超立方體堆砌
Template:NoteTA Template:Infobox polytope
在四維歐幾里得幾何空間中，超立方體堆砌（Template:Lang）[1]是三種正四維空間堆砌（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Template:Mset表示，透過超立方體胞填密4維空間構成[2]。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[3]，他們的形式皆為{4,3,...,3,4}[4]，為立方形堆砌家族的一部份，在這個家庭的鑲嵌都是自身对偶。
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[5]，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{4,3,31,1}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{∞}4。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]、Template:CDD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。
| 擴展 對稱群 |
擴展 标记 |
阶 | 蜂巢體 （堆砌） |
|---|---|---|---|
| [4,3,3,4]: | Template:CDD | ×1 |
Template:CDD 1,
Template:CDD 2,
Template:CDD 3,
Template:CDD 4, |
| [[4,3,3,4]] | Template:CDD | ×2 | Template:CDD (1), Template:CDD (2), Template:CDD (13), Template:CDD 18 Template:CDD (6), Template:CDD 19, Template:CDD 20 |
| [(3,3)[1+,4,3,3,4,1+]] = [(3,3)[31,1,1,1]] = [3,4,3,3] |
Template:CDD = Template:CDD = Template:CDD |
×6 |
Template:CDD 14, Template:CDD 15, Template:CDD 16, Template:CDD 17 |
參考文獻
- 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 [1] Template:Wayback
- (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
- Template:KlitzingPolytopes 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
- ↑ 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)
- ↑ Quaternionic modular groups Template:Wayback Submitted by C. DavisDedicated to the memory of John B. Wilker [2014-4-27]
- ↑ Barnes, John. "The Fourth Dimension." Gems of Geometry. Springer Berlin Heidelberg, 2009. 57-81.
- ↑ 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] Template:Wayback
- ↑ Template:Cite klitzing