答网友：足球的顶点坐标

足球和C60以及其他Fullerene结构的顶点坐标，计算方法都可以是一样的。

如果它们的对称性都是Ih点群，那么，只要用Ih的120个矩阵操作其中一个顶点的坐标，就可以求得其他59个坐标。

凡是坐标，都是有个参照坐标系的问题。也就是说，原点在哪里？X轴Y轴Z轴如何选取？

我把对称性最高的轴（过球心的两个五边形中心连线）给了Z轴，Y轴穿越6/6棱的中点，X轴被置于一个没有任何对称性的方向上。

Z轴穿越五边形中心，Y轴穿越6/6棱的中点，X轴穿越六边形而偏离中心

图 点群I_h对称性和笛卡儿坐标系的安放

网友要求：d56=d66=1，计算C60的顶点的坐标。结果如下：

1,":",.850651,0,2.327438,"    R=",2.478018
2,":",.262866,.809017,2.327438,"    R=",2.478018
3,":",.262866,-.809017,2.327438,"    R=",2.478018
4,":",-.688191,.5,2.327438,"    R=",2.478018
5,":",-.688191,-.5,2.327438,"    R=",2.478018
6,":",1.701301,0,1.801708,"    R=",2.478019
7,":",.52573,1.618035,1.801708,"    R=",2.47802
8,":",.52573,-1.618035,1.801708,"    R=",2.47802
9,":",-1.376383,.999999,1.801708,"    R=",2.478019
10,":",-1.376383,-.999999,1.801708,"    R=",2.478019
11,":",1.964166,.809017,1.275977,"    R=",2.478018
12,":",1.964166,-.809017,1.275977,"    R=",2.478018
13,":",1.376381,1.618035,1.275977,"    R=",2.478019
14,":",1.376381,-1.618035,1.275977,"    R=",2.478019
15,":",-.162461,2.118035,1.275977,"    R=",2.47802
16,":",-.162461,-2.118035,1.275977,"    R=",2.47802
17,":",-1.113517,1.809017,1.275977,"    R=",2.478019
18,":",-1.113517,-1.809017,1.275977,"    R=",2.478019
19,":",-2.064574,.5,1.275977,"    R=",2.47802
20,":",-2.064574,-.5,1.275977,"    R=",2.47802
21,":",2.389492,.5,.425326,"    R=",2.478018
22,":",2.389492,-.5,.425326,"    R=",2.478018
23,":",1.213921,2.118035,.425326,"    R=",2.478019
24,":",1.213921,-2.118035,.425326,"    R=",2.478019
25,":",.262865,2.427051,.425326,"    R=",2.478019
26,":",.262865,-2.427051,.425326,"    R=",2.478019
27,":",-1.639248,1.809017,.425326,"    R=",2.478019
28,":",-1.639248,-1.809017,.425326,"    R=",2.478019
29,":",-2.227033,.999999,.425326,"    R=",2.478019
30,":",-2.227033,-.999999,.425326,"    R=",2.478019
31,":",2.227033,.999999,-.425326,"    R=",2.478019
32,":",2.227033,-.999999,-.425326,"    R=",2.478019
33,":",1.639248,1.809017,-.425326,"    R=",2.478019
34,":",1.639248,-1.809017,-.425326,"    R=",2.478019
35,":",-.262865,2.427051,-.425326,"    R=",2.478019
36,":",-.262865,-2.427051,-.425326,"    R=",2.478019
37,":",-1.213921,2.118035,-.425326,"    R=",2.478019
38,":",-1.213921,-2.118035,-.425326,"    R=",2.478019
39,":",-2.389492,.5,-.425326,"    R=",2.478018
40,":",-2.389492,-.5,-.425326,"    R=",2.478018
41,":",2.064574,.5,-1.275977,"    R=",2.47802
42,":",2.064574,-.5,-1.275977,"    R=",2.47802
43,":",1.113517,1.809017,-1.275977,"    R=",2.478019
44,":",1.113517,-1.809017,-1.275977,"    R=",2.478019
45,":",.162461,2.118035,-1.275977,"    R=",2.47802
46,":",.162461,-2.118035,-1.275977,"    R=",2.47802
47,":",-1.376381,1.618035,-1.275977,"    R=",2.478019
48,":",-1.376381,-1.618035,-1.275977,"    R=",2.478019
49,":",-1.964166,.809017,-1.275977,"    R=",2.478018
50,":",-1.964166,-.809017,-1.275977,"    R=",2.478018
51,":",1.376383,.999999,-1.801708,"    R=",2.478019
52,":",1.376383,-.999999,-1.801708,"    R=",2.478019
53,":",-.52573,1.618035,-1.801708,"    R=",2.47802
54,":",-.52573,-1.618035,-1.801708,"    R=",2.47802
55,":",-1.701301,0,-1.801708,"    R=",2.478019
56,":",.688191,.5,-2.327438,"    R=",2.478018
57,":",.688191,-.5,-2.327438,"    R=",2.478018
58,":",-.262866,.809017,-2.327438,"    R=",2.478018
59,":",-.262866,-.809017,-2.327438,"    R=",2.478018
60,":",-.850651,0,-2.327438,"    R=",2.478018

