Triangular duotegum
Triangular duotegum  

Rank  4 
Type  Noble 
Notation  
Bowers style acronym  Triddit 
Coxeter diagram  m3o2m3o () 
Elements  
Cells  9 tetragonal disphenoids 
Faces  18 isosceles triangles 
Edges  6+9 
Vertices  6 
Vertex figure  Triangular tegum 
Measures (based on triangles of edge length 1)  
Edge lengths  Lacing (9): 
Base (6): 1  
Circumradius  
Inradius  
Hypervolume  
Central density  1 
Related polytopes  
Army  Triddit 
Regiment  Triddit 
Dual  Triangular duoprism 
Conjugate  None 
Abstract & topological properties  
Euler characteristic  0 
Orientable  Yes 
Skeleton  
Properties  
Symmetry  A_{2}≀S_{2}, order 72 
Convex  Yes 
Nature  Tame 
The triangular duotegum or triddit, also known as the triangulartriangular duotegum, the 3 duotegum, or the 33 duotegum, is a convex noble duotegum that consists of 9 tetragonal disphenoids and 6 vertices, with 6 cells joining at a vertex. It is the simplest possible duotegum, and is also the 62 step prism. It is the first in an infinite family of isogonal triangular hosohedral swirlchora and also the first in an infinite family of isochoric triangular dihedral swirlchora.
It shares the same vertex and edge configuration with the 5dimensional hexateron. In fact, it is the simplest polytope that is not a simplex, but every pair of vertices is joined by an edge. Every n2 step prism also has this property.
The ratio between the longest and shortest edges is 1: ≈ 1:1.22474.
It is notable as the Birkhoff polytope B_{3}.
Gallery[edit  edit source]

Wireframe, cell, net
Vertex coordinates[edit  edit source]
The vertices of a triangular duotegum based on 2 unitedge triangles, centered at the origin, are given by:
 ,
 ,
 ,
 .
Due to this polytope being a Birkhoff polytope, the vertices of a triangular duotegum can also be positioned in 9D by taking all 3 × 3 permutation matrices and unraveling them in reading order:
 (1, 0, 0, 0, 1, 0, 0, 0, 1)
 (1, 0, 0, 0, 0, 1, 0, 1, 0)
 (0, 0, 1, 1, 0, 0, 0, 1, 0)
 (0, 1, 0, 1, 0, 0, 0, 0, 1)
 (0, 1, 0, 0, 0, 1, 1, 0, 0)
 (0, 0, 1, 0, 1, 0, 1, 0, 0)
The longer edge length in this case is , and the shorter one 2. It is centered on .
External links[edit  edit source]
 Bowers, Jonathan. "Four Dimensional Dice Up To Twenty Sides".
 Klitzing, Richard. "triddit".
 Hi.gher.Space Wiki Contributors. "Duotrianglone".