截角正二十四胞體

截角正二十四胞體由48個三維胞組成: 24個立方體, 和24個截角八面體。每個頂點周圍環繞着三個截角八面體和一個立方體。

截角正二十四胞體
施萊格爾投影 (立方體胞在前)
類型	均勻多胞體
識別
名稱	截角正二十四胞體
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{3,4,3}
性質
胞	10 24 (4.4.4) 24 (4.6.6)
面	240 144 {4} 96 {6}
邊	384
頂點	144
組成與佈局
頂點圖	Irr. tetrahedron
對稱性
考克斯特群	F4, [3,4,3], order 1152
特性
convex, isogonal，環帶多胞體

構造

截角正二十四胞體的細胞可以通過在正二十四胞體的棱的三分點處截斷其頂點。截斷的24個正八面體變成新的截角八面體，並在原來的頂點處產生了24個新的立方體。

結合

截角八面體的六邊形面彼此結合在一起，而它們的正方形面則連接到立方體。

投影

正交投影

Fk 考克斯特平面	F4	B4	B3	B2
Graph
二面體群	[12]	[6]	[8]	[4]

• 
  展開圖
• 
  球極投影
  (對着一個 截角八面體胞)

坐標

一個棱長為2的截角正二十四胞體的144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，截角正二十四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