- Source: Inverted snub dodecadodecahedron
In geometry, the inverted snub dodecadodecahedron (or vertisnub dodecadodecahedron) is a nonconvex uniform polyhedron, indexed as U60. It is given a Schläfli symbol sr{5/3,5}.
Cartesian coordinates
Let
ξ
≈
2.109759446579943
{\displaystyle \xi \approx 2.109759446579943}
be the largest 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
≈
0.8516302281174128
{\displaystyle R={\frac {1}{2}}{\sqrt {\frac {2\xi -1}{\xi -1}}}\approx 0.8516302281174128}
Its midradius is
r
=
1
2
ξ
ξ
−
1
≈
0.6894012223976083
{\displaystyle r={\frac {1}{2}}{\sqrt {\frac {\xi }{\xi -1}}}\approx 0.6894012223976083}
The other real root of P plays a similar role in the description of the Snub dodecadodecahedron
Related polyhedra
= Medial inverted pentagonal hexecontahedron
=The medial inverted pentagonal hexecontahedron (or midly petaloid ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform inverted snub dodecadodecahedron. Its faces are irregular nonconvex pentagons, with one very acute angle.
Proportions
Denote the golden ratio by
ϕ
{\displaystyle \phi }
, and let
ξ
≈
−
0.236
993
843
45
{\displaystyle \xi \approx -0.236\,993\,843\,45}
be the largest (least negative) real zero of the polynomial
P
=
8
x
4
−
12
x
3
+
5
x
+
1
{\displaystyle P=8x^{4}-12x^{3}+5x+1}
. Then each face has three equal angles of
arccos
(
ξ
)
≈
103.709
182
219
53
∘
{\displaystyle \arccos(\xi )\approx 103.709\,182\,219\,53^{\circ }}
, one of
arccos
(
ϕ
2
ξ
+
ϕ
)
≈
3.990
130
423
41
∘
{\displaystyle \arccos(\phi ^{2}\xi +\phi )\approx 3.990\,130\,423\,41^{\circ }}
and one of
360
∘
−
arccos
(
ϕ
−
2
ξ
−
ϕ
−
1
)
≈
224.882
322
917
99
∘
{\displaystyle 360^{\circ }-\arccos(\phi ^{-2}\xi -\phi ^{-1})\approx 224.882\,322\,917\,99^{\circ }}
. Each face has one medium length edge, two short and two long ones. If the medium length is
2
{\displaystyle 2}
, then the short edges have length
1
−
1
−
ξ
ϕ
3
−
ξ
≈
0.474
126
460
54
,
{\displaystyle 1-{\sqrt {\frac {1-\xi }{\phi ^{3}-\xi }}}\approx 0.474\,126\,460\,54,}
and the long edges have length
1
+
1
−
ξ
ϕ
−
3
−
ξ
≈
37.551
879
448
54.
{\displaystyle 1+{\sqrt {\frac {1-\xi }{\phi ^{-3}-\xi }}}\approx 37.551\,879\,448\,54.}
The dihedral angle equals
arccos
(
ξ
/
(
ξ
+
1
)
)
≈
108.095
719
352
34
∘
{\displaystyle \arccos(\xi /(\xi +1))\approx 108.095\,719\,352\,34^{\circ }}
. The other real zero of the polynomial
P
{\displaystyle P}
plays a similar role for the medial pentagonal hexecontahedron.
See also
List of uniform polyhedra
Snub dodecadodecahedron
References
Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208 p. 124
External links
Weisstein, Eric W. "Medial inverted pentagonal hexecontahedron". MathWorld.
Weisstein, Eric W. "Inverted snub dodecadodecahedron". MathWorld.
Kata Kunci Pencarian:
- Daftar bentuk matematika
- Inverted snub dodecadodecahedron
- Snub dodecadodecahedron
- Compound of two inverted snub dodecadodecahedra
- List of mathematical shapes
- List of uniform polyhedra
- List of polygons, polyhedra and polytopes
- Snub polyhedron
- U60
- List of uniform polyhedra by Wythoff symbol
- List of Wenninger polyhedron models
