七维正八胞体

七维正八胞体共由8个顶点、28条边、56个三角形面、70个正四面体的三维胞、56个正五胞体的四维胞（英语：4-face）、28个五维正六胞体的五维胞（维基数据所列：Q18028552）和8个六维正七胞体的六维胞（维基数据所列：Q18028565）组成，其中六维正七胞体为七维正八胞体的维面。对于一个边长为a的七维正八胞体，其超胞积是${\displaystyle {\cfrac {a^{7}}{20160}}}$ ,表胞积是${\displaystyle {\cfrac {{\sqrt {7}}a^{6}}{720}}}$ ,高是${\displaystyle {\cfrac {2{\sqrt {7}}a}{7}}}$ 。若一个七维正八胞体的棱长为1，则其外接七维超球的半径为${\displaystyle {\frac {\sqrt {7}}{4}}}$ ,内切七维超球的半径为${\displaystyle {\frac {\sqrt {7}}{32}}}$ 。^[1]

边长为2的七维正八胞体可以内接于单位七维超立方体（英语：7-cube）中。^[4]下一个可以内接于单位超方形的最大单纯形为十一维正十二胞体。^[5]

作为一种排布

七维正八胞体的排布矩阵（英语：Configuration_(polytope)）为：^[1]

${\displaystyle {\begin{bmatrix}{\begin{matrix}8&7&21&35&35&21&7\\2&28&6&15&20&15&6\\3&3&56&5&10&10&5\\4&6&4&70&4&6&4\\5&10&10&5&56&3&3\\6&15&20&15&6&28&2\\7&21&35&35&21&7&8\end{matrix}}\end{bmatrix}}}$ 

行和列对应于七维正八胞体的顶点、边、面、胞、四维胞（英语：4-face）、五维胞（维基数据所列：Q18028552）和六维胞（维基数据所列：Q18028565）。对角线上的数字表示该元素在七维正八胞体中的数量。非对角线的数量表示对应行所代表的元素上有多少列所代表的元素交于该处。由于七维正八胞体是一种自身对偶的多胞体，因此这个排布矩阵旋转180度后会相同。^[6][7]

顶点坐标

若一个七维正八胞体几何中心位于原点，且边长为2单位长，则其顶点坐标为：

${\displaystyle \left({\sqrt {1/28}},\ {\sqrt {1/21}},\ {\sqrt {1/15}},\ {\sqrt {1/10}},\ {\sqrt {1/6}},\ {\sqrt {1/3}},\ \pm 1\right)}$ 

${\displaystyle \left({\sqrt {1/28}},\ {\sqrt {1/21}},\ {\sqrt {1/15}},\ {\sqrt {1/10}},\ {\sqrt {1/6}},\ -2{\sqrt {1/3}},\ 0\right)}$ 

${\displaystyle \left({\sqrt {1/28}},\ {\sqrt {1/21}},\ {\sqrt {1/15}},\ {\sqrt {1/10}},\ -{\sqrt {3/2}},\ 0,\ 0\right)}$ 

${\displaystyle \left({\sqrt {1/28}},\ {\sqrt {1/21}},\ {\sqrt {1/15}},\ -2{\sqrt {2/5}},\ 0,\ 0,\ 0\right)}$ 

${\displaystyle \left({\sqrt {1/28}},\ {\sqrt {1/21}},\ -{\sqrt {5/3}},\ 0,\ 0,\ 0,\ 0\right)}$ 

${\displaystyle \left({\sqrt {1/28}},\ -{\sqrt {12/7}},\ 0,\ 0,\ 0,\ 0,\ 0\right)}$ 

${\displaystyle \left(-{\sqrt {7/4}},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)}$ 

透过将七维正八胞体可以内接于七维超立方体（英语：7-cube）中可以获得更简单的坐标集合，其值为：^[1]

${\displaystyle \left({\frac {1}{4}},\,{\frac {1}{4}},\,{\frac {1}{4}},\,{\frac {1}{4}},\,{\frac {1}{4}},\,{\frac {1}{4}},\,{\frac {1}{4}}\right),}$ 

${\displaystyle \left({\frac {1}{4}},\,{\frac {1}{4}},\,{\frac {1}{4}},\,-{\frac {1}{4}},\,-{\frac {1}{4}},\,-{\frac {1}{4}},\,-{\frac {1}{4}}\right),}$ 

${\displaystyle \left({\frac {1}{4}},\,-{\frac {1}{4}},\,-{\frac {1}{4}},\,-{\frac {1}{4}},\,-{\frac {1}{4}},\,{\frac {1}{4}},\,{\frac {1}{4}}\right),}$ 

${\displaystyle \left({\frac {1}{4}},\,-{\frac {1}{4}},\,-{\frac {1}{4}},\,{\frac {1}{4}},\,{\frac {1}{4}},\,-{\frac {1}{4}},\,-{\frac {1}{4}}\right),}$ 

${\displaystyle \left(-{\frac {1}{4}},\,{\frac {1}{4}},\,-{\frac {1}{4}},\,{\frac {1}{4}},\,-{\frac {1}{4}},\,{\frac {1}{4}},\,-{\frac {1}{4}}\right),}$ 

${\displaystyle \left(-{\frac {1}{4}},\,{\frac {1}{4}},\,-{\frac {1}{4}},\,-{\frac {1}{4}},\,{\frac {1}{4}},\,-{\frac {1}{4}},\,{\frac {1}{4}}\right),}$ 

${\displaystyle \left(-{\frac {1}{4}},\,-{\frac {1}{4}},\,{\frac {1}{4}},\,{\frac {1}{4}},\,-{\frac {1}{4}},\,-{\frac {1}{4}},\,{\frac {1}{4}}\right),}$ 

${\displaystyle \left(-{\frac {1}{4}},\,-{\frac {1}{4}},\,{\frac {1}{4}},\,-{\frac {1}{4}},\,{\frac {1}{4}},\,{\frac {1}{4}},\,-{\frac {1}{4}}\right).}$ 

更简单地，七维正八胞体可以坐落于八维空间坐标(0,0,0,0,0,0,0,1)的排列。这个结构是基于八维正轴体（英语：8-orthoplex）的维面。