■サマーヴィルの等面四面体(その160)

  aj=√(1/2j(j+1))

6次元の場合,

P0(-a1,-a2,-a3,-a4,-a5,-a6)

P1(+a1,-a2,-a3,-a4,-a5,-a6)

P2( 0,+2a2,-a3,-a4,-a5,-a6)

P3( 0,  0,+3a3,-a4,-a5,-a6)

P4( 0,  0, 0,+4a4,-a5,-a6)

P5( 0,  0, 0,  0,+5a5,-a6)

P6( 0,  0, 0,  0,  0,+6a6)

対称超平面x=0での中心断面は

Q1( 0,-a2,-a3,-a4,-a5,-a6)

Q2( 0,+2a2,-a3,-a4,-a5,-a6)

Q3( 0,  0,+3a3,-a4,-a5,-a6)

Q4( 0,  0, 0,+4a4,-a5,-a6)

Q5( 0,  0, 0,  0,+5a5,-a6)

Q6( 0,  0, 0,  0,  0,+6a6)

Q1Q2^2=9a2^2=90/120

Q1Q3^2=a2^2+163^2=90/120

Q1Q4^2=a2^2+a3^2+25a4^2=90/120

Q1Q5^2=a2^2+a3^2+a4^2+36a5^2=180/240

Q1Q6^2=a2^2+a3^2+a4^2+a5^2+49a6^2=1/12+1/24+1/40+1/60+49/84=(140+70+42+28+980)/1680=1260/1680

Q2Q3^2=4a2^2+16a3^2=120/120

Q2Q4^2=4a2^2+a3^2+25a4^2=120/120

Q2Q5^2=4a2^2+a3^2+a4^2+36a5^2=240/240

Q2Q6^2=4a2^2+a3^2+a4^2+a5^2+49a6^2=(560+70+42+28+980)/1680=1680/1680

Q3Q4^2=9a3^2+25a4^2=120/120

Q3Q5^2=9a3^2+a4^2+36a5^2=240/240

Q3Q6^2=9a3^2+a4^2+a5^2+49a6^2=9/24+1/40+1/60+49/84=(630+42+28+960)/1680=1680/1680

Q4Q5^2=16a4^2+36a5^2=240/240

Q4Q6^2=16a4^2+a5^2+49a6^2=16/40+1/60+49/84=(672+28+980)1680/1680

Q5Q6^2=25a5^2+49a6^2=25/60+49/84=(700+980)/1680=1680/1680

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