■サマーヴィルの等面四面体(その844)

 α4について

  P0(0,0,0,0)

  P1(2,0,0,0)

  P2(1,√3,0,0)

  P3(1,√(1/3),√(8/3),0)

  P4(x,y,z,w)

とする.

 x=1

 y^2+z^2+w^2=3

 (y−√3)^2+z^2+w^2=4

 (y−√(1/3))^2+(z−√(8/3))^2+w^2=4

 (y−√3)^2+3−y^2=4

 −2y√3+3+3=4→y=1/√3

 z^2+w^2=3−1/3=8/3

 (y−√(1/3))^2+(z−√(8/3))^2+w^2=4

 (z−√(8/3))^2+w^2=4

 z^2−2z√(8/3)+8/3+w^2=4

 −2z√(8/3)+8/3+8/3=4

 16/3−4=2z√(8/3)

 2/3=z√(8/3)

 z=2/3√(3/8)=1/√6

 w^2=8/3−1/6=5/2

 P4(1,1/√3,1/√6,√(5/2))

  1+1/3+1/6+5/2=4

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