查看“︁截半正五胞体”︁的源代码

{{NoteTA|G1=Math}}
{{Infobox polychoron
| name = 截半正五胞体
| imagename = Schlegel half-solid rectified 5-cell.png
| caption = [[施莱格尔投影]]<BR>（显示5个[[正四面体]]胞）
| polytope = 截半正五胞体
| 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 [[正八面体|(''3.3.3.3'')]] <!-- 檔案不存在 [[Image:File:Octahedron.png|20px]] -->
| Face =30 {3}
| Edge =30
| Vertice =10
| Vertice_type =[[Image:Rectified 5-cell verf.png|80px]]<BR>[[三角柱]]
| Schläfli =t<sub>1</sub>{3,3,3}
| Symmetry_group = 
| dual = 
| Properties =[[Convex polytope|convex]], [[isogonal figure|isogonal]], [[isotoxal figure|isotoxal]]
| Index_references = ''[[正五胞体|1]]'' 2 ''[[截角正五胞体|3]]''
| Coxeter_group =A<sub>4</sub>, [3,3,3], order 120
| Coxeter_diagram = {{CDD|node|3|node_1|3|node|3|node}}
}}
[[Image:triangular prism.png|220px|thumb|right|[[顶点图]]: [[三角柱]]<BR>5个面:<BR>
[[Image:tetrahedron vertfig.svg|100px]][[Image:Octahedron vertfig.svg|100px]]<BR>
2([[正四面体|''3.3.3'']])和3([[正八面体|''3.3.3.3'']])
]]
在[[四维空间|四维]][[几何学]]中,'''截半正五胞体'''是一个由5个正四面体和5个正八面体[[胞]]组成的[[均匀多胞体]]。每条棱都连接到一个正四面体和两个正八面体。每个顶点周围环绕着两个[[正四面体]]和三个[[正八面体]]。它总共有30个三角形面，30条棱和10个顶点。它的[[顶点图]]是正[[三角柱]]。截半正五胞体是三个由两种或更多的[[正多面体]]胞组成的四维半正多胞体之一。

==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_t1.svg|150px]]
|[[File:4-simplex_t1_A3.svg|150px]]
|[[File:4-simplex_t1_A2.svg|150px]]
|- align=center
![[二面体群]]
|[5]
|[4]
|[3]
|}

{| class="wikitable" width=640
|align=center|[[Image:Rectified simplex stereographic.png|220px]]<BR>[[施莱格尔投影]]<BR>(对着一个[[正八面体]]胞)
|align=center|[[Image:Rectified 5-cell net.png|220px]]<BR>[[展开图]]
|-
|align=center|[[Image:Rectified 5cell-perspective-tetrahedron-first-01.gif]]
|[[正四面体]]为中心的3维透视投影，最接近的正四面体呈红色，周围的4个正八面体呈绿色。远端的胞清晰度降低(虽然可以从棱看出它们)。投影只是在三维空间中旋转，而不是在四维空间中旋转。
|}

==坐标==

一个棱长为2的截半正五胞体的顶点的[[笛卡儿坐标系]]坐标

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

更简单的，截半正五胞体的顶点是五维空间[[笛卡儿坐标系]]的(0,0,0,1,1)或(0,0,1,1,1)的全排列。

== 参考文献 ==
* [[Thorold Gosset|T. Gosset]]: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', [[Messenger of Mathematics]], Macmillan, 1900
* [[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]
* [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. (1966)

== 外部链接 ==
* [http://www.polytope.de/nr03.html Rectified 5-cell] {{Wayback|url=http://www.polytope.de/nr03.html |date=20201024171452 }} - data and images
** {{PolyCell | urlname = section1.html| title = 1. Convex uniform polychora based on the pentachoron - Model 2}}
* {{KlitzingPolytopes|polychora.htm|4D uniform polytopes (polychora)|x3o3o3o - rap}}

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