■等面単体の体積(その276)
n=4
P1P2=P2P3=P3P4=2
P1P3=P2P4=√6
P1P4=√6
P1(0,0,0,0)
P2(2,0,0,0)
P4(1,√5,0,0)
はこれを満たす.
P3(x,y,z,0)
とおくと,
x^2+y^2+z^2=6
(x−2)^2+y^2+z^2=4
(x−1)^2+(y−√5)^2+z^2=4
(x−2)^2+6−x^2=4
−4x+4+6=4,x=3/2
1/4+y^2+z^2=4
1/4+(y−√5)^2+z^2=4,y=(√5)/2,z=(√10)/2
P1(0,0,0,0)
P2(2,0,0,0)
P3(3/2,(√5)/2,(√10)/2,0)
P4(1,√5,0,0)
===================================
P0(x,y,z,w)
とおくと,
x^2+y^2+z^2+w^2=4
(x−2)^2+y^2+z^2+w^2=6
(x−3/2)^2+(y−(√5)/2)^2+(z−(√10)/2)^2+w^2=6
(x−1)^2+(y−√5)^2+z^2+w^2=4
(x−2)^2+4−x^2=6
−4x+4+4=6,x=1/2
y^2+z^2+w^2=4−1/4=15/4
1/4+(y−√5)^2+15/4−y^2=4
−2√5y+5=0,y=(√5)/2
z^2+w^2=10/4
1+0+(z−(√10)/2)^2+10/4−z^2=6
1−(√10)z+10/4+10/4=6,z=0,(√10)/2
P0(1/2,(√5)/2,0,(√10)/2)
===================================