1,"d2=",.9999999

"ATOM1=",.850651,0,2.327438

"ATOM2=",.262866,.809017,2.327438

2,"d2=",.9999999

"ATOM1=",.850651,0,2.327438

"ATOM2=",.262866,-.809017,2.327438

3,"d2=",.9999987

"ATOM1=",.850651,0,2.327438

"ATOM2=",1.701301,0,1.801708

4,"d2=",1

"ATOM1=",.262866,.809017,2.327438

"ATOM2=",-.688191,.5,2.327438

5,"d2=",.9999999

"ATOM1=",.262866,.809017,2.327438

"ATOM2=",.52573,1.618035,1.801708

6,"d2=",1

"ATOM1=",.262866,-.809017,2.327438

"ATOM2=",-.688191,-.5,2.327438

7,"d2=",.9999999

"ATOM1=",.262866,-.809017,2.327438

"ATOM2=",.52573,-1.618035,1.801708

8,"d2=",1

"ATOM1=",-.688191,.5,2.327438

"ATOM2=",-.688191,-.5,2.327438

9,"d2=",.9999996

"ATOM1=",-.688191,.5,2.327438

"ATOM2=",-1.376383,.999999,1.801708

10,"d2=",.9999996

"ATOM1=",-.688191,-.5,2.327438

"ATOM2=",-1.376383,-.999999,1.801708

11,"d2=",.9999998

"ATOM1=",1.701301,0,1.801708

"ATOM2=",1.964166,.809017,1.275977

12,"d2=",.9999998

"ATOM1=",1.701301,0,1.801708

"ATOM2=",1.964166,-.809017,1.275977

13,"d2=",1

"ATOM1=",.52573,1.618035,1.801708

"ATOM2=",1.376381,1.618035,1.275977

14,"d2=",1

"ATOM1=",.52573,1.618035,1.801708

"ATOM2=",-.162461,2.118035,1.275977

15,"d2=",1

"ATOM1=",.52573,-1.618035,1.801708

"ATOM2=",1.376381,-1.618035,1.275977

16,"d2=",1

"ATOM1=",.52573,-1.618035,1.801708

"ATOM2=",-.162461,-2.118035,1.275977

17,"d2=",1.000001

"ATOM1=",-1.376383,.999999,1.801708

"ATOM2=",-1.113517,1.809017,1.275977

18,"d2=",.9999995

"ATOM1=",-1.376383,.999999,1.801708

"ATOM2=",-2.064574,.5,1.275977

19,"d2=",1.000001

"ATOM1=",-1.376383,-.999999,1.801708

"ATOM2=",-1.113517,-1.809017,1.275977

20,"d2=",.9999995

"ATOM1=",-1.376383,-.999999,1.801708

"ATOM2=",-2.064574,-.5,1.275977

21,"d2=",1.000001

"ATOM1=",1.964166,.809017,1.275977

"ATOM2=",1.376381,1.618035,1.275977

22,"d2=",1

"ATOM1=",1.964166,.809017,1.275977

"ATOM2=",2.389492,.5,.425326

23,"d2=",1.000001

"ATOM1=",1.964166,-.809017,1.275977

"ATOM2=",1.376381,-1.618035,1.275977

24,"d2=",1

"ATOM1=",1.964166,-.809017,1.275977

"ATOM2=",2.389492,-.5,.425326

25,"d2=",1

"ATOM1=",1.376381,1.618035,1.275977

"ATOM2=",1.213921,2.118035,.425326

26,"d2=",1

"ATOM1=",1.376381,-1.618035,1.275977

"ATOM2=",1.213921,-2.118035,.425326

27,"d2=",.9999999

"ATOM1=",-.162461,2.118035,1.275977

"ATOM2=",-1.113517,1.809017,1.275977

28,"d2=",1

"ATOM1=",-.162461,2.118035,1.275977

"ATOM2=",.262865,2.427051,.425326

29,"d2=",.9999999

"ATOM1=",-.162461,-2.118035,1.275977

"ATOM2=",-1.113517,-1.809017,1.275977

30,"d2=",1

"ATOM1=",-.162461,-2.118035,1.275977

"ATOM2=",.262865,-2.427051,.425326

31,"d2=",1

"ATOM1=",-1.113517,1.809017,1.275977

