■等面単体の体積(その48)

 n=7のとき

  P0P1=P1P2=P2P3=P3P4=P4P5=P5P6=P6P7=√7

  P0P2=P1P3=P2P4=P3P5=P4P6=P5P7=√12

  P0P3=P1P4=P2P5=P3P6=P4P7=√15

  P0P4=P1P5=P2P6=P3P7=4

  P0P5=P1P6=P2P7=√15

  P0P6=P1P7=√12

  P0P7=√7

 基本単体の辺長は1,√2,√3,2,√5,√6,√7である.√12=2√3はよいが,√15はこれらの整数倍にはならない.

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 この空間充填等面単体は7次元直方体に内接する.等面体を直方体(a,b,c,d,e,f,g)に内接させる.

  a^2+b^2=7

  b^2+c^2=12

  c^2+d^2=15

  d^2+e^2=16

  e^2+f^2=15

  f^2+g^2=12

  g^2+a^2=7

  a^2=1,b^2=6,c^2=6,

  d^2=9,e^2=7,f^2=8,g^2=4→a^2=3 (NG)

  a^2=2,b^2=5,c^2=7,

  d^2=8,e^2=8,f^2=7,g^2=5→a^2=2 (OK)

  a^2=3,b^2=4,c^2=8,

  d^2=7,e^2=9,f^2=6,g^2=6→a^2=1 (NG)

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