数学中国

 找回密码
 注册
搜索
热搜: 活动 交友 discuz
查看: 7785|回复: 14

[原创]关于20面体

[复制链接]
发表于 2011-7-11 20:38 | 显示全部楼层 |阅读模式
[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]
发表于 2011-7-12 13:21 | 显示全部楼层

[原创]关于20面体

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!
发表于 2011-7-12 17:13 | 显示全部楼层

[原创]关于20面体

[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> ] {}
发表于 2011-7-12 17:39 | 显示全部楼层

[原创]关于20面体

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);
&#35; 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:


发表于 2011-7-12 17:43 | 显示全部楼层

[原创]关于20面体

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 时添加 -=-=-=-=-
http://www.hxu.edu.cn/partwebs/huaxuexi/qt/hxsj/newpage/jthx.htm
发表于 2011-7-12 20:22 | 显示全部楼层

[原创]关于20面体

不会群论,楼上高!
 楼主| 发表于 2011-7-13 17:33 | 显示全部楼层

[原创]关于20面体

下面引用由cjsh2011/07/12 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。
发表于 2011-7-13 22:31 | 显示全部楼层

[原创]关于20面体

[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)

本帖子中包含更多资源

您需要 登录 才可以下载或查看,没有帐号?注册

x
发表于 2011-7-14 06:22 | 显示全部楼层

[原创]关于20面体

下面引用由luyuanhong2011/07/13 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?
 楼主| 发表于 2011-7-15 21:21 | 显示全部楼层

[原创]关于20面体

下面引用由luyuanhong2011/07/13 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。
您需要登录后才可以回帖 登录 | 注册

本版积分规则

Archiver|手机版|小黑屋|数学中国 ( 京ICP备05040119号 )

GMT+8, 2025-7-11 16:05 , Processed in 0.097736 second(s), 16 queries .

Powered by Discuz! X3.4

Copyright © 2001-2020, Tencent Cloud.

快速回复 返回顶部 返回列表