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超立方体堆砌本文重定向自 超立方體堆砌

(重定向自正八胞體堆砌)
超立方体堆砌
Tesseractic tetracomb.png 一个3x3x3x3棋盘超立方体堆砌的透视投影|220px]]
类型正四维堆砌
家族立方形堆砌
维度4
四维{4,3,3} Schlegel wireframe 8-cell.png
{4,3} Hexahedron.png
{4} Kvadrato.svg
棱图Cubic honeycomb verf.png
8 {4,3}
顶点图4-cube t3.svg
16 {4,3,3}
施莱夫利符号{4,3,3,4}
t0,4{4,3,3,4}
{4,3,31,1}
{4,4}2
{4,3,4}x{∞}
{4,4}x{∞}2
{∞}4
欧拉示性数0
考克斯特记号英语Coxeter–Dynkin_diagramCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
类比立方体堆砌
考克斯特群, [4,3,3,4]
, [4,3,31,1]
对偶多胞体自身对偶
特性vertex-transitive, edge-transitive, face-transitive, cell-transitive

四维欧几里得几何空间中,超立方体堆砌Tesseractic Honeycomb[1]是三种四维空间堆砌(亦称为填充、镶嵌或蜂巢体)之一,由超立方体堆砌而成。它亦可被看作是五维空间中由无穷多个超立方体胞组成的二胞角为180°的五维正无穷胞体,因此在许多情况下它被算作是五维的多胞体。

超立方体堆砌在施莱夫利符号中,以{4,3,3,4}表示,透过超立方体胞填密4维空间构成[2]。其顶点图是一个正十六胞体,在每单位立方中,每相邻的两个超立方体胞有四个正方形相遇、八个边相遇、顶点则有16个相遇。超立方体堆砌是平面正方形镶嵌的类比、也是三维空间立方体堆砌在四维空间的类比[3],他们的形式皆为{4,3,...,3,4}[4],为立方形堆砌家族的一部分,在这个家庭的镶嵌都是自身对偶

坐标

此蜂巢体(即该堆砌的整体)的顶点皆位于四维空间中的整数点(i,j,k,l)上,对所有的i,j,k,l皆为超立方体边长的整数倍[5],因此边长为1超立方体堆砌也可以视为四维空间中的座标网格。

结构

超立方体堆砌有许多不同的Wythoff结构。最对称的形式是施莱夫利符号{4,3,3,4}表示正图形,另一种形式是有两种超立方体交替,有如棋盘一般,在施莱夫利符号中用{4,3,31,1}表示。最低的对称性Wythoff结构是在每个顶点附近有16个棱柱形,其施莱夫利符号表示为{∞}4。其可利用截胞(Sterication)来构造。

相关多面体和镶嵌

考克斯特群[4,3,3,4]、CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png产生了31个排列均匀的镶嵌,21具有独特的对称性和20具有独特的几何形状。扩展超立方体堆砌(也被称为截胞超立方体堆砌)其形状在几何上与超立方体堆砌相同。

扩展
对称群
扩展
标记
蜂巢体
(堆砌)
[4,3,3,4]: CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png ×1

CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 1, CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 2, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 3, CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 4,
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png 5, CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 6, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 7, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png 8,
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png 9, CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png 10, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png 11, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png 12,
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 13

[[4,3,3,4]] CDel node c3.pngCDel split1.pngCDel nodeab c2.pngCDel 4a4b.pngCDel nodeab c1.png ×2 CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png (1), CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h1.png (2), CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h.png (13), CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png 18
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h1.png (6), CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png 19, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png 20
[(3,3)[1+,4,3,3,4,1+]]
= [(3,3)[31,1,1,1]]
= [3,4,3,3]
CDel node c2.pngCDel split1.pngCDel nodeab c1.pngCDel 4a4b.pngCDel nodes.png
= CDel nodeab c1.pngCDel split2.pngCDel node c2.pngCDel split1.pngCDel nodeab c1.png
= CDel node c2.pngCDel 3.pngCDel node c1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
×6

CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 14, CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png 15, CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png 16, CDel node.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png 17

参考文献

  1. ^ John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, Architectonic and Catoptric tessellations, p 292-298, includes all the nonprismatic forms)
  2. ^ Quaternionic modular groups Submitted by C. DavisDedicated to the memory of John B. Wilker [2014-4-27]
  3. ^ Barnes, John. "The Fourth Dimension." Gems of Geometry. Springer Berlin Heidelberg, 2009. 57-81.
  4. ^ Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
  5. ^ test(tesseractic tetracomb) bendwavy.org [2014-4-27]
  • Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [2]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
  • George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs) - Model 1
  • Richard Klitzing, 4D, Euclidean tesselations x∞o x∞o x∞o x∞o, x∞x x∞o x∞o x∞o, x∞x x∞x x∞o x∞o, x∞x x∞x x∞x x∞o,x∞x x∞x x∞x x∞x, x∞o x∞o x4o4o, x∞o x∞o o4x4o, x∞x x∞o x4o4o, x∞x x∞o o4x4o, x∞o x∞o x4o4x, x∞x x∞x x4o4o, x∞x x∞x o4x4o, x∞x x∞o x4o4x, x∞x x∞x x4o4x, x4o4x x4o4x, x4o4x o4x4o, x4o4x x4o4o, o4x4o o4x4o, x4o4o o4x4o, x4o4o x4o4o, x∞x o3o3o *d4x, x∞o o3o3o *d4x, x∞x x4o3o4x, x∞o x4o3o4x, x∞x x4o3o4o, x∞o x4o3o4o, o3o3o *b3o4x, x4o3o3o4x, x4o3o3o4o - test - O1

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