimensional polyhedron, or any of the various generalizations thereof, including generalizations to higher dimensions and other abstractions ....
 (a polygon
Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
, polyhedron
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
 or tiling, for example) is isogonal or vertex-transitive if all its vertices
Vertex (geometry)

In geometry, a vertex is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces in 3d models, where each such point is given as a vector....
 are the same. That is, each vertex is surrounded by the same kinds of face
Face (geometry)

In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the square s that bound a cube is a face of the cube....
 in the same order, and with the same angles between corresponding faces.

Technically, we say that for any two vertices there exists a symmetry
Symmetry

Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance; such that it reflects beauty or perfection....
 of the polytope mapping the first isometrically
Isometry

In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces....
 onto the second. Other ways of saying this are that 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.






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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, a polytope
Polytope

In geometry, polytope is a generic term that can refer to a two-dimensional polygon, a three-dimensional polyhedron, or any of the various generalizations thereof, including generalizations to higher dimensions and other abstractions ....
 (a polygon
Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
, polyhedron
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
 or tiling, for example) is isogonal or vertex-transitive if all its vertices
Vertex (geometry)

In geometry, a vertex is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces in 3d models, where each such point is given as a vector....
 are the same. That is, each vertex is surrounded by the same kinds of face
Face (geometry)

In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the square s that bound a cube is a face of the cube....
 in the same order, and with the same angles between corresponding faces.

Technically, we say that for any two vertices there exists a symmetry
Symmetry

Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance; such that it reflects beauty or perfection....
 of the polytope mapping the first isometrically
Isometry

In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces....
 onto the second. Other ways of saying this are that 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 group
Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with Function composition as the operation. It is a subgroup of the isometry group of the space concerned....
s and graph theory
Graph theory

In mathematics and computer science, graph theory is the study of graph : mathematical structures used to model pairwise relations between objects from a certain collection....
.

Isogonal polygons

All regular polygons and regular star polygon
Star polygon

A star polygon is a non-convex polygon which looks in some way like a star. Only the regular ones have been studied in any depth; star polygons in general have never been formally defined....
s are isogonal.

Some even-sided polygons which alternate two edge lengths, for example rectangle
Rectangle

In geometry, a rectangle is a Closed set planar quadrilateral with four right angles. A rectangle with vertices ABCD would be denoted as .A rectangle with adjacent sides of lengths a and b has area ab and diagonals of equal length ....
, are isogonal.

All such 2n-gons have dihedral symmetry (Dn, n=2,3,...) with reflection lines across the mid-edge points.

Isogonal polyhedra


Isogonal polyhedra may be classified:
  • Regular
    Regular polyhedron

    A regular polyhedron is a polyhedron whose faces are Congruence regular polygons which are assembled in the same way around each vertex. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive - i.e....
     if it is also isohedral (face-transitive) and isotoxal (edge-transitive); this implies that every face is the same kind of regular polygon
    Regular polygon

    A regular polygon is a polygon which is Equiangular polygon and equilateral . Regular polygons may be convex or Star polygon....
    .
  • Quasi-regular
    Quasiregular polyhedron

    A polyhedron which has regular faces and is transitive on its edges but not transitive on its faces is said to be quasiregular.A quasiregular polyhedron can have faces of only two kinds and these must alternate around each vertex....
     if it is also isotoxal (edge-transitive) but not isohedral (face-transitive).
  • Semi-regular
    Semiregular polyhedron

    A semiregular polyhedron is a polyhedron with regular polygon faces and a symmetry group which is transitive on its vertices. Or at least, that is what follows from Thorold Gosset's 1900 definition of the more general semiregular polytope....
     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
    Uniform polyhedron

    A Uniform polytope polyhedron is a polyhedron which has regular polygons as Face and is transitive on its vertex . It follows that all vertices are Congruence , and the polyhedron has a high degree of reflectional and rotational symmetry....
     if every face is a regular polygon, i.e. it is regular, quasiregular or semi-regular.
  • Noble
    Noble polyhedron

    A noble polyhedron is one which is isohedral and isogonal . They were first studied in any depth by Hess and Bruckner around the turn of the century , and later by Branko Gr?nbaum....
     if it is also isohedral (face-transitive).


An isogonal polyhedron has a single kind of vertex figure
Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off....
. If the faces are regular (and the polyhedron is thus uniform) it can be represented by a vertex configuration
Vertex configuration

In polyhedral geometry a vertex configuration is a short-hand notation for representing a polyhedron vertex figure as the sequence of faces around a vertex....
 notation sequencing the faces around each vertex.

Isogonal polytopes and tessellations


These definitions can be extended to higher dimensional polytope
Polytope

In geometry, polytope is a generic term that can refer to a two-dimensional polygon, a three-dimensional polyhedron, or any of the various generalizations thereof, including generalizations to higher dimensions and other abstractions ....
s and tessellations
Honeycomb (geometry)

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....
. Most generally, all uniform polytope
Uniform polytope

A uniform polytope is a vertex-transitive polytope made from uniform polytope Facet . A uniform polytope must also have only regular polygon faces....
s are isogonal, for example, the uniform polychoron
Uniform polychoron

In geometry, a Uniform polytope polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedron.This article contains the complete list of 64 non-prismatic convex uniform polychora, and describes two infinite sets of convex prismatic forms....
s and convex uniform honeycomb
Convex uniform honeycomb

In geometry, a convex uniform honeycomb is a uniform space-filling tessellation in three-dimensional Euclidean space with non-overlapping convex uniform polyhedron cells....
s.

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

k-isogonal figures


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


This truncated rhombic dodecahedron
Truncated rhombic dodecahedron

The truncated rhombic dodecahedron is a convex polygon polyhedron constructed from the rhombic dodecahedron by Truncation the 6 vertices.The 6 vertices are truncated such that all edges are equal length....
 is 2-isogonal because it contains two transitivity classes of vertices. This polyhedron is made of squares
Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
 and flattened hexagon
Hexagon

In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....
s.
Dem3446bc

This demiregular tiling
Tiling by regular polygons

Plane Tessellation by regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Johannes Kepler in Harmonices Mundi....
 is also 2-isogonal. This tiling is made of equilateral triangle
Equilateral triangle

In geometry, an equilateral triangle is a triangle in which all three sides are equal. In traditional or Euclidean geometry, equilateral triangles are also Equiangular polygon; that is, all three internal angles are also congruent to each other and are each 60?....
, square and regular hexagon
Hexagon

In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....
al faces.


A more restrictive term, k-uniform figures is defined as an k-isogonal figure constructed only from regular polygon
Regular polygon

A regular polygon is a polygon which is Equiangular polygon and equilateral . Regular polygons may be convex or Star polygon....
s. They can be represented visually with colors by different uniform coloring
Uniform coloring

In geometry, a uniform coloring is a property of a uniform figure that is colored to be vertex-transitive. Different Symmetry can be expressed on the same geometric figure with the Face following different uniform color patterns....
s.

See also

  • Edge-transitive (Isotoxal figure)
  • Face-transitive (Isohedral figure)


External links

  • 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