In geometry, the great inverted snub icosidodecahedron (or great vertisnub icosidodecahedron) is a uniform star polyhedron, indexed as U69. It is given a Schläfli symbol and Coxeter-Dynkin diagram . In the book Polyhedron Models by Magnus Wenninger, the polyhedron is misnamed great snub icosidodecahedron, and vice versa.
Let
\xi ≈ -0.5055605785332548
x3+2x2-\phi-2
\phi
p
p= \begin{pmatrix} \xi\\ \phi-2-\phi-2\xi\\ -\phi-3+\phi-1\xi+2\phi-1\xi2 \end{pmatrix}
M
M= \begin{pmatrix} 1/2&-\phi/2&1/(2\phi)\\ \phi/2&1/(2\phi)&-1/2\\ 1/(2\phi)&1/2&\phi/2 \end{pmatrix}
M
(1,0,\phi)
2\pi/5
T0,\ldots,T11
(x,y,z)
(\pmx,\pmy,\pmz)
Ti
TiMj
(i=0,\ldots,11
j=0,\ldots,4)
TiMjp
-2\xi\sqrt{1-\xi}
-\xi\sqrt{2-\xi}
-\xi
For a great snub icosidodecahedron whose edge length is 1,the circumradius is
R=
| ||||
r= | 1 | \sqrt{ |
2 |
1 | |
1-\xi |
The four positive real roots of the sextic in,are the circumradii of the snub dodecahedron (U29), great snub icosidodecahedron (U57), great inverted snub icosidodecahedron (U69), and great retrosnub icosidodecahedron (U74).
The great inverted pentagonal hexecontahedron (or petaloidal trisicosahedron) is a nonconvex isohedral polyhedron. It is composed of 60 concave pentagonal faces, 150 edges and 92 vertices.
It is the dual of the uniform great inverted snub icosidodecahedron.
Denote the golden ratio by
\phi
\xi ≈ 0.25278028927
P=8x3-8x2+\phi-2
\arccos(\xi) ≈ 75.35790341742\circ
360\circ-\arccos(-\phi-1+\phi-2\xi) ≈ 238.56838633033\circ
l
l=
2-4\xi2 | |
1-2\xi |
≈ 3.52805303481
\arccos(\xi/(\xi+1)) ≈ 78.35919906062\circ
P