"ATOM2=",-1.639248,1.809017,.425326

32,"d2=",1

"ATOM1=",-1.113517,-1.809017,1.275977

"ATOM2=",-1.639248,-1.809017,.425326

33,"d2=",1

"ATOM1=",-2.064574,.5,1.275977

"ATOM2=",-2.064574,-.5,1.275977

34,"d2=",.9999995

"ATOM1=",-2.064574,.5,1.275977

"ATOM2=",-2.227033,.999999,.425326

35,"d2=",.9999995

"ATOM1=",-2.064574,-.5,1.275977

"ATOM2=",-2.227033,-.999999,.425326

36,"d2=",1

"ATOM1=",2.389492,.5,.425326

"ATOM2=",2.389492,-.5,.425326

37,"d2=",1

"ATOM1=",2.389492,.5,.425326

"ATOM2=",2.227033,.999999,-.425326

38,"d2=",1

"ATOM1=",2.389492,-.5,.425326

"ATOM2=",2.227033,-.999999,-.425326

39,"d2=",.9999992

"ATOM1=",1.213921,2.118035,.425326

"ATOM2=",.262865,2.427051,.425326

40,"d2=",1.000002

"ATOM1=",1.213921,2.118035,.425326

"ATOM2=",1.639248,1.809017,-.425326

41,"d2=",.9999992

"ATOM1=",1.213921,-2.118035,.425326

"ATOM2=",.262865,-2.427051,.425326

42,"d2=",1.000002

"ATOM1=",1.213921,-2.118035,.425326

"ATOM2=",1.639248,-1.809017,-.425326

43,"d2=",1

"ATOM1=",.262865,2.427051,.425326

"ATOM2=",-.262865,2.427051,-.425326

44,"d2=",1

"ATOM1=",.262865,-2.427051,.425326

"ATOM2=",-.262865,-2.427051,-.425326

45,"d2=",1.000001

"ATOM1=",-1.639248,1.809017,.425326

"ATOM2=",-2.227033,.999999,.425326

46,"d2=",1.000002

"ATOM1=",-1.639248,1.809017,.425326

"ATOM2=",-1.213921,2.118035,-.425326

47,"d2=",1.000001

"ATOM1=",-1.639248,-1.809017,.425326

"ATOM2=",-2.227033,-.999999,.425326

48,"d2=",1.000002

"ATOM1=",-1.639248,-1.809017,.425326

"ATOM2=",-1.213921,-2.118035,-.425326

49,"d2=",1

"ATOM1=",-2.227033,.999999,.425326

"ATOM2=",-2.389492,.5,-.425326

50,"d2=",1

"ATOM1=",-2.227033,-.999999,.425326

"ATOM2=",-2.389492,-.5,-.425326

51,"d2=",1.000001

"ATOM1=",2.227033,.999999,-.425326

"ATOM2=",1.639248,1.809017,-.425326

52,"d2=",.9999995

"ATOM1=",2.227033,.999999,-.425326

"ATOM2=",2.064574,.5,-1.275977

53,"d2=",1.000001

"ATOM1=",2.227033,-.999999,-.425326

"ATOM2=",1.639248,-1.809017,-.425326

54,"d2=",.9999995

"ATOM1=",2.227033,-.999999,-.425326

"ATOM2=",2.064574,-.5,-1.275977

55,"d2=",1

"ATOM1=",1.639248,1.809017,-.425326

"ATOM2=",1.113517,1.809017,-1.275977

56,"d2=",1

"ATOM1=",1.639248,-1.809017,-.425326

"ATOM2=",1.113517,-1.809017,-1.275977

57,"d2=",.9999992

"ATOM1=",-.262865,2.427051,-.425326

"ATOM2=",-1.213921,2.118035,-.425326

58,"d2=",1

"ATOM1=",-.262865,2.427051,-.425326

"ATOM2=",.162461,2.118035,-1.275977

59,"d2=",.9999992

"ATOM1=",-.262865,-2.427051,-.425326

"ATOM2=",-1.213921,-2.118035,-.425326

60,"d2=",1

"ATOM1=",-.262865,-2.427051,-.425326

"ATOM2=",.162461,-2.118035,-1.275977

61,"d2=",1

