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Inverted snub dodecadodecahedron

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(Redirected from Medial inverted pentagonal hexecontahedron) Polyhedron with 84 faces
Inverted snub dodecadodecahedron
Type Uniform star polyhedron
Elements F = 84, E = 150
V = 60 (χ = −6)
Faces by sides 60{3}+12{5}+12{5/2}
Coxeter diagram
Wythoff symbol | 5/3 2 5
Symmetry group I, , 532
Index references U60, C76, W114
Dual polyhedron Medial inverted pentagonal hexecontahedron
Vertex figure
3.3.5.3.5/3
Bowers acronym Isdid
3D model of an 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

Medial inverted pentagonal hexecontahedron
Type Star polyhedron
Face
Elements F = 60, E = 150
V = 84 (χ = −6)
Symmetry group I, , 532
Index references DU60
dual polyhedron Inverted snub dodecadodecahedron
3D model of a 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

References

  1. Roman, Maeder. "60: inverted snub dodecadodecahedron". MathConsult.

External links


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