正六邊形鑲嵌
在幾何學中,正六邊形鑲嵌是一種平面鑲嵌,由正六邊形重覆組合排列而成,且填滿整個平面,而且沒有任何空隙或重疊,由於皆由正多邊形組成,因此稱為正鑲嵌圖。正六邊形鑲嵌是三维欧几里得空间中三个正密铺之一。另外两个分别是正三角形镶嵌和正方形镶嵌。
康威將之稱為hextille。
由於正六邊形鑲嵌是由正六邊形組成,又因正六邊形內角為120°,因此每個頂點周圍都有3個正六邊形,且剛好占滿360°,才能填滿平面。
在施萊夫利符號中,正六邊形鑲嵌可用{6,3}或t{3,6}表示。
圆堆砌
正六边形镶嵌可以被用来进行圆堆砌,以其每个顶点为圆心放置等直径的圆。在这个堆砌里,每个圆都与3个相邻圆接触(接触数)。每个正六边形中间的部分实际上还可以再放入一个圆,这样我们就会得到二维最密圆堆砌——正三角形镶嵌式圆堆砌,这时接触数达到最大值6。
半正涂色
正六边形镶嵌共有3种不同的半正涂色,都可以由Wythoff镜面对称构造出来。(h,k)表示一种涂色的面周期性重复,以正六边形距离h、k计数,h在先,k在后。
k阶半正
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一阶半正
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二阶半正
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三阶半正
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图像
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颜色数
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1
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2
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3
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2
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4
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2
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7
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(h,k)
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(1,0)
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(1,1)
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(2,0)
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(2,1)
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施莱夫利符号
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{6,3}
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t{3,6}
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t0,1,2{3[3]}
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Wythoff符号
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3 | 6 2
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2 6 | 3
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3 3 3 |
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对称性
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*632 (p6m) [6,3]
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*333 (p3) [3[3]]
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*632 (p6m) [6,3]
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632 (p6) [6,3]+
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考克斯特符号
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![node_1](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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![node_1](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![node_1](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png)
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![node_1](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![split1](//upload.wikimedia.org/wikipedia/commons/a/a1/CDel_split1.png)
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康威多面体符号
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H
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tH
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teH
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t6daH
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t6dateH
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t6dsH
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其中三色正六边形镶嵌是一个由三阶全序多胞形产生的镶嵌。
相关半正镶嵌
正六边形镶嵌可以通过截角操作得到一系列与之相关的半正镶嵌,其与正六边形镶嵌拥有相似的对称性:
正三角形镶嵌家族的半正镶嵌
对称性: [6,3], (*632)
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[6,3]+, (632)
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[1+,6,3], (*333)
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[6,3+], (3*3)
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![node_1](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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![node_1](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node_1](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node_1](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node_1](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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![node_1](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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![node_1](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node_1](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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![node_h](//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node_h](//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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![node_h](//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node_h](//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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{6,3}
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t0,1{6,3}
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t1{6,3}
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t1,2{6,3}
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t2{6,3}
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t0,2{6,3}
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t0,1,2{6,3}
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s{6,3}
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h{6,3}
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h1,2{6,3}
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半正对偶
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![node_f1](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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![node_f1](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node_f1](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node_f1](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node_f1](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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![node_f1](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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![node_f1](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node_f1](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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![node_fh](//upload.wikimedia.org/wikipedia/commons/6/6f/CDel_node_fh.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node_fh](//upload.wikimedia.org/wikipedia/commons/6/6f/CDel_node_fh.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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![node_fh](//upload.wikimedia.org/wikipedia/commons/6/6f/CDel_node_fh.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node_fh](//upload.wikimedia.org/wikipedia/commons/6/6f/CDel_node_fh.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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V6.6.6
