■サマーヴィルの等面四面体(その375)

△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.

===================================

  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)

F7は

  P1P2=P2P3=P3P4=P4P5=P5P6=P6P7=√7

  P1P3=P2P4=P3P5=P4P6=P5P7=√12

  P1P4=P2P5=P3P6=P4P7=√15

  P1P5=P2P6=P3P7=4

  P1P6=P2P7=√15

  P1P7=√12

5m^2+h^2=7

8m^2+4h^2=36h^2=12

9m^2+9h^2=5m^2+25h^2=15

8m^2+16h^2=16

8m^2=32h^2

m^2=4h^2

  P0P1^2=21h^2

  P0P2^2=45h^2

  P0P3^2=48h^2

  P0P4^2=45h^2

  P0P5^2=36h^2

  P0P6^2=21h^2

  P1P2^2=36h^2

  P1P3^2=45h^2

  P1P4^2=48h^2

  P1P5^2=45h^2

  P1P6^2=36h^2

  P2P3^2=21h^2

  P2P4^2=36h^2

  P2P5^2=45h^2

  P2P6^2=48h^2

  P3P4^2=21h^2

  P3P5^2=36h^2

  P3P6^2=45h^2

  P4P5^2=21h^2

  P4P6^2=36h^2

  P5P6^2=21h^2

となって,√7:√12:√15:4となった.

===================================