胞 (結構)

立方体堆砌:每一邊有四個立方胞。
超立方體:每一邊有三個立方胞。

幾何學以及相關的晶體學材料學中,是指一個重複結構中的一個基本單位[1][2][3],如晶體結構中的晶胞[4]多胞形中的多維胞等。

幾何學

幾何學裡,是指高維物件中的三維或更高維度的元素[5]。一般稱胞為三維元素[6],更高維度的胞通常會以其維度稱呼,例如四維胞、五維胞等。[7][8]

多胞形的胞

一般而言,可以視為四維多胞形的邊界之一部份或更高維度幾何結構中三維或三維以上的元素[6],如多胞形[9]五維多胞體[10]四維凸正多胞體[11]堆砌體(三維空間填充結構)[12][13]

例如,立方體堆砌是由立方體形狀的三維胞所組成的,有時稱為立方胞。在這個胞上在每個邊上都有四個立方體。超立方體亦是由立方胞所組成的,但一邊只有三個立方體。[14]

是類比於胞之多面體密鋪[15]內的二維元素。[16][17]

三維胞的例子
四維多胞體 三維堆砌體
{4,3,3} {5,3,3} {4,3,4} {5,3,4}英语Order-4 dodecahedral honeycomb

超立方體的每條邊周圍都有3個立方體形狀的三維胞[14]

正一百二十胞体的每條邊周圍都有3個正十二面體形狀的三維胞[18][19]

立方體堆砌的每條邊周圍都有4個立方體形狀的三維胞[20]

{5,3,4}英语Order-4 dodecahedral honeycomb的每條邊周圍都有4個正十二面體形狀的三維胞[21]

四維元素(在五維多胞體及更高維度裡)會被稱為四維胞超胞4維面4-面。系統化地,n維面n-面為在(n+1)維多胞形或更高維多胞形內的元素[22][23][24]。例如在五維多胞體中存在有三維胞四維胞[25]

在英文中,胞稱為Cell,若在Cell詞彙前面加入一個數字則可以代表由該數量個胞組成的多胞形,例如24-Cell代表二十四胞體[6]。此外,在多胞形複形中,單一一個多胞形也稱為胞[26]

晶體學

氯化鈉的一個晶體,其中框出來的部分維一個晶胞。

晶體學中,為了探討原子於晶體中結構會將重複的單元拿出來討論,而一個重複的單元稱為一個,而組成晶體構造的基本胞稱為晶胞、若其同時能確保晶體結構的對稱性且體積又是最小的胞則稱為單位晶胞[27][28],且通常會將晶胞與幾何學一起討論[29]

此概念在幾何中也可以用於描述最密堆积的結構。[30]

單位晶胞

單位晶胞是晶體結構的基本結構單元,並且可以透過其幾何形狀以及其內部原子的排列結構來還原整個晶體結構,因此也可以視為定義晶體的方式。 [27][31]

參見

參考文獻

  1. ^ 龙四营, 冯毅雄, 高一聪, 谭建荣. 多面体体胞结构演变机理与抗撞性优化设计研究. 机械工程学报. 2014, 50 (11): 135––143. 
  2. ^ 顾璐英, 蒋高明, 缪旭红, 张爱军. 多轴向经编复合材料预制件的几何模型. 纺织学报. 2011, 32 (11): 42––48. 
  3. ^ 姜振益, 许小红, 武海顺, 张富强, 金志浩. SiC 多型体几何结构与电子结构研究. 物理学报. 2002, 51 (7). 
  4. ^ Williams, R. "The Unit Cell Concept." §2-4 in The Geometrical Foundation of Natural Structure: A Source Book of Design. New York: Dover, pp. 48-51, 1979.
  5. ^ Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n>=5)
  6. ^ 6.0 6.1 6.2 Weisstein, Eric W. (编). Cell. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. (英语). 
  7. ^ Guy Inchbald. Ditela, polytopes and dyads. Steelpillow.com. 2019-02-10 [2019-09-27]. (原始内容存档于2018-10-18). 
  8. ^ 施开达 and 马利庄. 正则多胞形和 N 维空间有限旋转群理论的一些新结果. 自然科学进展: 国家重点实验室通讯. 1999, 9 (A12): 1336––1341. 
  9. ^ N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
  10. ^ T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
  11. ^ 梭茨, 莎可娃, 黃俊瑋; et al, 有五階對稱的晶格嗎?, 國立交通大學, 2013 
  12. ^ Weisstein, Eric W. (编). Space-filling polyhedron. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. (英语). 
  13. ^ A. Boole Stott: Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
  14. ^ 14.0 14.1 Weisstein, Eric W. (编). Hypercube. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. (英语). 
  15. ^ 奧斯朋出版編輯群, 陳昭蓉譯. 《圖解數學辭典》. 台北市: 天下遠見出版社. 2006: P.36. ISBN 9864176145. 
  16. ^ Cromwell, Peter R., Polyhedra, Cambridge University Press: 13, 1999 [2019-09-16], (原始内容存档于2019-06-13) 
  17. ^ H.S.M. Coxeter, M.S. Longuet-Higgins und J.C.P. Miller: Uniform Polyhedra, Philosophical Transactions of the Royal Society of London, Londne, 1954
  18. ^ N.W. Johnson: Geometries and Transformations, (2018) ISBN 978-1-107-10340-5 Chapter 11: Finite Symmetry Groups, 11.5 Spherical Coxeter groups, p.249
  19. ^ Matila Ghyka, The Geometry of Art and Life (1977), p.68
  20. ^ 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)
  21. ^ Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
  22. ^ Matoušek, Jiří, Lectures in Discrete Geometry, Graduate Texts in Mathematics 212, Springer, 5.3 Faces of a Convex Polytope, p. 86, 2002 [2019-09-16], (原始内容存档于2019-06-10) 
  23. ^ Grünbaum, Branko, Convex Polytopes, Graduate Texts in Mathematics 221 2nd, Springer: 17, 2003 [2019-09-16], (原始内容存档于2013-10-31) 
  24. ^ Ziegler, Günter M., Lectures on Polytopes, Graduate Texts in Mathematics 152, Springer, Definition 2.1, p. 51, 1995 [2019-09-16], (原始内容存档于2019-06-12) 
  25. ^ H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
  26. ^ Polytopal Complexes. eg-models. [2019-09-23]. (原始内容存档于2020-01-30). 
  27. ^ 27.0 27.1 結晶固體之結構 (PDF). [2019-09-16]. (原始内容 (PDF)存档于2018-11-23). 
  28. ^ 徐恒均. 材料科学基础. 北京: 北京工业大学出版社. 2001: 24. ISBN 9787563909346. 
  29. ^ 礦物的結晶構造. [失效連結]
  30. ^ Weisstein, Eric W. (编). Unit Cell. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. (英语). 
  31. ^ 吴伟. 材料科学基础. 中国铁道出版社. ISBN 9787113197438. 

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