Polyhedron with 104 faces
3D model of a snub icosidodecadodecahedron
In geometry , the snub icosidodecadodecahedron is a nonconvex uniform polyhedron , indexed as U46 . It has 104 faces (80 triangles , 12 pentagons , and 12 pentagrams ), 180 edges, and 60 vertices.[ 1] As the name indicates, it belongs to the family of snub polyhedra .
Cartesian coordinates
Let
ρ ρ -->
≈ ≈ -->
1.3247179572447454
{\displaystyle \rho \approx 1.3247179572447454}
be the real zero of the polynomial
x
3
− − -->
x
− − -->
1
{\displaystyle x^{3}-x-1}
. The number
ρ ρ -->
{\displaystyle \rho }
is known as the plastic ratio . Denote by
ϕ ϕ -->
{\displaystyle \phi }
the golden ratio . Let the point
p
{\displaystyle p}
be given by
p
=
(
ρ ρ -->
ϕ ϕ -->
2
ρ ρ -->
2
− − -->
ϕ ϕ -->
2
ρ ρ -->
− − -->
1
− − -->
ϕ ϕ -->
ρ ρ -->
2
+
ϕ ϕ -->
2
)
{\displaystyle p={\begin{pmatrix}\rho \\\phi ^{2}\rho ^{2}-\phi ^{2}\rho -1\\-\phi \rho ^{2}+\phi ^{2}\end{pmatrix}}}
.
Let the matrix
M
{\displaystyle M}
be given by
M
=
(
1
/
2
− − -->
ϕ ϕ -->
/
2
1
/
(
2
ϕ ϕ -->
)
ϕ ϕ -->
/
2
1
/
(
2
ϕ ϕ -->
)
− − -->
1
/
2
1
/
(
2
ϕ ϕ -->
)
1
/
2
ϕ ϕ -->
/
2
)
{\displaystyle 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
{\displaystyle M}
is the rotation around the axis
(
1
,
0
,
ϕ ϕ -->
)
{\displaystyle (1,0,\phi )}
by an angle of
2
π π -->
/
5
{\displaystyle 2\pi /5}
, counterclockwise. Let the linear transformations
T
0
,
… … -->
,
T
11
{\displaystyle T_{0},\ldots ,T_{11}}
be the transformations which send a point
(
x
,
y
,
z
)
{\displaystyle (x,y,z)}
to the even permutations of
(
± ± -->
x
,
± ± -->
y
,
± ± -->
z
)
{\displaystyle (\pm x,\pm y,\pm z)}
with an even number of minus signs.
The transformations
T
i
{\displaystyle T_{i}}
constitute the group of rotational symmetries of a regular tetrahedron .
The transformations
T
i
M
j
{\displaystyle T_{i}M^{j}}
(
i
=
0
,
… … -->
,
11
{\displaystyle (i=0,\ldots ,11}
,
j
=
0
,
… … -->
,
4
)
{\displaystyle j=0,\ldots ,4)}
constitute the group of rotational symmetries of a regular icosahedron .
Then the 60 points
T
i
M
j
p
{\displaystyle T_{i}M^{j}p}
are the vertices of a snub icosidodecadodecahedron. The edge length equals
2
ϕ ϕ -->
2
ρ ρ -->
2
− − -->
2
ϕ ϕ -->
− − -->
1
{\displaystyle 2{\sqrt {\phi ^{2}\rho ^{2}-2\phi -1}}}
, the circumradius equals
(
ϕ ϕ -->
+
2
)
ρ ρ -->
2
+
ρ ρ -->
− − -->
3
ϕ ϕ -->
− − -->
1
{\displaystyle {\sqrt {(\phi +2)\rho ^{2}+\rho -3\phi -1}}}
, and the midradius equals
ρ ρ -->
2
+
ρ ρ -->
− − -->
ϕ ϕ -->
{\displaystyle {\sqrt {\rho ^{2}+\rho -\phi }}}
.
For a snub icosidodecadodecahedron whose edge length is 1,
the circumradius is
R
=
1
2
ρ ρ -->
2
+
ρ ρ -->
+
2
≈ ≈ -->
1.126897912799939
{\displaystyle R={\frac {1}{2}}{\sqrt {\rho ^{2}+\rho +2}}\approx 1.126897912799939}
Its midradius is
r
=
1
2
ρ ρ -->
2
+
ρ ρ -->
+
1
≈ ≈ -->
1.0099004435452335
{\displaystyle r={\frac {1}{2}}{\sqrt {\rho ^{2}+\rho +1}}\approx 1.0099004435452335}
3D model of a medial hexagonal hexecontahedron
The medial hexagonal hexecontahedron is a nonconvex isohedral polyhedron . It is the dual of the uniform snub icosidodecadodecahedron.
See also
References
External links
Kepler-Poinsot polyhedra (nonconvex regular polyhedra)Uniform truncations of Kepler-Poinsot polyhedra Nonconvex uniform hemipolyhedra Duals of nonconvex uniform polyhedra Duals of nonconvex uniform polyhedra with infinite stellations