■DE群多面体の面数公式(その375)

[1]221

 221の頂点間距離が2のとき,半径は√(8/3)

 R^2=1+1/3+1/6+1/10+1/15+a6^2=8/3

=1+1/3+1/6+1/10+2/5+b6^2

 1+1/3+1/6+1/10=(30+10+5+3)/30=8/5

 R^2=8/5+2/5+b6^2=8/5+1/15+a6^2=8/3

 a6^2=(40−24−1)/15=5/3

 b6^2=(40−24−6)/15=2/3

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[2]321

 頂点間距離が2のとき,半径は√3

 R^2=1+1/3+1/6+1/10+1/15+1/21+a7^2=3

=1+1/3+1/6+1/10+1/15+2/6+b7^2

 1+1/3+1/6+1/10+1/15=(30+10+5+3+2)/=5/3

 R^2=5/3+1/3+b7^2=5/3+1/21+a7^2=3

 a7^2=(63−35−1)/21=9/7

 b7^2=(9−5−1)/3=1

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[3]421

 頂点間距離が2のとき,半径は2

 R^2=1+1/3+1/6+1/10+1/15+1/21+1/28+a8^2=4

=1+1/3+1/6+1/10+1/15+1/21+2/7+b8^2

 R^2=12/7+2/7+b8^2=12/7+1/28+a8^2=4

 a8^2=(112−48−1)/28=9/4≧2

 b8^2=(28−12−2)/7=2≧2

 R^2=7/4+a8^2=2+b8^2

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 もしこれ以上を考えるのであれば,

 R^2=1+1/3+1/6+1/10+1/15+1/21+1/28+1/36+a9^2

=1+1/3+1/6+1/10+1/15+1/21+1/28+2/8+b9^2

1+1/3+1/6+1/10+1/15+1/21+1/28

=Σ2/j(j+1)

=2(1−1/8)=7/4

1+1/3+1/6+1/10+1/15+1/21+1/28+1/36+=2(1−1/9)=16/9

1+1/3+1/6+1/10+1/15+1/21+1/28+2/8=2=7/4+1/4=2≧2

 R^2=16/9+a9^2=2+b9^2

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