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    • Source: Great inverted snub icosidodecahedron

      • In geometry, the great inverted snub icosidodecahedron (or great vertisnub icosidodecahedron) is a uniform star polyhedron, indexed as U69. It is given a SchlƤfli symbol sr{5ā„3,3}, and Coxeter-Dynkin diagram . In the book Polyhedron Models by Magnus Wenninger, the polyhedron is misnamed great snub icosidodecahedron, and vice versa.


      Cartesian coordinates


      Let



      Ī¾
      ā‰ˆ
      āˆ’
      0.5055605785332548


      {\displaystyle \xi \approx -0.5055605785332548}

      be the largest (least negative) negative zero of the polynomial




      x

      3


      +
      2

      x

      2


      āˆ’

      Ļ•

      āˆ’
      2




      {\displaystyle x^{3}+2x^{2}-\phi ^{-2}}

      , where



      Ļ•


      {\displaystyle \phi }

      is the golden ratio. Let the point



      p


      {\displaystyle p}

      be given by




      p
      =


      (



      Ī¾





      Ļ•

      āˆ’
      2


      āˆ’

      Ļ•

      āˆ’
      2


      Ī¾




      āˆ’

      Ļ•

      āˆ’
      3


      +

      Ļ•

      āˆ’
      1


      Ī¾
      +
      2

      Ļ•

      āˆ’
      1



      Ī¾

      2





      )




      {\displaystyle p={\begin{pmatrix}\xi \\\phi ^{-2}-\phi ^{-2}\xi \\-\phi ^{-3}+\phi ^{-1}\xi +2\phi ^{-1}\xi ^{2}\end{pmatrix}}}

      .
      Let the matrix



      M


      {\displaystyle M}

      be given by




      M
      =


      (



      1

      /

      2


      āˆ’
      Ļ•

      /

      2


      1

      /

      (
      2
      Ļ•
      )




      Ļ•

      /

      2


      1

      /

      (
      2
      Ļ•
      )


      āˆ’
      1

      /

      2




      1

      /

      (
      2
      Ļ•
      )


      1

      /

      2


      Ļ•

      /

      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
      ,
      Ļ•
      )


      {\displaystyle (1,0,\phi )}

      by an angle of



      2
      Ļ€

      /

      5


      {\displaystyle 2\pi /5}

      , counterclockwise. Let the linear transformations




      T

      0


      ,
      ā€¦
      ,

      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



      (
      Ā±
      x
      ,
      Ā±
      y
      ,
      Ā±
      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
      ,
      ā€¦
      ,
      11


      {\displaystyle (i=0,\ldots ,11}

      ,



      j
      =
      0
      ,
      ā€¦
      ,
      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 great snub icosahedron. The edge length equals



      āˆ’
      2
      Ī¾


      1
      āˆ’
      Ī¾




      {\displaystyle -2\xi {\sqrt {1-\xi }}}

      , the circumradius equals



      āˆ’
      Ī¾


      2
      āˆ’
      Ī¾




      {\displaystyle -\xi {\sqrt {2-\xi }}}

      , and the midradius equals



      āˆ’
      Ī¾


      {\displaystyle -\xi }

      .
      For a great snub icosidodecahedron whose edge length is 1,
      the circumradius is




      R
      =


      1
      2






      2
      āˆ’
      Ī¾


      1
      āˆ’
      Ī¾




      ā‰ˆ
      0.6450202372957795


      {\displaystyle R={\frac {1}{2}}{\sqrt {\frac {2-\xi }{1-\xi }}}\approx 0.6450202372957795}


      Its midradius is




      r
      =


      1
      2





      1

      1
      āˆ’
      Ī¾




      ā‰ˆ
      0.4074936889340787


      {\displaystyle r={\frac {1}{2}}{\sqrt {\frac {1}{1-\xi }}}\approx 0.4074936889340787}


      The four positive real roots of the sextic in R2,




      4096

      R

      12


      āˆ’
      27648

      R

      10


      +
      47104

      R

      8


      āˆ’
      35776

      R

      6


      +
      13872

      R

      4


      āˆ’
      2696

      R

      2


      +
      209
      =
      0


      {\displaystyle 4096R^{12}-27648R^{10}+47104R^{8}-35776R^{6}+13872R^{4}-2696R^{2}+209=0}


      are the circumradii of the snub dodecahedron (U29), great snub icosidodecahedron (U57), great inverted snub icosidodecahedron (U69), and great retrosnub icosidodecahedron (U74).


      Related polyhedra




      = Great inverted pentagonal hexecontahedron

      =

      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.


      Proportions


      Denote the golden ratio by



      Ļ•


      {\displaystyle \phi }

      . Let



      Ī¾
      ā‰ˆ
      0.252

      780

      289

      27


      {\displaystyle \xi \approx 0.252\,780\,289\,27}

      be the smallest positive zero of the polynomial



      P
      =
      8

      x

      3


      āˆ’
      8

      x

      2


      +

      Ļ•

      āˆ’
      2




      {\displaystyle P=8x^{3}-8x^{2}+\phi ^{-2}}

      . Then each pentagonal face has four equal angles of



      arccos
      ā”
      (
      Ī¾
      )
      ā‰ˆ
      75.357

      903

      417


      42

      āˆ˜




      {\displaystyle \arccos(\xi )\approx 75.357\,903\,417\,42^{\circ }}

      and one angle of




      360

      āˆ˜


      āˆ’
      arccos
      ā”
      (
      āˆ’

      Ļ•

      āˆ’
      1


      +

      Ļ•

      āˆ’
      2


      Ī¾
      )
      ā‰ˆ
      238.568

      386

      330


      33

      āˆ˜




      {\displaystyle 360^{\circ }-\arccos(-\phi ^{-1}+\phi ^{-2}\xi )\approx 238.568\,386\,330\,33^{\circ }}

      . Each face has three long and two short edges. The ratio



      l


      {\displaystyle l}

      between the lengths of the long and the short edges is given by




      l
      =



      2
      āˆ’
      4

      Ī¾

      2




      1
      āˆ’
      2
      Ī¾



      ā‰ˆ
      3.528

      053

      034

      81


      {\displaystyle l={\frac {2-4\xi ^{2}}{1-2\xi }}\approx 3.528\,053\,034\,81}

      .
      The dihedral angle equals



      arccos
      ā”
      (
      Ī¾

      /

      (
      Ī¾
      +
      1
      )
      )
      ā‰ˆ
      78.359

      199

      060


      62

      āˆ˜




      {\displaystyle \arccos(\xi /(\xi +1))\approx 78.359\,199\,060\,62^{\circ }}

      . Part of each face lies inside the solid, hence is invisible in solid models. The other two zeroes of the polynomial



      P


      {\displaystyle P}

      play a similar role in the description of the great pentagonal hexecontahedron and the great pentagrammic hexecontahedron.


      See also


      List of uniform polyhedra
      Great snub icosidodecahedron
      Great retrosnub icosidodecahedron


      References


      Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208 p. 126


      External links


      Weisstein, Eric W. "Great inverted pentagonal hexecontahedron". MathWorld.
      Weisstein, Eric W. "Great inverted snub icosidodecahedron". MathWorld.

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