geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, there are seven uniform and uniform dual polyhedra named as ditrigonal.

Ditrigonal vertex figures

There are five uniform ditrigonal polyhedra, all with icosahedral symmetry.Har'El, 1993 The three

uniform star polyhedron In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting. Each polyhedron can contain either star polygon faces, star polygon vertex figures, ...

with

Wythoff symbol In geometry, the Wythoff symbol is a notation representing a Wythoff construction of a uniform polyhedron or plane tiling within a Schwarz triangle. It was first used by Coxeter, Longuet-Higgins and Miller in their enumeration of the uniform po ...

of the form 3 , ''p'' ''q'' or , ''p'' ''q'' are ditrigonal, at least if ''p'' and ''q'' are not 2. Each polyhedron includes two types of faces, being of

triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

pentagon In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°. A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...

s, or

pentagram A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle around ...

s. Their

vertex configuration In geometry, a vertex configuration is a shorthand notation for representing a polyhedron or Tessellation, tiling as the sequence of Face (geometry), faces around a Vertex (geometry), vertex. It has variously been called a vertex description, vert ...

s are of the form ''p''.''q''.''p''.''q''.''p''.''q'' or (''p''.''q'')³ with a symmetry of order 3. Here, term ditrigonal refers to a

hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A regular hexagon is de ...

having a symmetry of order 3 (triangular symmetry) acting with 2 rotational orbits on the 6 angles of the vertex figure (the word ''ditrigonal'' means "having two sets of 3 angles").Uniform Polyhedron
Mathworld (retrieved 10 June 2016)

Other uniform ditrigonal polyhedra

The

small ditrigonal dodecicosidodecahedron In geometry, the small ditrigonal dodecicosidodecahedron (or small dodekified icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U43. It has 44 faces (20 triangles, 12 pentagrams and 12 decagons), 120 edges, and 60 vertices. Its ver ...

and the

great ditrigonal dodecicosidodecahedron In geometry, the great ditrigonal dodecicosidodecahedron (or great dodekified icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U42. It has 44 faces (20 triangles, 12 pentagons, and 12 decagrams), 120 edges, and 60 vertices. Its v ...

are also uniform. Their duals are respectively the

small ditrigonal dodecacronic hexecontahedron In geometry, the small ditrigonal dodecacronic hexecontahedron (or fat star) is a nonconvex isohedral polyhedron. It is the dual of the uniform A uniform is a variety of costume worn by members of an organization while usually participating ...

and

great ditrigonal dodecacronic hexecontahedron In geometry, the great ditrigonal dodecacronic hexecontahedron (or great lanceal trisicosahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform great ditrigonal dodecicosidodecahedron. Its faces are kites A kite is a ...

References

Notes

Bibliography

* Coxeter, H.S.M., M.S. Longuet-Higgins and J.C.P Miller, Uniform Polyhedra, ''Phil. Trans.'' 246 A (1954) pp. 401–450. * Har'El, Z
''Uniform Solution for Uniform Polyhedra.''
Geometriae Dedicata 47, 57–110, 1993
Zvi Har'ElKaleido software

Ditrigonal vertex figures

Other uniform ditrigonal polyhedra

See also

References

Notes

Bibliography

Further reading