- Source: Snub dodecadodecahedron
In geometry, the snub dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U40. It has 84 faces (60 triangles, 12 pentagons, and 12 pentagrams), 150 edges, and 60 vertices. It is given a Schläfli symbol sr{5⁄2,5}, as a snub great dodecahedron.
Cartesian coordinates
Let
ξ
≈
1.2223809502469911
{\displaystyle \xi \approx 1.2223809502469911}
be the smallest real zero of the polynomial
P
=
2
x
4
−
5
x
3
+
3
x
+
1
{\displaystyle P=2x^{4}-5x^{3}+3x+1}
. Denote by
ϕ
{\displaystyle \phi }
the golden ratio. Let the point
p
{\displaystyle p}
be given by
p
=
(
ϕ
−
2
ξ
2
−
ϕ
−
2
ξ
+
ϕ
−
1
−
ϕ
2
ξ
2
+
ϕ
2
ξ
+
ϕ
ξ
2
+
ξ
)
{\displaystyle p={\begin{pmatrix}\phi ^{-2}\xi ^{2}-\phi ^{-2}\xi +\phi ^{-1}\\-\phi ^{2}\xi ^{2}+\phi ^{2}\xi +\phi \\\xi ^{2}+\xi \end{pmatrix}}}
.
Let the matrix
M
{\displaystyle M}
be given by
M
=
(
1
/
2
−
ϕ
/
2
1
/
(
2
ϕ
)
ϕ
/
2
1
/
(
2
ϕ
)
−
1
/
2
1
/
(
2
ϕ
)
1
/
2
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
)
{\displaystyle (1,0,\phi )}
by an angle of
2
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 dodecadodecahedron. The edge length equals
2
(
ξ
+
1
)
ξ
2
−
ξ
{\displaystyle 2(\xi +1){\sqrt {\xi ^{2}-\xi }}}
, the circumradius equals
(
ξ
+
1
)
2
ξ
2
−
ξ
{\displaystyle (\xi +1){\sqrt {2\xi ^{2}-\xi }}}
, and the midradius equals
ξ
2
+
ξ
{\displaystyle \xi ^{2}+\xi }
.
For a great snub icosidodecahedron whose edge length is 1,
the circumradius is
R
=
1
2
2
ξ
−
1
ξ
−
1
≈
1.2744398820380232
{\displaystyle R={\frac {1}{2}}{\sqrt {\frac {2\xi -1}{\xi -1}}}\approx 1.2744398820380232}
Its midradius is
r
=
1
2
ξ
ξ
−
1
≈
1.1722614951149297
{\displaystyle r={\frac {1}{2}}{\sqrt {\frac {\xi }{\xi -1}}}\approx 1.1722614951149297}
The other real root of P plays a similar role in the description of the Inverted snub dodecadodecahedron
Related polyhedra
= Medial pentagonal hexecontahedron
=The medial pentagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the snub dodecadodecahedron. It has 60 intersecting irregular pentagonal faces.
See also
List of uniform polyhedra
Inverted snub dodecadodecahedron
References
Wenninger, Magnus (1983), Dual Models, Cambridge University Press, doi:10.1017/CBO9780511569371, ISBN 978-0-521-54325-5, MR 0730208
External links
Weisstein, Eric W. "Medial pentagonal hexecontahedron". MathWorld.
Weisstein, Eric W. "Snub dodecadodecahedron". MathWorld.
Kata Kunci Pencarian:
- Daftar bentuk matematika
- Snub dodecadodecahedron
- Inverted snub dodecadodecahedron
- List of polygons, polyhedra and polytopes
- List of mathematical shapes
- List of uniform polyhedra
- Compound of two snub dodecadodecahedra
- Snub icosidodecadodecahedron
- Compound of two inverted snub dodecadodecahedra
- Icosidodecadodecahedron
- Snub polyhedron
