Dodecadodecahedron
Dodecadodecahedron | |
---|---|
Rank | 3 |
Type | Uniform |
Space | Spherical |
Notation | |
Bowers style acronym | Did |
Coxeter diagram | o5/2x5o () |
Schläfli symbol | |
Elements | |
Faces | 12 pentagons, 12 pentagrams |
Edges | 60 |
Vertices | 30 |
Vertex figure | Rectangle, edge lengths (1+√5)/2 and (√5–1)/2 |
Petrie polygons | 20 skew triambi |
Measures (edge length 1) | |
Circumradius | 1 |
Volume | 5 |
Dihedral angle | |
Central density | 3 |
Number of external pieces | 72 |
Level of complexity | 6 |
Related polytopes | |
Army | Id, edge length |
Regiment | Did |
Dual | Medial rhombic triacontahedron |
Petrie dual | Petrial dodecadodecahedron |
Conjugate | Dodecadodecahedron |
Convex core | Dodecahedron |
Abstract & topological properties | |
Flag count | 240 |
Euler characteristic | –6 |
Schläfli type | {5,4} |
Surface | Bring's surface |
Orientable | Yes |
Genus | 4 |
Properties | |
Symmetry | H_{3}, order 120 |
Convex | No |
Nature | Tame |
The dodecadodecahedron, or did, is a quasiregular uniform polyhedron. It consists of 12 pentagons and 12 pentagrams, with two of each joining at a vertex. It can be derived as a rectified small stellated dodecahedron or great dodecahedron.
This polyhedron is abstractly regular, being a quotient of the order-4 pentagonal tiling. Among the non-regular uniform polytopes, it shares this property with the ditrigonary dodecadodecahedron. Its realization may also be considered regular if one also counts conjugacies as symmetries.
Vertex coordinates[edit | edit source]
A dodecadodecahedron of side length 1 has vertex coordinates given by all permutations of
and even permutations of
The first set of vertices corresponds to a scaled octahedron which can be inscribed into the dodecadodecahedron.
Related polyhedra[edit | edit source]
The dodecadodecahedron is the colonel of a three-member regiment that also includes the small dodecahemicosahedron and the great dodecahemicosahedron.
Name | OBSA | Schläfli symbol | CD diagram | Picture |
---|---|---|---|---|
Great dodecahedron | gad | {5,5/2} | x5o5/2o () | |
Truncated great dodecahedron | tigid | t{5,5/2} | x5x5/2o () | |
Dodecadodecahedron | did | r{5,5/2} | o5x5/2o () | |
Truncated small stellated dodecahedron (degenerate, triple cover of doe) | t{5/2,5} | o5x5/2x () | ||
Small stellated dodecahedron | sissid | {5/2,5} | o5o5/2x () | |
Rhombidodecadodecahedron | raded | rr{5,5/2} | x5o5/2x () | |
Truncated dodecadodecahedron (degenerate, sird+12(10/2)) | tr{5,5/2} | x5x5/2x () | ||
Snub dodecadodecahedron | siddid | sr{5,5/2} | s5s5/2s () |
Gallery[edit | edit source]
A fundamental domain of the dodecadodecahedron in {5,4}.
External links[edit | edit source]
- Hartley, Michael. "{5,4}*240".
- Bowers, Jonathan. "Polyhedron Category 3: Quasiregulars" (#27).
- Bowers, Jonathan. "Batch 3: Id, Did, and Gid Facetings" (#1 under did).
- Klitzing, Richard. "did".
- Wikipedia Contributors. "Dodecadodecahedron".
- McCooey, David. "Dodecadodecahedron"