In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, 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 with
Wythoff symbol 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 edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colline ...
s,
pentagons, 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 arou ...
s. Their
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'' has ...
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 and the great ditrigonal dodecicosidodecahedron are also uniform.
Their duals are respectively the small ditrigonal dodecacronic hexecontahedron and great ditrigonal dodecacronic hexecontahedron.
See also
*Small complex icosidodecahedron
In geometry, the small complex icosidodecahedron is a degenerate uniform star polyhedron. Its edges are doubled, making it degenerate. The star has 32 faces (20 triangles and 12 pentagons), 60 (doubled) edges and 12 vertices and 4 sharing faces. ...
*Great complex icosidodecahedron
In geometry, the great complex icosidodecahedron is a degenerate uniform star polyhedron. It has 12 vertices, and 60 (doubled) edges, and 32 faces, 12 pentagrams and 20 triangles. All edges are doubled (making it degenerate), sharing 4 faces, but ...
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
Further reading
*Johnson, N.; ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 196
*
{{polyhedra
Polyhedra