"ATOM1=",-1.213921,2.118035,-.425326

"ATOM2=",-1.376381,1.618035,-1.275977

62,"d2=",1

"ATOM1=",-1.213921,-2.118035,-.425326

"ATOM2=",-1.376381,-1.618035,-1.275977

63,"d2=",1

"ATOM1=",-2.389492,.5,-.425326

"ATOM2=",-2.389492,-.5,-.425326

64,"d2=",1

"ATOM1=",-2.389492,.5,-.425326

"ATOM2=",-1.964166,.809017,-1.275977

65,"d2=",1

"ATOM1=",-2.389492,-.5,-.425326

"ATOM2=",-1.964166,-.809017,-1.275977

66,"d2=",1

"ATOM1=",2.064574,.5,-1.275977

"ATOM2=",2.064574,-.5,-1.275977

67,"d2=",.9999995

"ATOM1=",2.064574,.5,-1.275977

"ATOM2=",1.376383,.999999,-1.801708

68,"d2=",.9999995

"ATOM1=",2.064574,-.5,-1.275977

"ATOM2=",1.376383,-.999999,-1.801708

69,"d2=",.9999999

"ATOM1=",1.113517,1.809017,-1.275977

"ATOM2=",.162461,2.118035,-1.275977

70,"d2=",1.000001

"ATOM1=",1.113517,1.809017,-1.275977

"ATOM2=",1.376383,.999999,-1.801708

71,"d2=",.9999999

"ATOM1=",1.113517,-1.809017,-1.275977

"ATOM2=",.162461,-2.118035,-1.275977

72,"d2=",1.000001

"ATOM1=",1.113517,-1.809017,-1.275977

"ATOM2=",1.376383,-.999999,-1.801708

73,"d2=",1

"ATOM1=",.162461,2.118035,-1.275977

"ATOM2=",-.52573,1.618035,-1.801708

74,"d2=",1

"ATOM1=",.162461,-2.118035,-1.275977

"ATOM2=",-.52573,-1.618035,-1.801708

75,"d2=",1.000001

"ATOM1=",-1.376381,1.618035,-1.275977

"ATOM2=",-1.964166,.809017,-1.275977

76,"d2=",1

"ATOM1=",-1.376381,1.618035,-1.275977

"ATOM2=",-.52573,1.618035,-1.801708

77,"d2=",1.000001

"ATOM1=",-1.376381,-1.618035,-1.275977

"ATOM2=",-1.964166,-.809017,-1.275977

78,"d2=",1

"ATOM1=",-1.376381,-1.618035,-1.275977

"ATOM2=",-.52573,-1.618035,-1.801708

79,"d2=",.9999998

"ATOM1=",-1.964166,.809017,-1.275977

"ATOM2=",-1.701301,0,-1.801708

80,"d2=",.9999998

"ATOM1=",-1.964166,-.809017,-1.275977

"ATOM2=",-1.701301,0,-1.801708

81,"d2=",.9999996

"ATOM1=",1.376383,.999999,-1.801708

"ATOM2=",.688191,.5,-2.327438

82,"d2=",.9999996

"ATOM1=",1.376383,-.999999,-1.801708

"ATOM2=",.688191,-.5,-2.327438

83,"d2=",.9999999

"ATOM1=",-.52573,1.618035,-1.801708

"ATOM2=",-.262866,.809017,-2.327438

84,"d2=",.9999999

"ATOM1=",-.52573,-1.618035,-1.801708

"ATOM2=",-.262866,-.809017,-2.327438

85,"d2=",.9999987

"ATOM1=",-1.701301,0,-1.801708

"ATOM2=",-.850651,0,-2.327438

86,"d2=",1

"ATOM1=",.688191,.5,-2.327438

"ATOM2=",.688191,-.5,-2.327438

87,"d2=",1

"ATOM1=",.688191,.5,-2.327438

"ATOM2=",-.262866,.809017,-2.327438

88,"d2=",1

"ATOM1=",.688191,-.5,-2.327438

"ATOM2=",-.262866,-.809017,-2.327438

89,"d2=",.9999999

"ATOM1=",-.262866,.809017,-2.327438

"ATOM2=",-.850651,0,-2.327438

90,"d2=",.9999999

"ATOM1=",-.262866,-.809017,-2.327438

"ATOM2=",-.850651,0,-2.327438