查看“︁超立方體堆砌”︁的源代码

{{NoteTA|G1=Math}}
{{Infobox polytope
| name = 超立方體堆砌
| imagename =  Tesseractic tetracomb.png{{!}}280px{{LinkSymbol|right}}{{NewLine}}一個3x3x3x3[[紅]][[藍]][[棋盤]]超立方體堆砌的[[透視投影]]。<span style="color:white;">
| polytope = 超立方體堆砌
| Type = [[正四維堆砌]]
| group_type = [[立方形堆砌]]
| Dimension = [[四維|4]]
| dim = [[四維]]
| count = [[超立方體|{4,3,3}]] [[File:Schlegel_wireframe_8-cell.png|40px]]
| Cell = [[正方體|{4,3}]] [[File:Hexahedron.png|20px]]
| Face = [[正方形|{4}]] [[File:Kvadrato.svg|20px]]
| Edge_type = [[Image:Cubic honeycomb verf.svg|80px]]<br/>8 [[正八面体|{4,3}]]
| Vertice_type =[[Image:4-cube t3.svg|80px]]<br/>16 [[正十六胞體|{4,3,3}]]
| Coxeter_diagram = {{CDD|node_1|4|node|3|node|3|node|4|node}}<BR>{{CDD|node_1|4|node|3|node|3|node|4|node_1}}<BR>{{CDD|nodes|split2|node|3|node|4|node_1}}<BR>{{CDD|node_1|4|node|4|node|2|node_1|4|node|4|node}}<BR>{{CDD|node_1|4|node|4|node|2|node_1|infin|node|2|node_1|infin|node}}<BR>{{CDD|node_1|infin|node|2|node_1|infin|node|2|node_1|infin|node|2|node_1|infin|node}}
| Schläfli = {4,3,3,4}<BR>t<sub>0,4</sub>{4,3,3,4}<BR>{4,3,3<sup>1,1</sup>}<BR>{4,4}<sup>2</sup><BR>{4,3,4}x{&infin;}<BR>{4,4}x{&infin;}<sup>2</sup><BR>{&infin;}<sup>4</sup>
| Euler = 0
| analogy = [[立方体堆砌]]
| convex = 
| Symmetry_group = 
| Space_group = 
| Coxeter_group = <math>{\tilde{C}}_4</math>, [4,3,3,4]<BR><math>{\tilde{B}}_4</math>, [4,3,3<sup>1,1</sup>]
| dual = [[对偶多面体|自身对偶]]
| Properties = [[點可遞]]、 [[邊可遞]]、 [[面可遞]]、 [[胞可遞]]
}}

在[[四維]][[歐幾里得]]幾何空間中，'''超立方體堆砌'''（{{lang|en|Tesseractic Honeycomb}}）<ref>[[John Horton Conway|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)</ref>是三種[[正圖形|正]][[四維空間]][[堆砌]]（亦稱為[[填充]]、[[鑲嵌]]或[[蜂巢體]]）之一，由[[超立方體]]堆砌而成。它亦可被看作是五維空間中由無窮多個[[超立方體]]胞組成的二胞角為180°的五維正無窮胞體，因此在許多情況下它被算作是五維的多胞體。

超立方體堆砌在[[施萊夫利符號]]中，以{{mset|4,3,3,4}}表示，透過超立方體胞填密4維空間構成<ref>[http://www.sciencedirect.com/science/article/pii/S002437959900107X Quaternionic modular groups] {{Wayback|url=http://www.sciencedirect.com/science/article/pii/S002437959900107X |date=20150924154641 }} Submitted by C. DavisDedicated to the memory of John B. Wilker [2014-4-27]</ref>。其頂點圖是一個[[正十六胞體]]，在每單位立方中，每相鄰的兩個超立方體胞有四個正方形相遇、八個邊相遇、頂點則有16個相遇。超立方體堆砌是平面[[正方形鑲嵌]]的類比、也是三維空間[[立方體堆砌]]在四維空間的類比<ref>Barnes, John. "The Fourth Dimension." Gems of Geometry. Springer Berlin Heidelberg, 2009. 57-81.</ref>，他們的形式皆為{4,3,...,3,4}<ref>'''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 [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html] {{Wayback|url=http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html |date=20160711140441 }}</ref>，為[[立方形堆砌]]家族的一部份，在這個家庭的鑲嵌都是[[对偶多面体|自身对偶]]。

