■格子のボロノイ細胞(その2)

【1】E8無限鏡映群の基本単体の頂点座標

  (0,0,0,0,0,0,0,0)

  (0,0,0,0,0,0,1/2,1/2)

  (0,0,0,0,0,1/3,1/3,2/3)

  (0,0,0,0,1/4,1/4,1/4,3/4)

  (0,0,0,1/5,1/5,1/5,1/5,4/5)

  (0,0,1/6,1/6,1/6,1/6,1/6,5/6)

  (1/6,1/6,1/6,1/6,1/6,1/6,1/6,5/6)

  (−1/8,1/8,1/8,1/8,1/8,1/8,1/8,7/8)

  (0,0,0,0,0,0,0,1)

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【2】E8のボロノイ細胞

[1]x7+x8=1

[2]x1+x8=x2+x3+・・・+x7

[3]x1=x2

[4]x2=x3

[5]x3=x4

[6]x4=x5

[7]x5=x6

[8]x6=x7

[9]x1=−x2

[2]を外すと

 x1=x2=x3=x4=x5=x6=x7=0

 x8=1

[3]を外すと

 x1=1/8

 x2=x3=x4=x5=x6=x7=1/8

 x8=7/8

[4]を外すと

 x1=x2=0

 x3=x4=x5=x6=x7=1/6

 x8=5/6

[5]を外すと

 x1=x2=x3=0

 x4=x5=x6=x7=1/5

 x8=4/5

[6]を外すと

 x1=x2=x3=x4=0

 x5=x6=x7=1/4

 x8=3/4

[7]を外すと

 x1=x2=x3=x4=x5=0

 x6=x7=1/3

 x8=2/3

[8]を外すと

 x1=x2=x3=x4=x5=x6=0

 x7=1/2,x8=1/2

[9]を外すと

 x1=x2=x3=x4=x5=x6=x7=1/6

 x8=5/6

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【3】E8~について(521)

  (0,0,0,0,0,0,0,0)

  (1,0,0,0,0,0,0,0)

  (1,1/√3,0,0,0,0,0,0)

  (1,1/√3,1/√6,0,0,0,0,0)

  (1,1/√3,1/√6,1/√10,0,0,0,0)

  (1,1/√3,1/√6,1/√10,1/√15,0,0,0)

  (1,1/√3,1/√6,1/√10,1/√15,1/√12,0,0)

  (1,1/√3,1/√6,1/√10,1/√15,1/√12,1/2,0)

  (1,1/√3,1/√6,1/√10,1/√15,0,0,1/3)

>のすべての辺の長さが「SPLAG」第21章に一致することが確認された.

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