開羅五邊形鑲嵌

開羅五邊形鑲嵌
	歐幾里得平面
類別	半正鑲嵌對偶; 平面鑲嵌
對偶多面體	扭稜正方形鑲嵌
數學表示法
考克斯特符號; （英語：Coxeter-Dynkin diagram）	;
施萊夫利符號	dsr{6,3}
康威表示法	dsrS
性質
二面角	平角
組成與佈局
面的種類	五邊形
面的佈局; （英語：Face configuration）	V3.3.4.3.4
對稱性
對稱群	p4g, [4+,4], (4*2); p4, [4,4]+, (442)
旋轉對稱群; （英語：Rotation_groups）	p4, [4,4]+, (442)
特性
	面可遞
圖像
	; 扭稜正方形鑲嵌; （對偶多面體）
	; 扭稜正方形鑲嵌; （對偶多面體）
	閱; 論; 編;

在幾何學中，開羅五邊形鑲嵌是一種平面鑲嵌，其為半正鑲嵌扭稜正方形鑲嵌的對偶鑲嵌^[1]，密鋪於歐氏平面，其名為「開羅」是因為這種幾何圖形經常在埃及開羅的街道上出現^[2]^[3]，是15種已知的等面五邊形鑲嵌之一。

Quick Facts 類別, 對偶多面體 ...

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康威將之稱為4-fold pentille^[4]。

它也被稱為麥克馬洪網格（MacMahon's net）^[5]，出於珀西亞歷山大麥克馬洪1921年出版的《New Mathematical Pastimes》^[6]。

五邊石墨烯的化學結構與開羅五邊形鑲嵌接近^[7]這種形態建基於分析和模擬，在2014年被提出。^[7]進一步的計算顯示純粹以此形態存在的碳是不穩定的，^[8]但將其氫化後可變得穩定。^[9]由於其結構，它罕見地具有負值蒲松比，強度相信比石墨烯高，且據預測它能在高達1000K時仍為化學穩定。^[7]

[1]
Weisstein, Eric W. (編). Dual tessellation. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. （英語）.
[2]
Alsina, Claudi; Nelsen, Roger B., Charming proofs: a journey into elegant mathematics, Dolciani mathematical expositions 42, Mathematical Association of America: 164, 2010 [2014-06-11], ISBN 978-0-88385-348-1, （原始內容存檔於2014-07-05）.
[3]
Martin, George Edward, Transformation Geometry: An Introduction to Symmetry, Undergraduate Texts in Mathematics, Springer: 119, 1982 [2014-06-11], ISBN 978-0-387-90636-2, （原始內容存檔於2014-07-05）.
[4]
Conway, John H.; Burgiel, Heidi; Goodman-Strass, Chaim, The Symmetries of Things, AK Peters: 288, 2008, ISBN 978-1-56881-220-5
[5]
Plane nets in crystal chemistry. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences. 1980-02-29, 295 (1417): 553–618 [2020-05-04]. ISSN 0080-4614. doi:10.1098/rsta.1980.0150. （原始內容存檔於2019-10-24）（英語）.
[6]
Macmahon, Major P. A., New Mathematical Pastimes, University Press, 1921.
[7]
Shunhong Zhang, Jian Zhou, Qian Wang, Xiaoshuang Chen, Yoshiyuki Kawazoe, Puru Jena. Penta-graphene: A new carbon allotrope. Proceedings of the National Academy of Sciences. 2015-02-24, 112 (8): 2372–2377 [2020-05-04]. ISSN 0027-8424. PMC 4345574 . PMID 25646451. doi:10.1073/pnas.1416591112 （英語）.
[8]
Christopher P. Ewels, Xavier Rocquefelte, Harold W. Kroto, Mark J. Rayson, Patrick R. Briddon, Malcolm I. Heggie. Predicting experimentally stable allotropes: Instability of penta-graphene. Proceedings of the National Academy of Sciences. 2015-12-22, 112 (51): 15609–15612 [2020-05-04]. ISSN 0027-8424. PMC 4697406 . PMID 26644554. doi:10.1073/pnas.1520402112 （英語）.
[9]
Hamideh Einollahzadeh, Seyed Mahdi Fazeli, Reza Sabet Dariani. Studying the electronic and phononic structure of penta-graphane. Science and Technology of Advanced Materials. 2016-12, 17 (1): 610–617 [2020-05-04]. ISSN 1468-6996. PMC 5102001 . PMID 27877907. doi:10.1080/14686996.2016.1219970. （原始內容存檔於2020-08-15）（英語）.

Grünbaum, Branko ; and Shephard, G. C. Tilings and Patterns. New York: W. H. Freeman. 1987. ISBN 0-7167-1193-1. (Chapter 2.1: Regular and uniform tilings, p.58-65) (Page 480, Tilings by polygons, #24 of 24 polygonal isohedral types by pentagons)
Williams, Robert. The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. 1979: 38. ISBN 0-486-23729-X.
Wells, David, The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 23, 1991.

[3] [1]
Weisstein, Eric W. (編). Dual tessellation. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. （英語）.

[4] [2]
Alsina, Claudi; Nelsen, Roger B., Charming proofs: a journey into elegant mathematics, Dolciani mathematical expositions 42, Mathematical Association of America: 164, 2010 [2014-06-11], ISBN 978-0-88385-348-1, （原始內容存檔於2014-07-05）.

[5] [3]
Martin, George Edward, Transformation Geometry: An Introduction to Symmetry, Undergraduate Texts in Mathematics, Springer: 119, 1982 [2014-06-11], ISBN 978-0-387-90636-2, （原始內容存檔於2014-07-05）.

[conway-6] [4]
Conway, John H.; Burgiel, Heidi; Goodman-Strass, Chaim, The Symmetries of Things, AK Peters: 288, 2008, ISBN 978-1-56881-220-5

[7] [5]
Plane nets in crystal chemistry. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences. 1980-02-29, 295 (1417): 553–618 [2020-05-04]. ISSN 0080-4614. doi:10.1098/rsta.1980.0150. （原始內容存檔於2019-10-24）（英語）.

[8] [6]
Macmahon, Major P. A., New Mathematical Pastimes, University Press, 1921.

[pnas-9] [7]
Shunhong Zhang, Jian Zhou, Qian Wang, Xiaoshuang Chen, Yoshiyuki Kawazoe, Puru Jena. Penta-graphene: A new carbon allotrope. Proceedings of the National Academy of Sciences. 2015-02-24, 112 (8): 2372–2377 [2020-05-04]. ISSN 0027-8424. PMC 4345574 . PMID 25646451. doi:10.1073/pnas.1416591112 （英語）.

[pnas2-10] [8]
Christopher P. Ewels, Xavier Rocquefelte, Harold W. Kroto, Mark J. Rayson, Patrick R. Briddon, Malcolm I. Heggie. Predicting experimentally stable allotropes: Instability of penta-graphene. Proceedings of the National Academy of Sciences. 2015-12-22, 112 (51): 15609–15612 [2020-05-04]. ISSN 0027-8424. PMC 4697406 . PMID 26644554. doi:10.1073/pnas.1520402112 （英語）.

[stam-11] [9]
Hamideh Einollahzadeh, Seyed Mahdi Fazeli, Reza Sabet Dariani. Studying the electronic and phononic structure of penta-graphane. Science and Technology of Advanced Materials. 2016-12, 17 (1): 610–617 [2020-05-04]. ISSN 1468-6996. PMC 5102001 . PMID 27877907. doi:10.1080/14686996.2016.1219970. （原始內容存檔於2020-08-15）（英語）.

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開羅五邊形鑲嵌

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