截角正五胞体

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截角正五胞体
Schlegel half-solid truncated pentachoron.png
類型 均匀多胞体
10
5 (3.3.3) Tetrahedron.png
5 (3.6.6) Truncated tetrahedron.png
30
20 {3}
10 {6}
40
頂點 20
顶点图 Truncated 5-cell verf.png
Irr. tetrahedron
施萊夫利符號 t0,1{3,3,3}
考克斯特圖 CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
參考索引 2 3 4
考克斯特群 A4, [3,3,3], order 120
特性 convex, isogonal

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。

构造[编辑]

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[编辑]

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[编辑]

正交投影
Ak
考克斯特平面
A4 A3 A2
Graph 4-simplex t01.svg 4-simplex t01 A3.svg 4-simplex t01 A2.svg
二面体群 [5] [4] [3]

坐标[编辑]

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

\left( \frac{3}{\sqrt{10}},\  \sqrt{3 \over 2},\    \pm\sqrt{3},\         \pm1\right)
\left( \frac{3}{\sqrt{10}},\  \sqrt{3 \over 2},\    0,\                   \pm2\right)
\left( \frac{3}{\sqrt{10}},\  \frac{-1}{\sqrt{6}},\ \frac{2}{\sqrt{3}},\  \pm2\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}},\  \pm1\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}},\  \pm2\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}},\  \pm1\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[编辑]

  • 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]
      • (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.
  • Richard Klitzing, 4D, uniform polytopes (polychora) x3x3o3o - tip, o3x3x3o - deca