■サマーヴィルの等面四面体(その844)
α4について
P0(0,0,0,0)
P1(2,0,0,0)
P2(1,√3,0,0)
P3(1,√(1/3),√(8/3),0)
P4(x,y,z,w)
とする.
x=1
y^2+z^2+w^2=3
(y−√3)^2+z^2+w^2=4
(y−√(1/3))^2+(z−√(8/3))^2+w^2=4
(y−√3)^2+3−y^2=4
−2y√3+3+3=4→y=1/√3
z^2+w^2=3−1/3=8/3
(y−√(1/3))^2+(z−√(8/3))^2+w^2=4
(z−√(8/3))^2+w^2=4
z^2−2z√(8/3)+8/3+w^2=4
−2z√(8/3)+8/3+8/3=4
16/3−4=2z√(8/3)
2/3=z√(8/3)
z=2/3√(3/8)=1/√6
w^2=8/3−1/6=5/2
P4(1,1/√3,1/√6,√(5/2))
1+1/3+1/6+5/2=4
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