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Snub Disphenoid

\begin{figure}\BoxedEPSF{Deltahedron12_net.epsf scaled 900}\end{figure}

One of the convex Deltahedra also known as the Siamese Dodecahedron. It is Johnson Solid $J_{84}$.

\begin{figure}\begin{center}\BoxedEPSF{SnubDisphenoidCoords.epsf scaled 1000}\end{center}\end{figure}

The coordinates of the Vertices may be found by solving the set of four equations

$\displaystyle 1+{x_2}^2+{z_1}^2$ $\textstyle =$ $\displaystyle 4$  
$\displaystyle (x_2-1)^2+(z_3-z_1)^2$ $\textstyle =$ $\displaystyle 4$  
$\displaystyle 1+{x_2}^2+(z_3-z_2)^2$ $\textstyle =$ $\displaystyle 4$  
$\displaystyle {x_2}^2+{x_2}^2+(z_2-z_1)^2$ $\textstyle =$ $\displaystyle 4$  

for the four unknowns $x_2$, $z_1$, $z_2$, and $z_3$. Numerically,
$\displaystyle x_2$ $\textstyle =$ $\displaystyle 1.28917$  
$\displaystyle z_1$ $\textstyle =$ $\displaystyle 1.15674$  
$\displaystyle z_2$ $\textstyle =$ $\displaystyle 1.97898$  
$\displaystyle z_3$ $\textstyle =$ $\displaystyle 3.13572.$  

The analytic solution requires solving the Cubic Equation and gives
$\displaystyle x_2$ $\textstyle =$ $\displaystyle 1-7\cdot 2^{-2/3}(1-i\sqrt{3})\alpha^{-1}-{\textstyle{1\over 6}}\cdot 2^{-1/3}(1+i\sqrt{3})\alpha$  
$\displaystyle z_1$ $\textstyle =$ $\displaystyle {\textstyle{1\over 3}}\cdot 2^{-1/2} [-48+6\beta(1+i\sqrt{3})+\beta^2(1-i\sqrt{3})$  
  $\textstyle \phantom{=}$ $\displaystyle +147\beta\gamma(\sqrt{3}-i)+42\beta^2\gamma(\sqrt{3}+i)]^{1/2},$  

$\displaystyle \alpha$ $\textstyle \equiv$ $\displaystyle (12i\sqrt{237}-54)^{1/3}$  
$\displaystyle \beta$ $\textstyle \equiv$ $\displaystyle 3^{1/3}(2i\sqrt{237}-9)^{1/3}$  
$\displaystyle \gamma$ $\textstyle \equiv$ $\displaystyle (9i+2\sqrt{237}\,)^{-1}.$  

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© 1996-9 Eric W. Weisstein