在几何学中,截角十二面体是一种由正十边形和正三角形组成的三十二面体[1],是一种阿基米德立体[2]。其每个顶点都是1个三角形和2个十边形的公共顶点,具有每个顶角相等的性质,因此截角十二面体是一种半正多面体[3]。
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截角十二面体共有32个面、90条边和60个顶点[4],每个顶点都是1个三角形和2个十边形的公共顶点,其顶点图可以用3.10.10来表示,也可以简写为3.102[5]。
截角十二面体可以经由正十二面体透过截角变换构造而成。截角变换使得正十二面体原本的正五边形面变成正十边形面,并在原本的顶点处形成正三角形。
边长为a的截角十二面体体积V和表面积A分别为:
![{\displaystyle A=5\left({\sqrt {3}}+6{\sqrt {5+2{\sqrt {5}}}}\right)a^{2}\approx 100.990\,76a^{2}}](//wikimedia.org/api/rest_v1/media/math/render/svg/5dd9cc515f2a675924c04f92427e5ba80d18c440)
![{\displaystyle V={\frac {5}{12}}\left(99+47{\sqrt {5}}\right)a^{3}\approx 85.039\,6646a^{3}}](//wikimedia.org/api/rest_v1/media/math/render/svg/6abed1446f6c6e4894333c182326a43aa1ef40df)
边长为2φ − 2且几何中心位于原点的截角十二面体[6]其顶点坐标为[7]:
、
、
- (±φ, ±2, ±(φ + 1))。
其中φ =
,为黄金比例.
截角十二面体对应的结构也可以构建成球面镶嵌,并以球极平面投影的方式呈现。
有一些多面体与截角十二面体具有相同的顶点布局,换句话说,及他们与截角十二面体共用顶点、或者可以具有相同的顶点坐标。这些多面体有[8][9][10]:
截角二十面体是正二十面体经过截半变换后的结果,其他也是由正二十面体透过康威变换得到的多面体有:
More information [5,3]+, (532), 半正多面体对偶 ...
正二十面体家族半正多面体
对称群: [5,3], (*532)
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[5,3]+, (532)
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{5,3}
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t0,1{5,3}
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t1{5,3}
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t0,1{3,5}
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{3,5}
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t0,2{5,3}
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t0,1,2{5,3}
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s{5,3}
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半正多面体对偶
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V5.5.5
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V3.10.10
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V3.5.3.5
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V5.6.6
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V3.3.3.3.3
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V3.4.5.4
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V4.6.10
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V3.3.3.3.5
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截角二十面体可以独立填满双曲仿紧三维空间,这种由几何结构称为截角十二面体堆砌[11]。
![Thumb](//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/H3_353-0110_center_ultrawide.png/640px-H3_353-0110_center_ultrawide.png)
- Williams, Robert. The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. 1979. ISBN 0-486-23729-X. (Section 3-9)
- Cromwell, P. Polyhedra. United Kingdom: Cambridge. 1997: 79-86 Archimedean solids. ISBN 0-521-55432-2.
Cromwell, P. Polyhedra, CUP hbk (1997), pbk. (1999). Ch.2 p.79-86 Archimedean solids
Cundy, H. and Rollett, A. "Truncated Dodecahedron. 3.102." §3.7.9 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 109, 1989. ISBN 978-0906212202
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966