■可積分系とテータ関数(その20)
順番が逆になったが,計算の概要を示したい.
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vol(P)=ΣNjhj/n・Vn-1(j)
U(P)=ΣNjhj/(n+2)・{hj^2Vn-1(j)+Un-1(j)}
G(P)=1/n・U(P)/vol(P)^(1+2/n)
=1/n・I(P)/vol(P)^(2/n) (無次元化慣性
[1]切頂八面体
正方形面:N1=6
六角形面:N2=8
辺長:2l
vol(P)=6l√8/3・4l^2+8l√6/3・6√3l^2=64√2l^3
U(P)=6l√8/3・{8l^2・4l^2+8l^4/3}+8l√6/3・{6l^2・6√3l^2+10√3l^4}=304√2l^5
G(P)=1/3・I(P)/vol(P)^(2/3)
=19/192(2)^1/3
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
[2]正p角形
vol(P)=pl^2cot(π/p)
I(P)=l^2/6・{1+3cot^2(π/p)}
G(P)=1/6p・I(cosec(2π/p)+cot(π/p))
p=3→G(P)=1/(6√3)
p=4→G(P)=1/12=0.0833333
p=6→G(P)=5/(36√3)
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
[3]正20面体
vol(P)=20l^3τ^2/3
G(P)=1/20・(6τ/5)^2/3=0.0778185
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
[4]正12面体
vol(P)=4√5l^3τ^4
G(P)=(11τ+17)/300・(2/τ√5)^2/3=0.0781285
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
[5]正24胞体
vol(P)=32l^4
G(P)=13/120√2=0.0766032
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
[6]正120胞体
vol(P)=120√5l^4τ^8
G(P)=(43τ+13)/300√6(5)^1/4=0.0751470
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
[7]正600胞体
vol(P)=100l^4τ^3
G(P)=(3τ+4)τ^1/2/150=0.0750839
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