정이십면체 뫼비우스 변환군

개요

정이십면체의 대칭은 교대군 <math>A_5</math>
<math>G_{60}=\langle S,T|S^5=T^2=(TS)^3=1\rangle\subset \operatorname{PSL}(2,\mathbb{C})</math>

<math>S=\left( \begin{array}{cc} \zeta ^3 & 0 \\ 0 & \zeta ^2 \end{array} \right)</math> order 5

<math>\sqrt{5}T=\left( \begin{array}{cc} \zeta -\zeta ^4 & \zeta ^3-\zeta ^2 \\ \zeta ^3-\zeta ^2 & \zeta ^4-\zeta \end{array} \right)</math> order 2

<math>W=TS</math> : order 3

vertex points
- <math>V=F_1=z_1z_2(z_1^{10}+11z_1^5z_2^5-z_2^{10})</math>
face points
- <math>F=F_2=-(z_1^{20}+z_2^{20})+228(z_1^{15}z_2^{5}-z_1^{5}z_2^{15})-494z_1^{10}z_2^{10}</math>
edge points
- <math>E=F_3=(z_1^{30}+z_2^{30})+522(z_1^{25}z_2^{5}-z_1^{5}z_2^{25})-10005(z_1^{20}z_2^{10}+z_1^{10}z_2^{20})</math>
syzygy relation:<math>1728F_1^5-F_2^3-F_3^2=0</math> 또는 <math>1728V^5-E^2-F^3=0</math>
<math>F_2=HF_1</math>
<math>F_3=JF_1</math>