Inverted snub dodecadodecahedron
| Inverted snub dodecadodecahedron | |
|---|---|
 | |
| Rank | 3 |
| Type | Uniform |
| Space | Spherical |
| Notation | |
| Bowers style acronym | Isdid |
| Coxeter diagram | s5/3s5s (    ) |
| Elements | |
| Faces | 60 triangles, 12 pentagons, 12 pentagrams |
| Edges | 60+60+30 |
| Vertices | 60 |
| Vertex figure | Irregular pentagon, edge lengths 1, 1, (√5–1)/2, 1, (1+√5)/2  |
| Measures (edge length 1) | |
| Circumradius | ≈ 0.85163 |
| Volume | ≈ 4.61431 |
| Dihedral angles | 3–3: ≈ 130.49074° |
| 5–3: ≈ 68.64088° | |
| 5/2–3: ≈ 11.12448° | |
| Central density | 9 |
| Number of external pieces | 372 |
| Level of complexity | 39 |
| Related polytopes | |
| Army | Non-uniform snid |
| Regiment | Isdid |
| Dual | Medial inverted pentagonal hexecontahedron |
| Conjugate | Snub dodecadodecahedron |
| Convex core | Dodecahedron |
| Abstract & topological properties | |
| Flag count | 600 |
| Euler characteristic | -6 |
| Orientable | Yes |
| Genus | 4 |
| Properties | |
| Symmetry | H3+, order 60 |
| Convex | No |
| Nature | Tame |
The inverted snub dodecadodecahedron or isdid, is a uniform polyhedron. It consists of 60 snub triangles, 12 pentagrams, and 12 pentagons. Three triangles, 1 pentagon, and one pentagram meeting at each vertex. It can be constructed by alternation of the quasitruncated dodecadodecahedron and then setting all edge lengths to be equal.
Measures[edit | edit source]
The circumradius R ≈ 0.85163 of the inverted snub dodecadodecahedron with unit edge length is the smallest positive real root of:
Its volume V ≈ 4.61431 is given by the smallest positive real root of:
These same polynomials define the circumradius and volume of the snub dodecadodecahedron.
Related polyhedra[edit | edit source]
The inverted disnub dodecadodecahedron is a uniform polyhedron compound composed of the 2 opposite chiral forms of the inverted snub dodecadodecahedron.
| Name | OBSA | Schläfli symbol | CD diagram | Picture |
|---|---|---|---|---|
| Small stellated dodecahedron | sissid | {5/3,5} | x5/3o5o (  )
|
 |
| Quasitruncated small stellated dodecahedron | quit sissid | t{5/3,5} | x5/3x5o ()
|
 |
| Dodecadodecahedron | did | r{5,5/3} | o5/3x5o ()
|
 |
| Truncated great dodecahedron | tigid | t{5,5/3} | o5/3x5x ()
|
 |
| Great dodecahedron | gad | {5,5/3} | o5/3o5x ()
|
 |
| Complex ditrigonal rhombidodecadodecahedron (degenerate, ditdid+rhom) | cadditradid | rr{5,5/3} | x5/3o5x ()
|
|
| Quasitruncated dodecadodecahedron | quitdid | tr{5,5/3} | x5/3x5x ()
|
 |
| Inverted snub dodecadodecahedron | isdid | sr{5,5/3} | s5/3s5s ()
|
|
External links[edit | edit source]
- Bowers, Jonathan. "Polyhedron Category 6: Snubs" (#69).
- Klitzing, Richard. "isdid".
- Wikipedia Contributors. "Inverted snub dodecadodecahedron".
- McCooey, David. "Inverted Snub Dodecadodecahedron"