截角正五胞体

截角正五胞体由十个三维胞组成: 五个正四面体, 和五个截角四面体。每个顶点周围环绕着三个截角四面体和一个正四面体。截角正五胞体是截角四面体的四维类比。

截角正五胞体
施莱格尔投影 (正四面体胞在前)
類型	均匀多胞体
對偶多胞形	四角化正五胞體
識別
名稱	截角正五胞体
參考索引	2 3 4
數學表示法
考克斯特符號 （英语：Coxeter-Dynkin diagram）
施萊夫利符號	t0,1{3,3,3}
性質
胞	10 5 (3.3.3) 5 (3.6.6)
面	30 20 {3} 10 {6}
邊	40
頂點	20
組成與佈局
顶点图	Irr. tetrahedron
對稱性
考克斯特群	A4, [3,3,3], order 120
特性
convex, isogonal
查 论 编

构造

截角正五胞体的细胞可以通过在正五胞体的棱的三分点处截断其顶点。截断的五个正四面体变成新的截角四面体，并在原来的顶点处产生了五个新的正四面体。

结合

截角四面体的六边形面彼此结合在一起，而它们的三角形面则连接到正四面体。

投影

正交投影

Ak 考克斯特平面	A4	A3	A2
Graph
二面体群	[5]	[4]	[3]

•  

  展开图

•  

  球极投影
  (对着一个 截角四面体胞)

坐标

一个棱长为2的截角正五胞体的20个顶点的笛卡儿坐标系坐标

${\displaystyle \left({\frac {3}{\sqrt {10}}},\ {\sqrt {3 \over 2}},\ \pm {\sqrt {3}},\ \pm 1\right)}$  ${\displaystyle \left({\frac {3}{\sqrt {10}}},\ {\sqrt {3 \over 2}},\ 0,\ \pm 2\right)}$  ${\displaystyle \left({\frac {3}{\sqrt {10}}},\ {\frac {-1}{\sqrt {6}}},\ {\frac {2}{\sqrt {3}}},\ \pm 2\right)}$  ${\displaystyle \left({\frac {3}{\sqrt {10}}},\ {\frac {-1}{\sqrt {6}}},\ {\frac {4}{\sqrt {3}}},\ 0\right)}$  ${\displaystyle \left({\frac {3}{\sqrt {10}}},\ {\frac {-5}{\sqrt {6}}},\ {\frac {1}{\sqrt {3}}},\ \pm 1\right)}$  ${\displaystyle \left({\frac {3}{\sqrt {10}}},\ {\frac {-5}{\sqrt {6}}},\ {\frac {-2}{\sqrt {3}}},\ 0\right)}$

${\displaystyle \left(-{\sqrt {2 \over 5}},\ {\sqrt {2 \over 3}},\ {\frac {2}{\sqrt {3}}},\ \pm 2\right)}$  ${\displaystyle \left(-{\sqrt {2 \over 5}},\ {\sqrt {2 \over 3}},\ {\frac {-4}{\sqrt {3}}},\ 0\right)}$  ${\displaystyle \left(-{\sqrt {2 \over 5}},\ -{\sqrt {6}},\ 0,\ 0\right)}$  ${\displaystyle \left({\frac {-7}{\sqrt {10}}},\ {\frac {1}{\sqrt {6}}},\ {\frac {1}{\sqrt {3}}},\ \pm 1\right)}$  ${\displaystyle \left({\frac {-7}{\sqrt {10}}},\ {\frac {1}{\sqrt {6}}},\ {\frac {-2}{\sqrt {3}}},\ 0\right)}$  ${\displaystyle \left({\frac {-7}{\sqrt {10}}},\ -{\sqrt {3 \over 2}},\ 0,\ 0\right)}$

更简单的，截角正五胞体的顶点是五维空间笛卡儿坐标系的（0,0,0,1,2）或（0,1,2,2,2）的全排列。

参考文献

• H.S.M. Coxeter:
  • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1] （页面存档备份，存于互联网档案馆）
    • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
    • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, ISBN 0-486-40919-8 p.88 (Chapter 5: Regular Skew Polyhedra in three and four dimensions and their topological analogues, Proceedings of the London Mathematics Society, Ser. 2, Vol 43, 1937.)
  • Coxeter, H. S. M. Regular Skew Polyhedra in Three and Four Dimensions. Proc. London Math. Soc. 43, 33-62, 1937.
• Norman Johnson Uniform Polytopes, Manuscript (1991)
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
• Olshevsky, George, Pentachoron at Glossary for Hyperspace.
  • 1. Convex uniform polychora based on the pentachoron - Model 3, George Olshevsky.
• Klitzing, Richard. 4D uniform polytopes (polychora). bendwavy.org.  x3x3o3o - tip, o3x3x3o - deca