■基本単体の二面角(その273)

 有限群

 αn:aj=(2/j(j+1))^1/2

 βn:bj=(2/j(j+1))^1/2,bn=√(2/n)

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[1]An

 正単体において,R=nrはよく知られている.

 (R/ρ)^2=n^2

 R^2=1+1/3+・・・+2/n(n−1)+an^2=2−2/n+an^2

 ρ^2=an^2

 (2−2/n+an^2)/an^2=n^2→an^2=2/n(n+1)  (OK)

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

[2]BCn

 R=r√nより

 (R/ρ)^2=n

 R^2=1+1/3+・・・+2/n(n−1)+an^2=2−2/n+an^2

 ρ^2=an^2

 (2−2/n+an^2)/an^2=n→an^2=2/n  (OK)

 R^2=1+1+・・・+1+an^2=(n−1)+an^2

 ρ^2=an^2

 (n−1+an^2)/an^2=n→an^2=1  (OK)

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