[原创]关于20面体

技术员 · 发表于 2011-7-11 20:38

[watermark]用5种颜色给正20面体每个面上色，要求每个边相邻两个面不同色，如何填充？有几种方式？[/watermark]

ysr · 发表于 2011-7-12 13:21

4色定理(已有机器证明)是可平面图,立体的行吗?5色定理也是平面图,这个可以吗?画不出图,无法试!

cjsh · 发表于 2011-7-12 17:13

[这个贴子最后由cjsh在 2011/07/12 05:18pm 第 1 次编辑] 群方程就专解这种问题的---------24阶群同构正6、8面体变换群类方程: |G| = |Z(G)| + ∑i [G : Hi] 看下面24阶群轨道稳定子信息，对着书上这类题正6、8面体变换群能很快的出答案，这种问题阶大了只有计算机才能很好的解决和彩珠串一样这是一大类问题，深了可和组合图论，算法图论连着 Symmetric group S acting on a set of cardinality 4 Order = 24 = 2^3 * 3 Permutation group G acting on a set of cardinality 4 Order = 24 = 2^3 * 3 (1, 2, 3, 4) (1, 2) { (1, 2, 3, 4), (1, 2) } 2 GSet{@ 1, 2, 3, 4 @} Conjugacy Classes of group G ---------------------------- [1] Order 1 Length 1 Rep Id(G) [2] Order 2 Length 3 Rep (1, 2)(3, 4) [3] Order 2 Length 6 Rep (1, 2) [4] Order 3 Length 8 Rep (1, 2, 3) [5] Order 4 Length 6 Rep (1, 2, 3, 4) Conjugacy Classes of group G ---------------------------- [1] Order 1 Length 1 Rep Id(G) [2] Order 2 Length 3 Rep (1, 2)(3, 4) [3] Order 2 Length 6 Rep (1, 2) [4] Order 3 Length 8 Rep (1, 2, 3) [5] Order 4 Length 6 Rep (1, 2, 3, 4) Mapping from: GrpPerm: G to { 1 .. 5 } 5 Conjugacy classes of subgroups ------------------------------ [ 1] Order 1 Length 1 Permutation group acting on a set of cardinality 4 Order = 1 [ 2] Order 2 Length 3 Permutation group acting on a set of cardinality 4 Order = 2 (1, 4)(2, 3) [ 3] Order 2 Length 6 Permutation group acting on a set of cardinality 4 Order = 2 (3, 4) [ 4] Order 3 Length 4 Permutation group acting on a set of cardinality 4 Order = 3 (2, 3, 4) [ 5] Order 4 Length 1 Permutation group acting on a set of cardinality 4 Order = 4 = 2^2 (1, 4)(2, 3) (1, 3)(2, 4) [ 6] Order 4 Length 3 Permutation group acting on a set of cardinality 4 Order = 4 = 2^2 (1, 4, 2, 3) (1, 2)(3, 4) [ 7] Order 4 Length 3 Permutation group acting on a set of cardinality 4 Order = 4 = 2^2 (3, 4) (1, 2)(3, 4) [ 8] Order 6 Length 4 Permutation group acting on a set of cardinality 4 Order = 6 = 2 * 3 (3, 4) (2, 3, 4) [ 9] Order 8 Length 3 Permutation group acting on a set of cardinality 4 Order = 8 = 2^3 (3, 4) (1, 4)(2, 3) (1, 3)(2, 4) [10] Order 12 Length 1 Permutation group acting on a set of cardinality 4 Order = 12 = 2^2 * 3 (2, 3, 4) (1, 4)(2, 3) (1, 3)(2, 4) [11] Order 24 Length 1 Permutation group acting on a set of cardinality 4 Order = 24 = 2^3 * 3 (3, 4) (2, 3, 4) (1, 4)(2, 3) (1, 3)(2, 4) Partially ordered set of subgroup classes ----------------------------------------- [11] Order 24 Length 1 Maximal Subgroups: 8 9 10 --- [10] Order 12 Length 1 Maximal Subgroups: 4 5 [ 9] Order 8 Length 3 Maximal Subgroups: 5 6 7 --- [ 8] Order 6 Length 4 Maximal Subgroups: 3 4 [ 7] Order 4 Length 3 Maximal Subgroups: 2 [ 6] Order 4 Length 3 Maximal Subgroups: 2 3 [ 5] Order 4 Length 1 Maximal Subgroups: 2 --- [ 4] Order 