■サマーヴィルの等面四面体(その532)

n=6の場合

P0(m√(1/2),0,m√(1/2),m,m√3,h)

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

P2(0,0,0,0,0,6h)

P3(m√2,m√3,0,0,0,5h)

P4(m√8,0,0,0,0,4h)

P5(m√(9/2),0,m√(9/2),0,0,3h)

P6(m√2,0,m√2,2m,0,2h)

としてみる.

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  P0P1^2=5m^2+h^2

  P0P2^2=5m^2+25h^2

  P0P3^2=8m^2+16h^2

  P0P4^2=9m^2+9h^2

  P0P5^2=8m^2+4h^2

  P0P6^2=5m^2+h^2

  P1P2^2=36h^2

  P1P3^2=5m^2+25h^2

  P1P4^2=8m^2+16h^2

  P1P5^2=9m^2+9h^2

  P1P6^2=8m^2+4h^2

  P2P3^2=5m^2+h^2

  P2P4^2=8m^2+4h^2

  P2P5^2=9m^2+9h^2

  P2P6^2=8m^2+16h^2

  P3P4^2=5m^2+h^2

  P3P5^2=8m^2+4h^2

  P3P6^2=9m^2+9h^2

  P4P5^2=5m^2+h^2

  P4P6^2=8m^2+4h^2

  P5P6^2=5m^2+h^2

h^2=1/6,m^2=7/6は

  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

これらを満たす.

  h^2=1/6,6h=√6  (△6の最短辺)

  m^2の値からスケーリングができる.

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