■DE群多面体の面数公式(その521)

 E5=D5より

 R^2=1+c2^2+c3^2=7/3

 R^2=1+1/3+c3^2+c4^2=12/5

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

 R^2=1+1/3+1/6+1/10+c5^2+c6^2=8/3

 R^2=1+1/3+1/6+1/10+1/15+c6^2+c7^2=3

 R^2=1+1/3+1/6+1/10+1/15+1/21+c7^2+c8^2=4

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 E3〜E8が入れ子構造をなしていると仮定して計算してみたい.

 R^2=1+1/2+c3^2=7/3→c3^2=5/6

 R^2=1+1/3+1/2+c4^2=12/5

   =2{1−1/3)+1/2+c4^2=12/5

 c4^2=12/5−4/3−1/2=(72−40−15)/30=17/30

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

   =2(1−1/4)+1/2+1/2=5/2

 c5^2=5/2−3/2−1/2=1/2

 R^2=1+1/3+1/6+1/10+1/2+c6^2=8/3

   =2(1−1/5)+1/2+c6^2=8/3

 c6^2=8/3−8/5−1/2=(80−48−15)/30=17/30

 R^2=1+1/3+1/6+1/10+1/15+1/2+c7^2=3

   =2(1−1/6)+1/2+c7^2=3

 c7^2=3−10/6−1/2=(18−10−3)/6=5/6

 R^2=1+1/3+1/6+1/10+1/15+1/21+1/2+c8^2=4

   =2(1−1/7)+1/2+c8^2=4

 c7^2=4−12/7−1/2=(56−24−7)/14=25/14

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

   =2(1−1/4)+1/2+1/2=5/2

 c5^2=5/2−3/2−1/2=1/2

について対称な形になっている.偶然の一致とは考えにくいが・・・

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