■格子のボロノイ細胞(その104)

【6】E8格子の場合

  P0(0,0,0,0,0,0,0,0)

  P1(1,0,0,0,0,0,0,0)

  P2(1,1/√3,0,0,0,0,0,0)

  P3(1,1/√3,1/√6,0,0,0,0,0)

  P4(1,1/√3,1/√6,1/√10,0,0,0,0)

  P5(1,1/√3,1/√6,1/√10,1/√15,0,0,0)

  P6(1,1/√3,1/√6,1/√10,1/√15,1/√12,0,0)

  P7(1,1/√3,1/√6,1/√10,1/√15,1/√12,1/2,0)

  P8(1,1/√3,1/√6,1/√10,1/√15,0,0,1/3)

超平面をax+by+cz+dw+ev+fu+gt+is=jとする.

[1]P1P2P3P4P5P6P7P8を通る超平面:x=1

[2]P0P2P3P4P5P6P7P8を通る超平面

  j=0

  a+b/√3=0,a=1,b=−√3

  1−1+c/√6=0,c=0

  1−1+c/√6+d/√6=0,d=0,e=0,f=0

g=0,h=0,i=0

[3]P0P1P3P4P5P6P7P8を通る超平面

  j=0,a=0

  b/√3+c/√6=0,b=1,c=−√2

  b/√3+c/√6+d/√6=0,d=0,e=0,f=0

g=0,h=0,i=0

[4]P0P1P2P4P5P6P7P8を通る超平面

  j=0,a=0,b=0

  c/√6+d/√10=0,c=√3,d=−√5

  e=0,f=0,g=0,h=0,i=0

[5]P0P1P2P3P5P6P7P8を通る超平面

  j=0,a=0,b=0,c=0

  d/√10+e/√15=0,d=√2,e=−√3

  f=0,g=0,h=0,i=0

[6]P0P1P2P3P4P6P7P8を通る超平面

  j=0,a=0,b=0,c=0,d=0

e/√15+f/√12=0,e=√5,f=−2

  g=0,h=0

e/√15+i/3=0,i=−3e/√15=−√3

[7]P0P1P2P3P4P5P7P8を通る超平面

  j=0,a=0,b=0,c=0,d=0,e=0

f/√12+g/2=0,f=√3,g=−1

  h=0,i=0

[8]P0P1P2P3P4P5P6P8を通る超平面:t=0

[9]P0P1P2P3P4P5P6P7を通る超平面:s=0

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  a=(1,0,0,0,0,0,0,0)

  b=(1,−√3,0,0,0,0,0,0)

  c=(0,1,−√2,0,0,0,0,0)

  d=(0,0,√3,−√5,0,0,0,0)

  e=(0,0,0,√2,−√3,0,0,0)

  f=(0,0,0,0,√5,−2,0,−√3)

  g=(0,0,0,0,0,√3,−1,0)

  h=(0,0,0,0,0,0,1,0)

  i=(0,0,0,0,0,0,0,1)

を正規化すると

  a=(1,0,0,0,0,0,0,0)

  b=(1/2,−√3/2,0,0,0,0,0,0)

  c=(0,1/√3,−√(2/3),0,0,0,0,0)

  d=(0,0,√(3/8),−√(5/8),0,0,0,0)

  e=(0,0,0,√(2/5),−√(3/5),0,0,0)

  f=(0,0,0,0,√(5/12),−1/√3,0,−1/2)

  g=(0,0,0,0,0,√3/2,−1/2,0)

  h=(0,0,0,0,0,0,1,0)

  i=(0,0,0,0,0,0,0,1)

a・b=1/2

b・c=−1/2

c・d=−1/2

d・e=−1/2

d・h=−1/2

e・f=−1/2

f・g=−1/2

f・i=−1/2

g・h=−1/2

h・i=0

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