■サマーヴィルの等面四面体(その375)
△5の場合,
P0(√(1/2),0,√(1/2),1,√3)
P1(0,0,0,0,0)
P2(√2,√3,0,0,0)
P3(√8,0,0,0,0)
P4(√(9/2),0,√(9/2),0,0)
P5(√2,0,√2,2,0)
を
P0(m√(1/2),0,m√(1/2),m,m√3,h)
P1(0,0,0,0,0,0)
P2(0,0,0,0,0,6h)
P3(m√2,m√3,0,0,0,5h)
P4(m√8,0,0,0,0,4h)
P5(m√(9/2),0,m√(9/2),0,0,3h)
P6(m√2,0,m√2,2m,0,2h)
としてみる.
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P0P1^2=5m^2+h^2
P0P2^2=5m^2+25h^2
P0P3^2=8m^2+16h^2
P0P4^2=9m^2+9h^2
P0P5^2=8m^2+4h^2
P0P6^2=5m^2+h^2
P1P2^2=36h^2
P1P3^2=5m^2+25h^2
P1P4^2=8m^2+16h^2
P1P5^2=9m^2+9h^2
P1P6^2=8m^2+4h^2
P2P3^2=5m^2+h^2
P2P4^2=8m^2+4h^2
P2P5^2=9m^2+9h^2
P2P6^2=8m^2+16h^2
P3P4^2=5m^2+h^2
P3P5^2=8m^2+4h^2
P3P6^2=9m^2+9h^2
P4P5^2=5m^2+h^2
P4P6^2=8m^2+4h^2
P5P6^2=5m^2+h^2
5m^2+h^2(6)<5m^2+25h^2(2)
8m^2+4h^2(5)<8m^2+16h^2(3)
9m^2+9h^2(4)
36h^2(1)
F7は
P1P2=P2P3=P3P4=P4P5=P5P6=P6P7=√7
P1P3=P2P4=P3P5=P4P6=P5P7=√12
P1P4=P2P5=P3P6=P4P7=√15
P1P5=P2P6=P3P7=4
P1P6=P2P7=√15
P1P7=√12
5m^2+h^2=7
8m^2+4h^2=36h^2=12
9m^2+9h^2=5m^2+25h^2=15
8m^2+16h^2=16
8m^2=32h^2
m^2=4h^2
P0P1^2=21h^2
P0P2^2=45h^2
P0P3^2=48h^2
P0P4^2=45h^2
P0P5^2=36h^2
P0P6^2=21h^2
P1P2^2=36h^2
P1P3^2=45h^2
P1P4^2=48h^2
P1P5^2=45h^2
P1P6^2=36h^2
P2P3^2=21h^2
P2P4^2=36h^2
P2P5^2=45h^2
P2P6^2=48h^2
P3P4^2=21h^2
P3P5^2=36h^2
P3P6^2=45h^2
P4P5^2=21h^2
P4P6^2=36h^2
P5P6^2=21h^2
となって,√7:√12:√15:4となった.
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