超立方體堆砌
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超立方體堆砌 | |
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類型 | 正四維堆砌 |
家族 | 立方形堆砌 |
維度 | 4 |
對偶多胞形 | 自身对偶 |
類比 | 立方体堆砌 |
數學表示法 | |
考克斯特符號 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
施萊夫利符號 | {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 |
性質 | |
四維胞 | {4,3,3} ![]() |
胞 | {4,3} ![]() |
面 | {4} ![]() |
歐拉示性數 | 0 |
組成與佈局 | |
棱圖 | ![]() 8 {4,3} |
顶点图 | ![]() 16 {4,3,3} |
對稱性 | |
考克斯特群 | , [4,3,3,4] , [4,3,31,1] |
特性 | |
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(Tesseractic Honeycomb)[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{4,3,3,4}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]、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) |
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[4,3,3,4]: | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
×1 | |
[[4,3,3,4]] | ![]() ![]() ![]() ![]() ![]() |
×2 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[(3,3)[1+,4,3,3,4,1+]] = [(3,3)[31,1,1,1]] = [3,4,3,3] |
![]() ![]() ![]() ![]() ![]() = ![]() ![]() ![]() ![]() ![]() = ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
×6 |
參考文獻
- ^ 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 (页面存档备份,存于互联网档案馆) 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 [1] (页面存档备份,存于互联网档案馆)
- ^ Klitzing, Richard. test(tesseractic tetracomb). bendwavy.org. [2014-04-27].
- 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] (页面存档备份,存于互联网档案馆)
- (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
- Klitzing, Richard. 4D Euclidean tesselations. bendwavy.org. 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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