过截角正五胞体

过截角正五胞体（又叫正十胞体）是一个四维多胞体, 由10个相同的三维胞截角四面体组成。每条边连接到两个六边形和一个三角形。

过截角正五胞体
施莱格尔投影
类型	均匀多胞体
识别
名称	过截角正五胞体
参考索引	5 6 7
数学表示法
考克斯特符号	or
施莱夫利符号	t1,2{3,3,3}
性质
胞	10 (3.6.6)
面	20 {3} 20 {6}
边	60
顶点	30
组成与布局
顶点图	(锲形体)
对称性
考克斯特群	A4, [[3,3,3]], order 240
特性
convex, isogonal isotoxal, isochoric

过截角正五胞体的五维类比是过截角五维正六胞体。它的n维类比的考克斯特-迪金点图都是中间的一个或两个点有环。

过截角正五胞体是两个由一种三维胞所组成的半正多胞体之一。另一个是过截角正二十四胞体，它由48个截角立方体组成。

投影

正射投影

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

球极投影 (对着一个六边形面)

展开图

坐标

一个棱长为2的过截角正五胞体的20个顶点的笛卡儿坐标系坐标

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

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