■サマーヴィルの等面四面体(その340)

[1]G5

  Q1(0,0,0)

  Q2(9/5√2,√3,3/5√2)

  Q3(18/5√2,0,6/5√2)

Q1Q2^2=81/50+3+9/50=90/50+3=24/5

Q1Q3^2=324/50+36/50=360/50=36/5

Q2Q3^2=81/50+3+9/50=90/50+3=24/5

 2:2:√6で,△4の2次元面G4と一致した.

  (n^2−1)/n=24/5・・・(OK)

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[2]G6

 Q2(0,0,0,0)=Q1

 Q3(7√3/12,7√7/12,−5√14/12,0)

 Q4(14√3/12,2√7/12,−4√14/12,6√14/12)

 Q5(21√3/12,−3√7/12,−3√14/12,0)

Q2Q3^2=(147+343+350)/12^2=840/12^2

Q2Q4^2=(588+28+224+504)/12^2=1344/12^2

Q2Q5^2=(1323+63+126)/12^2=1512/12^2

Q3Q4^2=(147+175+14+504)/12^2=840^2

Q3Q5^2=(588+700+56)/12^2=1344/12^2

Q4Q5^2=(147+175+14+504)/12^2=840/12^2

840:1344:1512=5:8:9

 n=5のとき

  P0P1=P1P2=P2P3=P3P4=P4P5=√5

  P0P2=P1P3=P2P4=P3P5=√8

  P0P3=P1P4=P2P5=3

  P0P4=P1P5=√8

  P0P5=√5

に一致.

  (n^2−1)/n=35/6=840/12^2・・・(OK)

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[まとめ]Gnの断面の最短辺の長さはe^2=(n^2−1)/nで表される.

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