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V3.12.12
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V3.6.3.6
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V6.6.6
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V3.3.3.3.3.3
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V3.4.12.4
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V.4.6.12
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V3.3.3.3.6
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V3.3.3.3.3.3
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正六边形镶嵌在拓扑上与一系列一直延伸到双曲镶嵌的顶点图为n3的(广义)多面体相关:
(三阶)正六边形镶嵌在拓扑上与一系列面为正六边形的密铺相关联,这些镶嵌都可称之为“正六边形镶嵌”,所以我们以“n 阶”来区分,其施莱夫利符号为{6,n},考克斯特符号![node_1](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![n](//upload.wikimedia.org/wikipedia/commons/4/46/CDel_n.png) ,一直到n = ∞:
这个镶嵌还是一系列有考克斯特对称群[n,3]对称性的(半)截角菱形多面体或镶嵌的一员。立方体可以被看作是“菱形六面体”,这里菱形就是正方形。它们的截角形在原顶点处有正的多边形,而原来的菱形面则被截成了非正六边形。这一系列多面体或镶嵌有两种顶点图:(n.6.6)和(6,6,6)。
正六边形镶嵌亦可被看作延长菱形镶嵌,菱形镶嵌的每一个顶点都被延长成了新的棱。这类似于三维空间中的菱形十二面体堆砌和菱形六角化十二面体堆砌之间的关系。
基于正六边形镶嵌和正三角形镶嵌的Wythoff构建
就像半正多面体一样,这里也有8个基于正六边形镶嵌(和正三角形镶嵌)的半正镶嵌。在以下的图片中,原有面对应的面被涂成了红色,原有顶点所对应的面被涂成了黄色,原有棱对应的面被涂成了蓝色。这8个半正镶嵌中,只有7个是拓扑上相异的。(截顶正三角形镶嵌与正六边形镶嵌在拓扑上相同)
正六边形镶嵌/正三角形镶嵌
对称性: [6,3], (*632)
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[6,3]+ (632)
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[1+,6,3] (*333)
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[6,3+] (3*3)
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{6,3}
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t{6,3}
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r{6,3} r{3[3]}
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t{3,6} t{3[3]}
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{3,6} {3[3]}
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rr{6,3} s2{6,3}
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tr{6,3}
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sr{6,3}
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h{6,3} {3[3]}
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h2{6,3} r{3[3]}
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s{3,6} s{3[3]}
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![node_1](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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![node_1](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node_1](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node_1](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node_1](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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![node_1](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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![node_1](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node_1](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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![node_h](//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node_h](//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node_h](//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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![node_h0](//upload.wikimedia.org/wikipedia/commons/5/5d/CDel_node_h0.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node_1](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) = ![branch_11](//upload.wikimedia.org/wikipedia/commons/f/fc/CDel_branch_11.png) ![split2](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_split2.png)
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![node_h0](//upload.wikimedia.org/wikipedia/commons/5/5d/CDel_node_h0.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node_1](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![node_1](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) = ![branch_11](//upload.wikimedia.org/wikipedia/commons/f/fc/CDel_branch_11.png) ![split2](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_split2.png)
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![node_h0](//upload.wikimedia.org/wikipedia/commons/5/5d/CDel_node_h0.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![node_1](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) = ![branch](//upload.wikimedia.org/wikipedia/commons/4/43/CDel_branch.png) ![split2](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_split2.png)
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![node_1](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node_h](//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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![node_h1](//upload.wikimedia.org/wikipedia/commons/4/48/CDel_node_h1.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) =
![branch_10ru](//upload.wikimedia.org/wikipedia/commons/c/c2/CDel_branch_10ru.png) ![split2](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_split2.png) or ![branch_01rd](//upload.wikimedia.org/wikipedia/commons/7/74/CDel_branch_01rd.png) ![split2](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_split2.png)
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![node_h1](//upload.wikimedia.org/wikipedia/commons/4/48/CDel_node_h1.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) =
![branch_10ru](//upload.wikimedia.org/wikipedia/commons/c/c2/CDel_branch_10ru.png) ![split2](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_split2.png) or ![branch_01rd](//upload.wikimedia.org/wikipedia/commons/7/74/CDel_branch_01rd.png) ![split2](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_split2.png)
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![node_h0](//upload.wikimedia.org/wikipedia/commons/5/5d/CDel_node_h0.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node_h](//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![node_h](//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png) = ![branch_hh](//upload.wikimedia.org/wikipedia/commons/6/63/CDel_branch_hh.png) ![split2](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_split2.png)
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![](//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Uniform_tiling_63-t1.png/40px-Uniform_tiling_63-t1.png)
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![](//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Uniform_tiling_63-t12.png/40px-Uniform_tiling_63-t12.png)
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![](//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Uniform_tiling_63-t2.png/40px-Uniform_tiling_63-t2.png)
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![](//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Uniform_tiling_63-t02.png/40px-Uniform_tiling_63-t02.png)
