In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling 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, one says 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 acts transitively 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.[1]
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.
polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex...
Isogonal, a mathematical term meaning "having similar angles", may refer to: Isogonalfigure or polygon, polyhedron, polytope or tiling Isogonal trajectory...
isotopic figure is an n-dimensional polytope or honeycomb with its facets ((n−1)-faces) congruent and transitive. The dual of an isotope is an isogonal polytope...
other isogonal (vertex-transitive) polytopes because one vertex figure can define the entire polytope. For polyhedra with regular faces, a vertex figure can...
equilateral polygons are isotoxal. The duals of isotoxal polygons are isogonal polygons. Isotoxal 4 n {\displaystyle 4n} -gons are centrally symmetric...
If there are k orbits of vertices, a tiling is known as k-uniform or k-isogonal; if there are t orbits of tiles, as t-isohedral; if there are e orbits...
Deeper truncations of the regular dodecagon and dodecagrams can produce isogonal (vertex-transitive) intermediate star polygon forms with equal spaced vertices...
isogon may refer to: Isogonalfigure - a polygon or polyhedron with all of its vertices equivalent under the symmetries of the figure. A type of contour...
truncations of the regular pentadecagon and pentadecagrams can produce isogonal (vertex-transitive) intermediate star polygon forms with equal spaced vertices...
letter and group order. r12 is full symmetry, and a1 is no symmetry. p6, an isogonal hexagon constructed by three mirrors can alternate long and short edges...
}}\triangle BCG)={\tfrac {1}{3}}({\text{Area of }}\triangle ABC).} The isogonal conjugate of a triangle's centroid is its symmedian point. Any of the three...
B, C respectively. The isogonal conjugate of P is sometimes denoted by P*. The isogonal conjugate of P* is P. A self-isogonal cubic is a triangle cubic...
rhombicosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon...
self-dual. The dual of an isogonal (vertex-transitive) polygon is an isotoxal (edge-transitive) polygon. For example, the (isogonal) rectangle and (isotoxal)...
A hexagram (Greek) or sexagram (Latin) is a six-pointed geometric star figure with the Schläfli symbol {6/2}, 2{3}, or {{3}}. Since there are no true regular...
as directed edges. The highest symmetry irregular decagons are d10, an isogonal decagon constructed by five mirrors which can alternate long and short...
symmetry is labeled a1. The most common high symmetry octagons are p8, an isogonal octagon constructed by four mirrors can alternate long and short edges...
omnitruncated icosahedron is an Archimedean solid, one of thirteen convex, isogonal, non-prismatic solids constructed by two or more types of regular polygon...
equiangular: all its corner angles are equal (each of 90 degrees). It is isogonal or vertex-transitive: all corners lie within the same symmetry orbit. It...
tilings are isogonal and monohedral. A semi-regular (or Archimedean) tessellation uses more than one type of regular polygon in an isogonal arrangement...
circumcircle. Tangential: all sides are tangent to an inscribed circle. Isogonal or vertex-transitive: all corners lie within the same symmetry orbit. The...
polychoron is isogonal and has 2p p-gonal prisms and p2 rectangular trapezoprisms (a cube with D2d symmetry) but cannot be made uniform. The vertex figure is a...
polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows...
intersecting line segments are removed from a regular star n-gon, the resulting figure is no longer regular, but can be seen as an isotoxal concave simple 2n-gon...
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