# 正六邊形鑲嵌

(點選檢視大圖)

t0,1{3,6}

3 | 6 2
2 6 | 3
3 3 3 |

63

p6, [6,3]+, (632)

 6.6.6 (or 63)（頂點圖） 正三角形鑲嵌(對偶多面體)

## 半正涂色

k阶半正 一阶半正 二阶半正 三阶半正

(h,k) (1,0) (1,1) (2,0) (2,1)

3 | 6 2 2 6 | 3 3 3 3 |

(p6m)
[6,3]
*333
(p3)
[3[3]]
*632
(p6m)
[6,3]
632
(p6)
[6,3]+

## 相关半正镶嵌

{6,3} t0,1{6,3} t1{6,3} t1,2{6,3} t2{6,3} t0,2{6,3} t0,1,2{6,3} s{6,3} h{6,3} h1,2{6,3}

V6.6.6 V3.12.12 V3.6.3.6 V6.6.6 V3.3.3.3.3.3 V3.4.12.4 V.4.6.12 V3.3.3.3.6 V3.3.3.3.3.3

{2,3}

{3,3}

{4,3}

{5,3}

{6,3}

{7,3}

{8,3}

...
{∞,3}

（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}，     ，一直到n = ∞：

{6,2}

{6,3}

{6,4}

{6,5}

{6,6}

{6,7}

{6,8}

...
{6,∞}

[3,3] [4,3] [5,3] [6,3] [7,3] [8,3]

 菱形镶嵌 正六边形镶嵌 利用这一关系的栅栏

## 基于正六边形镶嵌和正三角形镶嵌的Wythoff构建

(632)
[1+,6,3]
(*333)
[6,3+]
(3*3)
{6,3} t{6,3} r{6,3}
r{3[3]}
t{3,6}
t{3[3]}
{3,6}
{3[3]}
rr{6,3}
s2{6,3}
tr{6,3} sr{6,3} h{6,3}
{3[3]}
h2{6,3}
r{3[3]}
s{3,6}
s{3[3]}

=

=

=
=
or
=
or

=

V63 V3.122 V(3.6)2 V63 V36 V3.4.12.4 V.4.6.12 V34.6 V36 V(3.6)2 V36

a1 [3[3]]     ×1 (None)
i2 <[3[3]]>
= [6,3]

=
×2     1,     2
r6 [3[3[3]]]
= [6,3]

=
×6     3,     (1)
3 | 3 3 3 3 | 3 3 | 3 3 3 3 | 3 3 | 3 3 3 3 | 3 3 3 3 | | 3 3 3

(3.3)3

3.6.3.6

(3.3)3

3.6.3.6

(3.3)3

3.6.3.6

6.6.6

3.3.3.3.3.3

## 拓扑相同的镶嵌

p6m (*632) p6 (632)

## 參考文獻

1. ^ Weisstein, Eric W. Honeycomb Conjecture. MathWorld. [27 Dec 2010].
2. ^ Hales, Thomas C. The Honeycomb Conjecture. Discrete and Computational Geometry. 8 Jun 1999, 25: 1–22 (2001). arXiv:math/9906042.
1. Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs
2. 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)
3. Richard Klitzing, 2D Euclidean tilings, o3o6x - hexat - O3
4. Williams, Robert. The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. 1979: 35. ISBN 0-486-23729-X.
5. John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 [1]