== 坐標 ==
此蜂巢體（即該堆砌的整體）的頂點皆位於四維空間中的整數點(i,j,k,l)上，對所有的i,j,k,l皆為超立方體邊長的整數倍<ref>{{cite klitzing|test.htm|rooturl=incmats|title=test('''tes'''seractic '''t'''etracomb)|accessdate=2014-04-27}}</ref>，因此邊長為1超立方體堆砌也可以視為四維空間中的座標網格。
== 結構 ==
超立方體堆砌有許多不同的Wythoff結構。最對稱的形式是[[施萊夫利符號]]{4,3,3,4}表示[[正圖形]]，另一種形式是有兩種超立方體交替，有如棋盤一般，在[[施萊夫利符號]]中用{4,3,3<sup>1,1</sup>}表示。最低的對稱性Wythoff結構是在每個頂點附近有16個稜柱形，其[[施萊夫利符號]]表示為{&infin;}<sup>4</sup>。其可利用截胞（Sterication）來構造。
== 相關多面體和鑲嵌 ==
考克斯特群[4,3,3,4]、{{CDD|node|4|node|3|node|3|node|4|node}}產生了31個排列均勻的鑲嵌，21具有獨特的對稱性和20具有獨特的幾何形狀。擴展超立方體堆砌（也被稱為截胞超立方體堆砌）其形狀在幾何上與超立方體堆砌相同。
{| class=wikitable
![[考克斯特标记|擴展]]<BR>[[對稱群]]
!擴展<BR>[[考克斯特标记|标记]]
![[阶（群论）|阶]]
!蜂巢體<BR>（堆砌）
|- align=center
![4,3,3,4]:
!{{CDD|node_1|4|node|3|node|3|node|4|node}}
!×1
|
{{CDD|node_1|4|node|3|node|3|node|4|node}} <sub>[[Tesseractic honeycomb|1]]</sub>,
{{CDD|node|4|node_1|3|node|3|node|4|node}} <sub>[[Rectified tesseractic honeycomb|2]]</sub>,
{{CDD|node_1|4|node_1|3|node|3|node|4|node}} <sub>[[Truncated tesseractic honeycomb|3]]</sub>,
{{CDD|node_1|4|node|3|node_1|3|node|4|node}} <sub>[[Cantellated tesseractic honeycomb|4]]</sub>,<BR>
{{CDD|node_1|4|node|3|node|3|node_1|4|node}} <sub>[[Runcinated tesseractic honeycomb|5]]</sub>,
{{CDD|node|4|node_1|3|node_1|3|node|4|node}} <sub>[[Bitruncated tesseractic honeycomb|6]]</sub>, 
{{CDD|node_1|4|node_1|3|node_1|3|node|4|node}} <sub>[[Cantitruncated tesseractic honeycomb|7]]</sub>, 
{{CDD|node_1|4|node_1|3|node|3|node_1|4|node}} <sub>[[Runcitruncated tesseractic honeycomb|8]]</sub>,<BR>
{{CDD|node_1|4|node_1|3|node|3|node|4|node_1}} <sub>[[Steritruncated tesseractic honeycomb|9]]</sub>, 
{{CDD|node_1|4|node|3|node_1|3|node_1|4|node}} <sub>[[Runcicantellated tesseractic honeycomb|10]]</sub>, 
{{CDD|node_1|4|node_1|3|node_1|3|node_1|4|node}} <sub>[[Runcicantitruncated tesseractic honeycomb|11]]</sub>, 
{{CDD|node_1|4|node_1|3|node_1|3|node|4|node_1}} <sub>[[Stericantitruncated tesseractic honeycomb|12]]</sub>,<BR>
{{CDD|node_h1|4|node|3|node|3|node|4|node}} <sub>[[16-cell honeycomb|13]]</sub>

|- align=center
!<nowiki>[[</nowiki>4,3,3,4]]
!{{CDD|node_c3|split1|nodeab_c2|4a4b||nodeab_c1}}
!×2
|{{CDD|node_1|4|node|3|node|3|node|4|node_1}} <sub>[[Tesseractic honeycomb|(1)]]</sub>, {{CDD|node_h1|4|node|3|node|3|node|4|node_h1}} <sub>[[Rectified tesseractic honeycomb|(2)]]</sub>, {{CDD|node_h|4|node|3|node|3|node|4|node_h}} <sub>[[16-cell honeycomb|(13)]]</sub>, {{CDD|node_1|4|node|3|node_1|3|node|4|node_1}} <sub>[[Stericantellated tesseractic honeycomb|18]]</sub><BR>{{CDD|node_h1|4|node|3|node_1|3|node|4|node_h1}} <sub>[[Bitruncated tesseractic honeycomb|(6)]]</sub>, {{CDD|node_1|4|node_1|3|node|3|node_1|4|node_1}} <sub>[[Steriruncitruncated tesseractic honeycomb|19]]</sub>, {{CDD|node_1|4|node_1|3|node_1|3|node_1|4|node_1}} <sub>[[Omnitruncated tesseractic honeycomb|20]]</sub>

|- align=center
![(3,3)[1<sup>+</sup>,4,3,3,4,1<sup>+</sup>]]<BR>= [(3,3)[3<sup>1,1,1,1</sup>]]<BR>= [3,4,3,3]
!{{CDD|node_c2|split1|nodeab_c1|4a4b||nodes}}<BR>= {{CDD|nodeab_c1|split2|node_c2|split1|nodeab_c1}}<BR>= {{CDD|node_c2|3|node_c1|4|node|3|node|3|node}}
!×6
|
{{CDD|node|4|node|3|node_1|3|node|4|node}} <sub>[[24-cell honeycomb|14]]</sub>, 
{{CDD|node|4|node_1|3|node|3|node_1|4|node}} <sub>[[Rectified 24-cell honeycomb|15]]</sub>, 
{{CDD|node|4|node_1|3|node_1|3|node_1|4|node}} <sub>[[Truncated 24-cell honeycomb|16]]</sub>,
{{CDD|node|4|node_h|3|node_h|3|node_h|4|node}} <sub>[[Snub 24-cell honeycomb|17]]</sub>

|}

== 參考文獻 ==
{{reflist}}
* Coxeter, H.S.M. ''Regular Polytopes'', (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p.&nbsp;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 [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html] {{Wayback|url=http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html |date=20160711140441 }}
** (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
* {{KlitzingPolytopes|flat.htm|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

{{正鑲嵌}}
[[Category:多胞形]]
[[Category:多胞体]]
[[Category:堆砌 (幾何)]]