■等面単体の体積(その388)

 n=5の場合,

  P0(1/2,(√5)/2,0,(√10)/2)

  P1(0,0,0,0)

  P2(2,0,0,0)

  P3(3/2,(√5)/2,(√10)/2,0)

  P4(1,√5,0,0)

  P0(m/2,m√5/2,0,m√10/2,h)

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

  P2(0,0,0,0,5h)

  P3(2m,0,0,0,4h)

  P4(3m/2,m√5/2,m√10/2,0,3h)

  P5(m,m√5,0,0,2h)

としてみる.

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

  P0P1^2=4m^2+h^2

  P0P2^2=4m^2+16h^2

  P0P3^2=6m^2+9h^2

  P0P4^2=6m^2+4h^2

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

  P1P2^2=25h^2

  P1P3^2=4m^2+16h^2

  P1P4^2=6m^2+9h^2

  P1P5^2=6m^2+4h^2

  P2P3^2=4m^2+h^2

  P2P4^2=6m^2+4h^2

  P2P5^2=6m^2+9h^2

  P3P4^2=4m^2+h^2

  P3P5^2=6m^2+4h^2

  P4P5^2=4m^2+h^2

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

6m^2+4h^2(4)<6m^2+9h^2(3)

25h^2(1)

  P0P1=P1P2=P2P3=P3P4=P4P5=√5

  P0P2=P1P3=P2P4=P3P5=√8

  P0P3=P1P4=P2P5=3

  P0P4=P1P5=√8

  P0P5=√5

4m^2+h^2=25h^2=5,h^2=1/5,m^2=6/5

6m^2+4h^2=8

6m^2+9h^2=9

4m^2+16h^2=8

h^2=1/5,m^2=6/5はこれらを満たす.

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