五邊形鑲嵌

在幾何學中，五邊形鑲嵌是指用五邊形鑲嵌平面。

正五邊形不能鑲嵌平面，因為其內角是108°，不能整除360°。截至2015年^[update]，已知有15种凸五边形鑲嵌平面。2017年5月，里昂高等师范学校Michaël Rao宣称已证明只存在上述的15种凸五边形鑲嵌平面情况。^[1]

Reinhardt (1918)發現了「鑲嵌塊遞移」（tile transitive）的5種五邊形鑲嵌，即是說鑲嵌的對稱性可以將任何一塊帶到任何另一塊（用數學語言描述，鑲嵌的自同構群作用在鑲嵌塊上是可遞的。）Kershner (1968)發現了3種新的五邊形鑲嵌，都不是鑲嵌遞移的；他錯誤聲稱已經找出所有的五邊形鑲嵌。1975年Richard E. James III找到第9種。Schattschneider (1978)描述業餘數學家瑪喬里·賴斯在1976至1977年間找到新的4種五邊形鑲嵌。Schattschneider (1985)描述Rolf Stein在1985年找到的第14種五邊形鑲嵌。Bagina (2011)證明邊對邊（edge-to-edge）的凸五邊形鑲嵌只有8種，Sugimoto (2012)獨立證出同一結果。2015年，华盛顿大学數學家Casey Mann、Jennifer McLoud和David Von Derau發現了第15種五邊形鑲嵌，使用了電腦算法搜尋。^[2]

15種凸五邊形鑲嵌平面
1	2	3	4	5
B+C=180° A+D+E=360°	c=e B+D=180°	a = b, d = c + e A = C = D = 120°	b = c, d = e B = D = 90°	a = b, d = e A = 60°, D = 120°
6	7	8	9	10
a = d = e, b = c B + D = 180°, 2B = E	b = c = d = e B + 2E = 2C + D = 360°	b = c = d = e 2B + C = D + 2E = 360°	b = c = d = e 2A + C = D + 2E = 360°	a = b = c + e A = 90°, B + E = 180°, B + 2C = 360°
11	12	13	14	15
2a + c = d = e A = 90°, 2B + C = 360° C + E = 180°	2a = d = c + e A = 90°, 2B + C = 360° C + E = 180°	d = 2a = 2e B = E = 90°, 2A + D = 360°	2a = 2c = d = e A = 90°, B ≈ 145.34°, C ≈ 69.32°, D ≈ 124.66°, E ≈ 110.68° (2B + C = 360°, C + E = 180°).	a = c = e, b = 2a A = 150°, B = 60°, C = 135°, D = 105°, E = 90°

引用

[1]
Exhaustive search of convex pentagons which tile the plane (PDF). [2017-07-19]. （原始内容存档 (PDF)于2020-11-12）.
[2]
Bellos, Alex. Attack on the pentagon results in discovery of new mathematical tile. The Guardian. 11 August 2015 [2015-08-18]. （原始内容存档于2015-08-18）.

引用错误：在<references>标签中name属性为“NPR”的参考文献没有在文中使用

来源

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