查看“︁截角正五胞体”︁的源代码

{{NoteTA|G1=Math}}
{{Infobox polychoron
| name = 截角正五胞体
| imagename = Schlegel half-solid truncated pentachoron.png
| caption = [[施莱格尔投影]]<BR>([[正四面体]]胞在前)
| polytope = 截角正五胞体
| Type = [[均匀多胞体]]
| group_type = 
| Cell =10<br/>5 [[正四面体|(''3.3.3'')]] [[Image:Tetrahedron.png|20px]]<BR>5 [[截角四面体|(''3.6.6'')]] [[Image:Truncated tetrahedron.png|20px]]
| Face =30<br/>20 {3}<BR>10 {6}
| Edge =40
| Vertice =20
| Vertice_type =[[Image:Truncated 5-cell verf.png|80px]]<BR>Irr. tetrahedron
| Schläfli =t<sub>0,1</sub>{3,3,3}
| Symmetry_group = 
| dual = [[四角化正五胞體]]
| Properties =[[Convex polytope|convex]], [[isogonal figure|isogonal]]
| Index_references = ''[[截半正五胞体|2]]'' 3 ''[[Cantellated 5-cell|4]]''
| Coxeter_group =A<sub>4</sub>, [3,3,3], order 120
| Coxeter_diagram = {{CDD|node_1|3|node_1|3|node|3|node}}
}}
'''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==

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==

[[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 ==
{| class=wikitable
|+ [[正交投影]]
|- align=center
!A<sub>k</sub><BR>[[考克斯特平面]]
!A<sub>4</sub>
!A<sub>3</sub>
!A<sub>2</sub>
|- align=center
!Graph
|[[File:4-simplex_t01.svg|150px]]
|[[File:4-simplex_t01_A3.svg|150px]]
|[[File:4-simplex_t01_A2.svg|150px]]
|- align=center
![[二面体群]]
|[5]
|[4]
|[3]
|}

<gallery>
Image:Truncated pentachoron net.png|[[展开图]]
Image:Truncated simplex stereographic.png|[[球极投影]]<BR>(对着一个 [[截角四面体]]胞)
</gallery>

==坐标==

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

{| 
|
:<math>\left( \frac{3}{\sqrt{10}},\  \sqrt{3 \over 2},\    \pm\sqrt{3},\         \pm1\right)</math>
:<math>\left( \frac{3}{\sqrt{10}},\  \sqrt{3 \over 2},\    0,\                   \pm2\right)</math>
:<math>\left( \frac{3}{\sqrt{10}},\  \frac{-1}{\sqrt{6}},\ \frac{2}{\sqrt{3}},\  \pm2\right)</math>
:<math>\left( \frac{3}{\sqrt{10}},\  \frac{-1}{\sqrt{6}},\ \frac{4}{\sqrt{3}},\  0   \right)</math>
:<math>\left( \frac{3}{\sqrt{10}},\  \frac{-5}{\sqrt{6}},\ \frac{1}{\sqrt{3}},\  \pm1\right)</math>
:<math>\left( \frac{3}{\sqrt{10}},\  \frac{-5}{\sqrt{6}},\ \frac{-2}{\sqrt{3}},\ 0   \right)</math>
|
:<math>\left( -\sqrt{2 \over 5},\    \sqrt{2 \over 3},\    \frac{2}{\sqrt{3}},\  \pm2\right)</math>
:<math>\left( -\sqrt{2 \over 5},\    \sqrt{2 \over 3},\    \frac{-4}{\sqrt{3}},\ 0   \right)</math>
:<math>\left( -\sqrt{2 \over 5},\    -\sqrt{6},\           0,\                   0   \right)</math>
:<math>\left( \frac{-7}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\  \frac{1}{\sqrt{3}},\  \pm1\right)</math>
:<math>\left( \frac{-7}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\  \frac{-2}{\sqrt{3}},\ 0   \right)</math>
:<math>\left( \frac{-7}{\sqrt{10}},\ -\sqrt{3 \over 2},\   0,\                   0   \right)</math>
|}

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 ==
* [[Harold Scott MacDonald Coxeter|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 [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html] {{Wayback|url=http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html |date=20160711140441 }}
*** (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 (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. (1966)
*{{GlossaryForHyperspace | anchor=Pentachoron | title=Pentachoron}}
** {{PolyCell | urlname = section1.html| title = 1. Convex uniform polychora based on the pentachoron - Model 3}}
* {{KlitzingPolytopes|polychora.htm|4D|uniform polytopes (polychora)}} x3x3o3o - tip, o3x3x3o - deca

[[Category:四维几何]]
[[Category:四维多胞体]]
[[Category:多胞体]]