■等面単体の体積(その308)

P0(2/√3,0,0,0,√(7/6),√(7/6))

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

P2((√3)/2,(√7)/2,(√14)/2,0,0,0)

P3(√3,√7,0,0,0,0)

P4(9/√12,(√7)/2,0,(√14)/2,0,0)

P5(√12,0,0,0,0,0)

P6(4/√3,0,0,0,√(14/3),0)

  P0P1=P1P2=P2P3=P3P4=P4P5=P5P6=√6

  P0P2=P1P3=P2P4=P3P5=P4P6=√10

  P0P3=P1P4=P2P5=P3P6=√12

  P0P4=P1P5=P2P6=√12

  P0P5=P1P6=√10

  P0P6=√6

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 P0との距離が最短なのはP1かP6であるが,ここではP1を外してみる.新たなP1(a,b,c,d,e,f)を

P1=P2+sP0P1=((√3)/2,(√7)/2,(√14)/2,0,0,0)+s(2/√3,0,0,0,√(7/6),√(7/6))

P1=P3+sP0P1=(√3,√7,0,0,0,0)+s(2/√3,0,0,√(7/6),√(7/6))

P1=P4+sP0P1=(9/√12,(√7)/2,0,(√14)/2,0,0)+s(2/√3,0,0,0,√(7/6),√(7/6))

P1=P5+sP0P1=(√12,0,0,0,0,0)+s(2/√3,0,0,0,√(7/6),√(7/6))

P1=P6+sP0P1=(4/√3,0,0,0,√(14/3),0)+s(2/√3,0,0,0,√(7/6),√(7/6))

とおいて,

(a−2/√3)^2+b^2+c^2+d^2+(e−√(7/6))^2+(f−√(7/6))^2=6

(a−√3)/2)^2+(b−(√7)/2)^2+(c−(√14)/2)+d^2+e^2+f^2=6

(a−√3)^2+(b−√7)^2+c^2+d^2+e^2+f^2=10

(a−9/√12)^2+(b−(√7)/2)^2+c^2+(d−(√14)/2)^2+e^2+f^2=12

(a−√12)^2+b^2+c^2+d^2+e^2+f^2=12

(a−4/√3)+b^2+c^2+d^2+(e−√(14/3))^2+f^2=10を満たすものを探す.

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