■多面体のコマ(その4)

 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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