截角正五胞體

截角正五胞體由十個三維胞組成: 五個正四面體, 和五個截角四面體。每個頂點周圍環繞著三個截角四面體和一個正四面體。截角正五胞體是截角四面體的四維類比。

截角正五胞體
施萊格爾投影 (正四面體胞在前)
類型	均勻多胞體
對偶多胞形	四角化正五胞體
識別
名稱	截角正五胞體
參考索引	2 3 4
數學表示法
考克斯特符號
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
頂點圖	Irr. tetrahedron
對稱性
考克斯特群	A4, [3,3,3], order 120
特性
convex, isogonal

構造

截角正五胞體的細胞可以通過在正五胞體的棱的三分點處截斷其頂點。截斷的五個正四面體變成新的截角四面體，並在原來的頂點處產生了五個新的正四面體。

結合

截角四面體的六邊形面彼此結合在一起，而它們的三角形面則連接到正四面體。

投影

正交投影

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

• 
  展開圖
• 
  球極投影
  (對著一個 截角四面體胞)

坐標

一個棱長為2的截角正五胞體的20個頂點的笛卡兒坐標系坐標

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

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

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