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![](//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Uniform_tiling_333-t0.png/30px-Uniform_tiling_333-t0.png)
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![](//upload.wikimedia.org/wikipedia/commons/thumb/0/01/Uniform_tiling_333-t02.png/30px-Uniform_tiling_333-t02.png)
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![](//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Uniform_tiling_63-h12.png/40px-Uniform_tiling_63-h12.png)
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半正对偶
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V63
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V3.122
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V(3.6)2
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V63
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V36
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V3.4.12.4
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V.4.6.12
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V34.6
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V36
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V(3.6)2
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V36
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![node_f1](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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![node_f1](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node_f1](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node_f1](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node_f1](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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![node_f1](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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![node_f1](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node_f1](//upload.wikimedia.org/wikipedia/commons/9/9b/CDel_node_f1.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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![node_fh](//upload.wikimedia.org/wikipedia/commons/6/6f/CDel_node_fh.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node_fh](//upload.wikimedia.org/wikipedia/commons/6/6f/CDel_node_fh.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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![node_fh](//upload.wikimedia.org/wikipedia/commons/6/6f/CDel_node_fh.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
|
![node_fh](//upload.wikimedia.org/wikipedia/commons/6/6f/CDel_node_fh.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
|
![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node_fh](//upload.wikimedia.org/wikipedia/commons/6/6f/CDel_node_fh.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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|
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三角形 对称性
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拓展对称性
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拓展 符号
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拓展 阶
|
堆砌符号
|
a1
|
[3[3]]
|
![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![split1](//upload.wikimedia.org/wikipedia/commons/a/a1/CDel_split1.png)
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×1
|
(None)
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i2
|
<[3[3]]> = [6,3]
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![node_c1](//upload.wikimedia.org/wikipedia/commons/8/80/CDel_node_c1.png) ![split1](//upload.wikimedia.org/wikipedia/commons/a/a1/CDel_split1.png) ![branch_c2](//upload.wikimedia.org/wikipedia/commons/6/65/CDel_branch_c2.png) = ![node_c1](//upload.wikimedia.org/wikipedia/commons/8/80/CDel_node_c1.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![node_c2](//upload.wikimedia.org/wikipedia/commons/4/4d/CDel_node_c2.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png)
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×2
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![node_1](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![split1](//upload.wikimedia.org/wikipedia/commons/a/a1/CDel_split1.png) 1, ![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![split1](//upload.wikimedia.org/wikipedia/commons/a/a1/CDel_split1.png) 2
|
r6
|
[3[3[3]]] = [6,3]
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![node_c1](//upload.wikimedia.org/wikipedia/commons/8/80/CDel_node_c1.png) ![split1](//upload.wikimedia.org/wikipedia/commons/a/a1/CDel_split1.png) ![branch_c1](//upload.wikimedia.org/wikipedia/commons/8/83/CDel_branch_c1.png) = ![node_c1](//upload.wikimedia.org/wikipedia/commons/8/80/CDel_node_c1.png) ![6](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_6.png) ![node](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![3](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png)
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×6
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![node_1](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![split1](//upload.wikimedia.org/wikipedia/commons/a/a1/CDel_split1.png) 3, ![node_h](//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png) ![split1](//upload.wikimedia.org/wikipedia/commons/a/a1/CDel_split1.png) (1)
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拓扑相同的镶嵌
正六边形镶嵌是有着{6,3}拓扑的一种特殊的正的镶嵌,而实际上,这里有12种类型的非正但是面全同且顶点全同的六边形镶嵌,前7种可以被认为是没有边对边正好对上的四边形镶嵌,也可被认为是有两对共线边的六边形镶嵌。这里的“对称性”假定所有的面都是相同的。
-
-
平行四边形 pmg对称
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平行四边形 pgg对称
-
-
-
矩形 pgg对称
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矩形 cmm对称
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六边形 p2对称
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六边形 pgg对称
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六边形 pmg对称
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展长六边形 cmm对称
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正六边形镶嵌也可被变形为一种手征性的四填充色三向同性的编织图案。其中部分正六边形被扭曲成了平行四边形。这一图案有着旋转632 (p6) 对称性。
四色正六边形镶嵌
正六边形
|
六边形编织
|
p6m (*632)
|
p6 (632)
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|
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應用
正六邊形鑲嵌是二維空間最密的排列方式。在蜂窩猜想中,正六邊形鑲嵌是使用最少的總周長將該表面劃分成面積相等的區域的最佳方法。[1][2]最佳的三維結構由開爾文勳爵(Lord Kelvin)提出,他認為,開爾文結構(體心立方晶格)是最佳的結構(最佳結構可能出現於肥皂泡)。然而,一个更加不对称的韋爾—費倫結構要比它好一些。
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
- 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)
- 埃里克·韦斯坦因. Hexagonal Grid. MathWorld.
- Klitzing, Richard. 2D Euclidean tilings o3o6x - hexat - O3. bendwavy.org.
- Williams, Robert. The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. 1979: 35. ISBN 0-486-23729-X.
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 [1]
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