■サマーヴィルの等面四面体(その384)

△5の場合,

P0(√(1/2),0,√(1/2),1,√3)

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

P2(√2,√3,0,0,0)

P3(√8,0,0,0,0)

P4(√(9/2),0,√(9/2),0,0)

P5(√2,0,√2,2,0)

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

5m^2+h^2(6)<5m^2+25h^2(2)

8m^2+4h^2(5)<8m^2+16h^2(3)

9m^2+9h^2(4)

36h^2(1)

H9は

  P3P4=P4P5=P5P6=P6P7=P7P8=P8P9=√9

  P3P5=P4P6=P5P7=P6P8=P7P9=√16

  P3P6=P4P7=P5P8=P6P9=√21

  P3P7=P4P8=P5P9=√24

  P3P8=P4P9=√25

  P3P9=√24

5m^2+h^2=9

8m^2+4h^2=16

9m^2+9h^2=21

36h^2=24

5m^2+25h^2=25

8m^2+16h^2=24

h^2=2/3,m^2=5/3

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