過截角正五胞體

過截角正五胞體（又叫正十胞體）是一個四維多胞體, 由10個相同的三維胞截角四面體組成。每條邊連接到兩個六邊形和一個三角形。

過截角正五胞體
施萊格爾投影
類型	均勻多胞體
識別
名稱	過截角正五胞體
參考索引	5 6 7
數學表示法
考克斯特符號 （英語：Coxeter-Dynkin diagram）	or
施萊夫利符號	t1,2{3,3,3}
性質
胞	10 (3.6.6)
面	20 {3} 20 {6}
邊	60
頂點	30
組成與佈局
頂點圖	(鍥形體)
對稱性
考克斯特群	A4, [[3,3,3]], order 240
特性
convex, isogonal isotoxal, isochoric

過截角正五胞體的五維類比是過截角五維正六胞體。它的n維類比的考克斯特-迪金點圖都是中間的一個或兩個點有環。

過截角正五胞體是兩個由一種三維胞所組成的半正多胞體之一。另一個是過截角正二十四胞體，它由48個截角立方體組成。

投影

正射投影

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

球極投影 (對著一個六邊形面)

展開圖

坐標

一個棱長為2的過截角正五胞體的20個頂點的笛卡兒坐標系坐標

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

更簡單的，過截角正五胞體的頂點是五維空間笛卡兒坐標系的（0,0,1,2,2）或(1,0,0,0,-1)的全排列。

參考文獻

• 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] （頁面存檔備份，存於網際網路檔案館）
    • (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