■DE群多面体の面数公式(その520)

 D群ではR^2=n/2となっている.D3〜D8が入れ子構造をなしていると仮定して計算してみたい.

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[1]D3

 R^2=1+1/3+a3^3=3/2

 a3^2=(9−8)/6=1/6

 R^2=1+b2^2+b3^2=3/2

 b3^2=1/2

 b2^2=(9−6−3)/6=0

[2]D4

 R^2=1+1/3+1/6+a4^2=4/2

 a4^2=(12−6−2−1)/6=3/6

 R^2=1+1/3+b3^2+b4^2=4/2

 b4^2=1/2

 b3^2=(12−6−2−3)/6=1/6

[3]D5

 R^2=1+1/3+1/6+1/10+a5^2=5/2

=1+1/3+1/6+2/4+b5^2

 R^2=9/6+2/4+b5^2=9/6+1/10+a5^2=5/2

 a5^2=(150−90−6)/60=54/60

 b5^2=(150−90−30)/60=30/60=(1/√2)^2

[4]D6

 R^2=1+1/3+1/6+1/10+1/15+a6^2=6/2

=1+1/3+1/6+1/10+b5^2+b6^2

 b6^2=1/2と考えられる.

 R^2=48/30+1/2+b6^2=48/30+1/15+a6^2=6/2

 a6^2=(90−48−2)/30=40/30

 b5^2=(90−48−15)/30=27/30

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