■サマーヴィルの等面四面体(その533)

 A=a^2,B=b^2,C=c^2,・・・

 以下の付帯条件をもっと一般化できるだろうか?

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[1]3次元の場合,3A−3B+C=0

3A−3B<0

  P0P1=P1P2=P2P3=a

  P0P2=P1P3=b

  P0P3=c

[2]4次元の場合,4A−6B+4C−D=0

4A−6B+4C>0

  P0P1=P1P2=P2P3=P3P4=a

  P0P2=P1P3=P2P4=b

  P0P3=P1P4=c

  P0P4=d

[3]5次元の場合,5A−10B+10C−5D+E=0

5A−10B+10C−5D<0

  P0P1=P1P2=P2P3=P3P4=P4P5=a

  P0P2=P1P3=P2P4=P3P5=b

  P0P3=P1P4=P2P5=c

  P0P4=P1P5=d

  P0P5=e

[4]6次元の場合,6A−15B+20C−15D+6E−F=0

6A−15B+20C−15D+6E>0

  P0P1=P1P2=P2P3=P3P4=P4P5=P5P6=a

  P0P2=P1P3=P2P4=P3P5=P4P6=b

  P0P3=P1P4=P2P5=P3P6=c

  P0P4=P1P5=P2P6=d

  P0P5=P1P6=e

  P0P6=f

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