■サマーヴィルの等面四面体(その184)

  P1P2=P2P3=P3P4=P4P5=√6

  P1P3=P2P4=P3P5=√10

  P1P4=P2P5=√12

  P1P5=√12

 最短辺の

  P1P2=P2P3=P3P4=P4P5=√6

だけを検討したが,伸長する方向はそれ以外にあるのだろうか?

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 P1から,P2P4,P3P5,P2P5方向に伸長させた点をP0とする.

[1]P1+P2P4方向(6/2√3,0,−√14/2,√14/2)

P0(6/2√3,0,−√14/2,√14/2)

P1(0,0,0,0)

P2(3/2√3,(√7)/2,(√14)/2,0)

P3(6/2√3,√7,0,0)

P4(9/2√3,(√7)/2,0,(√14)/2)

P5(12/2√3,0,0,0)

  P1P0^2=36/12+14/4+14/4=10

  P2P0^2=9/12+7/4+14  (NG)

[2]P1−P2P4方向(−6/2√3,0,√14/2,−√14/2)

P0(−6/2√3,0,√14/2,−√14/2)

P1(0,0,0,0)

P2(3/2√3,(√7)/2,(√14)/2,0)

P3(6/2√3,√7,0,0)

P4(9/2√3,(√7)/2,0,(√14)/2)

P5(12/2√3,0,0,0)

  P1P0^2=36/12+14/4+14/4=10

  P2P0^2=81/12+7/4+14/4=12

  P3P0^2=144/12+7+14/4+14/4  (NG)

[3]P1+P3P5方向(6/2√3,−√7,0,0)

P0(6/2√3,−√7,0,0)

P1(0,0,0,0)

P2(3/2√3,(√7)/2,(√14)/2,0)

P3(6/2√3,√7,0,0)

P4(9/2√3,(√7)/2,0,(√14)/2)

P5(12/2√3,0,0,0)

  P1P0^2=36/12+7=10

  P2P0^2=9/12+7/4+14/4=6

  P3P0^2=28  (NG)

[4]P1−P3P5方向(−6/2√3,√7,0,0)

P0(−6/2√3,√7,0,0)

P1(0,0,0,0)

P2(3/2√3,(√7)/2,(√14)/2,0)

P3(6/2√3,√7,0,0)

P4(9/2√3,(√7)/2,0,(√14)/2)

P5(12/2√3,0,0,0)

  P1P0^2=36/12+7=10

  P2P0^2=81/12+7/4+14/4=12

  P3P0^2=144/12=12

  P3P0^2=225/12+7/4+14/4  (NG)

[5]P1+P2P5方向(9/2√3,−√7/2,−√14/2,0)

P0(9/2√3,−√7/2,−√14/2,0)

P1(0,0,0,0)

P2(3/2√3,(√7)/2,(√14)/2,0)

P3(6/2√3,√7,0,0)

P4(9/2√3,(√7)/2,0,(√14)/2)

P5(12/2√3,0,0,0)

  P1P0^2=81/12+7/2+14/2=12

  P2P0^2=36/12+7+14  (NG)

[6]P1−P2P5方向(−9/2√3,√7/2,√14/2,0)

P0(−9/2√3,√7/2,√14/2,0)

P1(0,0,0,0)

P2(3/2√3,(√7)/2,(√14)/2,0)

P3(6/2√3,√7,0,0)

P4(9/2√3,(√7)/2,0,(√14)/2)

P5(12/2√3,0,0,0)

  P1P0^2=81/12+7/2+14/2=12

  P2P0^2=144/12=12

  P3P0^2=225/12+7/4+14/4  (NG)

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