D Canonical-Basis Relationships
217
Fig. D.2 Icosahedron with
x,y,z coordinates in D 2
setting
|T 2a =
1
√
3
|T 2ξ +|T 2η +|T 2ζ
= d z ′2
|T 2θ =
1
√
6
|T 2ξ +|T 2η −2|T 2ζ
=
1
√
3
(
√
2d x ′2 −y ′2 − d y ′ z ′ )
|T 2ǫ =
1
√
2
|T 2η −|T 2ξ
=
1
√
3
(−
√
2d x ′ y ′ + d x ′ z ′ )
O (D 3 basis)
D(C
z ′
3 )
D(C x ′
2 )
|E θ , |E ǫ
⎛
⎝
−
1
2
−
√
3
2
+
√
3
2
−
1
2
⎞
⎠
10
0 −1
|T 1a , |T 1θ , |T 1ǫ
⎛
⎜
⎝
10
0
0 −
1
2
−
√
3
2
0 +
√
3
2
−
1
2
⎞
⎟
⎠
⎛
⎝
−10 0
010
00 −1
⎞
⎠
|T 2a , |T 2θ , |T 2ǫ
⎛
⎜
⎝
10
0
0 −
1
2
−
√
3
2
0 +
√
3
2
−
1
2
⎞
⎟
⎠
⎛
⎝
10 0
01 0
00−1
⎞
⎠