Inverted snub dodecadodecahedron

Let $\backslash xi\backslash approx\; 2.109759446579943$ be the largest real zero of the polynomial $P=2x^4-5x^3+3x+1$. Denote by $\backslash phi$ the golden ratio. Let the point $p$ be given by

:$p=$

\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 $M$ be given by

:$M=$

\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}

.

$M$ is the rotation around the axis $(1,\; 0,\; \backslash phi)$ by an angle of $2\backslash pi/5$, counterclockwise. Let the linear transformations $T\_0,\; \backslash ldots,\; T\_\{11\}$

be the transformations which send a point $(x,\; y,\; z)$ to the even permutations of $(\backslash pm\; x,\; \backslash pm\; y,\; \backslash pm\; z)$ with an even number of minus signs.

The transformations $T\_i$ constitute the group of rotational symmetries of a regular tetrahedron.

The transformations $T\_i\; M^j$ $(i\; =\; 0,\backslash ldots,\; 11$, $j\; =\; 0,\backslash ldots,\; 4)$ constitute the group of rotational symmetries of a regular icosahedron.

Then the 60 points $T\_i\; M^j\; p$ are the vertices of a snub dodecadodecahedron. The edge length equals $2(\backslash xi+1)\backslash sqrt\{\backslash xi^2-\backslash xi\}$, the circumradius equals $(\backslash xi+1)\backslash sqrt\{2\backslash xi^2-\backslash xi\}$, and the midradius equals $\backslash xi^2+\backslash xi$.

For a great snub icosidodecahedron whose edge length is 1,

the circumradius is

:$R\; =\; \backslash frac12\backslash sqrt\{\backslash frac\{2\backslash xi-1\}\{\backslash xi-1\}\}\; \backslash approx\; 0.8516302281174128$

Its midradius is

:$r=\backslash frac\{1\}\{2\}\backslash sqrt\{\backslash frac\{\backslash xi\}\{\backslash xi-1\}\}\; \backslash approx\; 0.6894012223976083$

The other real root of P plays a similar role in the description of the Snub dodecadodecahedron

{{Uniform polyhedra db|Uniform dual polyhedron stat table|Isdid}}

File:Medial inverted pentagonal hexecontahedron.stl

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.

Denote the golden ratio by $\backslash phi$, and let $\backslash xi\backslash approx\; -0.236\backslash ,993\backslash ,843\backslash ,45$ be the largest (least negative) real zero of the polynomial $P=8x^4-12x^3+5x+1$. Then each face has three equal angles of $\backslash arccos(\backslash xi)\backslash approx\; 103.709\backslash ,182\backslash ,219\backslash ,53^\{\backslash circ\}$, one of $\backslash arccos(\backslash phi^2\backslash xi+\backslash phi)\backslash approx\; 3.990\backslash ,130\backslash ,423\backslash ,41^\{\backslash circ\}$ and one of $360^\{\backslash circ\}-\backslash arccos(\backslash phi^\{-2\}\backslash xi-\backslash phi^\{-1\})\backslash approx\; 224.882\backslash ,322\backslash ,917\backslash ,99^\{\backslash circ\}$. Each face has one medium length edge, two short and two long ones. If the medium length is $2$, then the short edges have length

$$1-\backslash sqrt\{\backslash frac\{1-\backslash xi\}\{\backslash phi^3-\backslash xi\}\}\backslash approx\; 0.474\backslash ,126\backslash ,460\backslash ,54,$$

and the long edges have length

$$1+\backslash sqrt\{\backslash frac\{1-\backslash xi\}\{\backslash phi^\{-3\}-\backslash xi\}\}\; \backslash approx\; 37.551\backslash ,879\backslash ,448\backslash ,54.$$

The dihedral angle equals $\backslash arccos(\backslash xi/(\backslash xi+1))\backslash approx\; 108.095\backslash ,719\backslash ,352\backslash ,34^\{\backslash circ\}$. The other real zero of the polynomial $P$ plays a similar role for the medial pentagonal hexecontahedron.

• List of uniform polyhedra
• Snub dodecadodecahedron

• {{Citation | last1=Wenninger | first1=Magnus | author1-link=Magnus Wenninger | title=Dual Models | publisher=Cambridge University Press | isbn=978-0-521-54325-5 |mr=730208 | year=1983}} p. 124

{{Reflist}}

• {{mathworld | urlname = MedialInvertedPentagonalHexecontahedron| title =Medial inverted pentagonal hexecontahedron}}
• {{mathworld | urlname = InvertedSnubDodecadodecahedron | title = Inverted snub dodecadodecahedron}}

Category:Uniform polyhedra

{{Polyhedron-stub}}