加長型球狀屋頂

若一個加長型球狀屋頂邊長為${\displaystyle a}$，則其表面積${\displaystyle A}$為：[9]

${\displaystyle A=\left(2+4{\sqrt {3}}\right)a^{2}\approx 8.92820a^{2},}$[10]

而其體積${\displaystyle V}$為：

${\displaystyle V=\xi a^{3}\approx 1.94811a^{3},}$

其中的常數${\displaystyle \xi }$由 A334114給出[11]，其為下列多項式的其中一個實根，約為1.948108228859[11]：

${\displaystyle {\begin{aligned}&1680x^{16}-4800x^{15}-3712x^{14}+17216x^{13}+1568x^{12}-24576x^{11}+2464x^{10}+17248x^{9}\\&\quad {}-3384x^{8}-5584x^{7}+2000x^{6}+240x^{5}-776x^{4}+304x^{3}+200x^{2}-56x-23.\end{aligned}}}$

邊長為2的加長型球狀屋頂的頂點座標為：

${\displaystyle \left(\pm 1,\,0,\,2B\right)}$

${\displaystyle \left(\pm 1,\,\pm 2k,\,0\right)}$

${\displaystyle \left(\pm {\frac {C+B}{B}},\,0,\,{\frac {D}{B}}\right)}$

${\displaystyle \left(0,\,\pm 1,\,-{\sqrt {2+4k-4k^{2}}}\right)}$

${\displaystyle \left(\pm \left(1+{\frac {CD}{B^{3}}}\right),\,0,\,{\frac {2k^{4}-1}{B^{3}}}\right)}$

其中，${\displaystyle B}$、${\displaystyle C}$和${\displaystyle D}$為：

${\displaystyle B={\sqrt {1-k^{2}}}}$

${\displaystyle C={\sqrt {3-4k^{2}}}}$

${\displaystyle D=1-2k^{2}}$

其中，${\displaystyle k}$ ≈ 0.59463是下列多項式的做小實根：

${\displaystyle {\begin{aligned}&1680x^{16}-4800x^{15}-3712x^{14}+17216x^{13}+1568x^{12}-24576x^{11}+2464x^{10}+17248x^{9}\\&\quad {}-3384x^{8}-5584x^{7}+2000x^{6}+240x^{5}-776x^{4}+304x^{3}+200x^{2}-56x-23.\end{aligned}}}$

這些座標也可以由下列頂點的軌道的並集在沿xz平面和yz平面鏡射所產生的空間對稱群之群作用下給出：[12]

${\displaystyle {\begin{aligned}&\left(0,1,2{\sqrt {1-k^{2}}}\right),\,(2k,1,0),\,\left(0,{\frac {\sqrt {3-4k^{2}}}{\sqrt {1-k^{2}}}}+1,{\frac {1-2k^{2}}{\sqrt {1-k^{2}}}}\right),\\&\left(1,0,-{\sqrt {2+4k-4k^{2}}}\right),\,\left(0,{\frac {{\sqrt {3-4k^{2}}}\left(2k^{2}-1\right)}{\left(k^{2}-1\right){\sqrt {1-k^{2}}}}}+1,{\frac {2k^{4}-1}{\left(1-k^{2}\right)^{\frac {3}{2}}}}\right)\end{aligned}}}$