3 Length 4 Maximal Subgroups: 1 [ 3] Order 2 Length 6 Maximal Subgroups: 1 [ 2] Order 2 Length 3 Maximal Subgroups: 1 --- [ 1] Order 1 Length 1 Maximal Subgroups: [ GSet{@ 1, 2, 3, 4 @} ] [ <4, 1> ] {} Symmetric group S acting on a set of cardinality 4 Order = 24 = 2^3 * 3 Permutation group G acting on a set of cardinality 4 Order = 24 = 2^3 * 3 (1, 2, 3, 4) (1, 2) { (1, 2, 3, 4), (1, 2) } 2 GSet{@ 1, 2, 3, 4 @} Conjugacy Classes of group G ---------------------------- [1] Order 1 Length 1 Rep Id(G) [2] Order 2 Length 3 Rep (1, 2)(3, 4) [3] Order 2 Length 6 Rep (1, 2) [4] Order 3 Length 8 Rep (1, 2, 3) [5] Order 4 Length 6 Rep (1, 2, 3, 4) Conjugacy Classes of group G ---------------------------- [1] Order 1 Length 1 Rep Id(G) [2] Order 2 Length 3 Rep (1, 2)(3, 4) [3] Order 2 Length 6 Rep (1, 2) [4] Order 3 Length 8 Rep (1, 2, 3) [5] Order 4 Length 6 Rep (1, 2, 3, 4) Mapping from: GrpPerm: G to { 1 .. 5 } 5 Conjugacy classes of subgroups ------------------------------ [ 1] Order 1 Length 1 Permutation group acting on a set of cardinality 4 Order = 1 [ 2] Order 2 Length 3 Permutation group acting on a set of cardinality 4 Order = 2 (1, 4)(2, 3) [ 3] Order 2 Length 6 Permutation group acting on a set of cardinality 4 Order = 2 (3, 4) [ 4] Order 3 Length 4 Permutation group acting on a set of cardinality 4 Order = 3 (2, 3, 4) [ 5] Order 4 Length 1 Permutation group acting on a set of cardinality 4 Order = 4 = 2^2 (1, 4)(2, 3) (1, 3)(2, 4) [ 6] Order 4 Length 3 Permutation group acting on a set of cardinality 4 Order = 4 = 2^2 (1, 4, 2, 3) (1, 2)(3, 4) [ 7] Order 4 Length 3 Permutation group acting on a set of cardinality 4 Order = 4 = 2^2 (3, 4) (1, 2)(3, 4) [ 8] Order 6 Length 4 Permutation group acting on a set of cardinality 4 Order = 6 = 2 * 3 (3, 4) (2, 3, 4) [ 9] Order 8 Length 3 Permutation group acting on a set of cardinality 4 Order = 8 = 2^3 (3, 4) (1, 4)(2, 3) (1, 3)(2, 4) [10] Order 12 Length 1 Permutation group acting on a set of cardinality 4 Order = 12 = 2^2 * 3 (2, 3, 4) (1, 4)(2, 3) (1, 3)(2, 4) [11] Order 24 Length 1 Permutation group acting on a set of cardinality 4 Order = 24 = 2^3 * 3 (3, 4) (2, 3, 4) (1, 4)(2, 3) (1, 3)(2, 4) Partially ordered set of subgroup classes ----------------------------------------- [11] Order 24 Length 1 Maximal Subgroups: 8 9 10 --- [10] Order 12 Length 1 Maximal Subgroups: 4 5 [ 9] Order 8 Length 3 Maximal Subgroups: 5 6 7 --- [ 8] Order 6 Length 4 Maximal Subgroups: 3 4 [ 7] Order 4 Length 3 Maximal Subgroups: 2 [ 6] Order 4 Length 3 Maximal Subgroups: 2 3 [ 5] Order 4 Length 1 Maximal Subgroups: 2 --- [ 4] Order 3 Length 4 Maximal Subgroups: 1 [ 3] Order 2 Length 6 Maximal Subgroups: 1 [ 2] Order 2 Length 3 Maximal Subgroups: 1 --- [ 1] Order 1 Length 1 Maximal Subgroups: [ GSet{@ 1, 2, 3, 4 @} ] [ <4, 1> ] {}

