斜方星形二十面體

斜方星形二十面體的二面角為[6]：

${\displaystyle \cos ^{-1}(-{\frac {5}{7}})\approx 2.366399\approx 135.58469^{\circ }}$

如果斜方星形二十面體的幾何中心位於原點且其對偶邊長為1單位長，則其頂點座標為[7]：

${\displaystyle (0,\pm {\frac {{\sqrt {5}}-1}{2}},\pm {\frac {1+{\sqrt {5}}}{2}})}$、

${\displaystyle (\pm {\frac {1+{\sqrt {5}}}{2}},0,\pm {\frac {{\sqrt {5}}-1}{2}})}$、

${\displaystyle (\pm {\frac {{\sqrt {5}}-1}{2}},\pm {\frac {1+{\sqrt {5}}}{2}},0)}$、

${\displaystyle (0,0,\pm {\frac {3{\sqrt {5}}}{5}})}$、

${\displaystyle (\pm {\frac {3{\sqrt {5}}}{5}},0,0)}$、

${\displaystyle (0,\pm {\frac {3{\sqrt {5}}}{5}},0)}$、

${\displaystyle (\pm {\frac {3\left(5-{\sqrt {5}}\right)}{20}},\pm {\frac {3{\sqrt {5}}}{10}},\pm {\frac {3\left(5+{\sqrt {5}}\right)}{20}})}$、

${\displaystyle (\pm {\frac {3\left(5+{\sqrt {5}}\right)}{20}},\pm {\frac {3\left(5-{\sqrt {5}}\right)}{20}},\pm {\frac {3{\sqrt {5}}}{10}})}$、

${\displaystyle (\pm {\frac {3{\sqrt {5}}}{10}},\pm {\frac {3\left(5+{\sqrt {5}}\right)}{20}},\pm {\frac {3\left(5-{\sqrt {5}}\right)}{20}})}$、

${\displaystyle (\pm 1,\pm 1,\pm 1)}$。

斜方星形二十面體由60個全等的反平行四邊形組成，每個反平行四邊形有四個角，其中兩個銳角為${\displaystyle \arccos({\frac {3}{4}})\approx 41.409\,622\,109\,27^{\circ }}$，兩個鈍角為${\displaystyle \arccos(-{\frac {1}{6}})\approx 99.594\,068\,226\,86^{\circ }}$，交叉邊的交角為${\displaystyle \arccos({\frac {1}{8}}+{\frac {7{\sqrt {5}}}{24}})\approx 38.996\,309\,663\,87^{\circ }}$。在整個立體中，長邊（反平行四邊形的交叉邊）與短邊（反平行四邊形外側的腰）各有60條。[6]長邊和短邊的比為${\displaystyle {\frac {3}{2}}+{\frac {1}{2}}{\sqrt {5}}}$，恰好是黃金比例的平方。

斜方星形二十面體的面有跟其他的面相交的性質，其互相相交的部分恰好位於反平行四邊形自相交的交點處。[5]

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  斜方星形二十面體的其中三個面，分別以紅色、藍色和綠色表示
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  斜方星形二十面體之面的形狀