# 七維正八胞體

verse-and-dimensions的wikiaBowers acronym
oca

{36}

70個正四面體
56個正三角形
28

## 性質

### 作為一種排佈

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

### 頂點座標

${\displaystyle \left({\sqrt {1/28}},\ {\sqrt {1/21}},\ {\sqrt {1/15}},\ {\sqrt {1/10}},\ {\sqrt {1/6}},\ {\sqrt {1/3}},\ \pm 1\right)}$
${\displaystyle \left({\sqrt {1/28}},\ {\sqrt {1/21}},\ {\sqrt {1/15}},\ {\sqrt {1/10}},\ {\sqrt {1/6}},\ -2{\sqrt {1/3}},\ 0\right)}$
${\displaystyle \left({\sqrt {1/28}},\ {\sqrt {1/21}},\ {\sqrt {1/15}},\ {\sqrt {1/10}},\ -{\sqrt {3/2}},\ 0,\ 0\right)}$
${\displaystyle \left({\sqrt {1/28}},\ {\sqrt {1/21}},\ {\sqrt {1/15}},\ -2{\sqrt {2/5}},\ 0,\ 0,\ 0\right)}$
${\displaystyle \left({\sqrt {1/28}},\ {\sqrt {1/21}},\ -{\sqrt {5/3}},\ 0,\ 0,\ 0,\ 0\right)}$
${\displaystyle \left({\sqrt {1/28}},\ -{\sqrt {12/7}},\ 0,\ 0,\ 0,\ 0,\ 0\right)}$
${\displaystyle \left(-{\sqrt {7/4}},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)}$

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

## 圖像

 三維空間的七維正八胞體 三角化四面體包络中的球棍模型 以振幅多面体（英语：Amplituhedron）表面呈現的七維正八胞體 七維正八胞體投影到三維，再用相機投影示意其皮特里投影

## 正交投影

Ak A7 A6 A5

[8] [7] [6]
Ak考克斯特平面 A4 A3 A2

## 參考文獻

1. Klitzing, Richard. octaexon. bendwavy.org. [2022-12-19]. （原始内容存档于2022-12-19）.
2. ^ French, K.L. The Hidden Geometry of Life: The Science and Spirituality of Nature. Gateway series. Watkins Media Limited. 2014 [2022-12-19]. ISBN 9781780288451. （原始内容存档于2023-01-09）.
3. ^ Klitzing, Richard. 7D uniform polytopes (polyexa) x3o3o3o3o3o3o — oca. bendwavy.org.
4. ^ Adams, Joshua; Zvengrowski, Peter; Laird, Philip. Vertex Embeddings of Regular Polytopes. Expositiones Mathematicae. 2003.
5. ^ Sloane, N.J.A. (编). Sequence A019442. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Integers n such that a simplex of dimension n-1 can be inscribed in a hypercube of dimension n-1
6. ^ Coxeter, H.S.M. §1.8 Configurations. 3rd. Dover. 1973. ISBN 0-486-61480-8.
7. ^ Coxeter, H.S.M. Regular Complex Polytopes 2nd. Cambridge University Press. 1991: 117 [2022-12-19]. ISBN 9780521394901. （原始内容存档于2023-01-09）.