cjsh · 发表于 2011-7-12 17:39

正十二面体群或正二十面体群与5次交错群同构
A5 := Alt(5);
A5;
Order(A5);
# A5 ;
Generators(A5);
NumberingMap(A5);
ConjugacyClasses(A5);
Nclasses(A5);
Fix(A5);
Orbits(A5) ;
Transitivity(A5);
Center(A5);
NormalLattice(A5) ;

Permutation group A5 acting on a set of cardinality 5
Order = 60 = 2^2 * 3 * 5
(3, 4, 5)
(1, 2, 3)
60
60
{
(1, 2, 3),
(3, 4, 5)
}
Mapping from: GrpPerm: A5 to { 1 .. 60 }
Conjugacy Classes of group A5
-----------------------------
[1]    Order 1    Length 1
      Rep Id(A5)
[2]    Order 2    Length 15
      Rep (1, 2)(3, 4)
[3]    Order 3    Length 20
      Rep (1, 2, 3)
[4]    Order 5    Length 12
      Rep (1, 2, 3, 4, 5)
[5]    Order 5    Length 12
      Rep (1, 3, 4, 5, 2)

5
{}
[
GSet{@ 1, 2, 3, 4, 5 @}
]
3
Permutation group acting on a set of cardinality 5
Order = 1
Normal subgroup lattice
-----------------------
[2]  Order 60  Length 1  Maximal Subgroups: 1
---
[1]  Order 1 Length 1  Maximal Subgroups:

cjsh · 发表于 2011-7-12 17:43

32种晶体学点群的记号

http://www.hxu.edu.cn/partwebs/huaxuexi/qt/hxsj/newpage/jthxdq.htm[br][br]-=-=-=-=- 以下内容由 cjsh 在时添加 -=-=-=-=-
http://www.hxu.edu.cn/partwebs/huaxuexi/qt/hxsj/newpage/jthx.htm

ysr · 发表于 2011-7-12 20:22

不会群论,楼上高!

技术员 · 发表于 2011-7-13 17:33

下面引用由cjsh在 2011/07/12 05:13pm 发表的内容： 群方程就专解这种问题的---------24阶群同构正6、8面体变换群
类方程:
|G| = |Z(G)| + ∑i 看下面24阶群轨道稳定子信息，对着书上这类题正6、8面体变换群能很快的出答案，这种问题阶大了只有计算机才能很　...

能否把立体展成平面，以图示人。

luyuanhong · 发表于 2011-7-13 22:31

[这个贴子最后由luyuanhong在 2011/07/14 07:58am 第 2 次编辑]

对正 20 面体表面涂色，只需要 3 种颜色就够了：
（下面的图示，把立体展成平面形式，区域的外部，代表一个白色的三角形面）

尚九天 · 发表于 2011-7-14 06:22

下面引用由luyuanhong在 2011/07/13 10:31pm 发表的内容：

:em05: 对 20 面体表面涂色，只需要 3 种颜色就够了：
（下面的图示，把立体展成平面形式，区域的外部，代表一个白色的三角形面）
:em05: 此主题相关图片如下：

:em05: 20面体，三色足矣。40面体、80面体、120面体呢？ 121、123面体呢？是否三色亦足矣？

技术员 · 发表于 2011-7-15 21:21

下面引用由luyuanhong在 2011/07/13 10:31pm 发表的内容： 对正 20 面体表面涂色，只需要 3 种颜色就够了：
（下面的图示，把立体展成平面形式，区域的外部，代表一个白色的三角形面）

陆教授，我就要5种色，是要求不相邻。

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[原创]关于20面体