■等面単体の体積(その170)
▲=a^2b^2c^2{1−(a/2)^2}{1−(b/2)^2}{1−(c/2)^2}
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[左辺]
▲+{(1−c^2/2)−(1−a^2/2)(1−b^2/2)}{(1−b^2/2)−(1−c^2/2)(1−a^2/2)}{(1−a^2/2)−(1−b^2/2)(1−c^2/2)}
=▲+(1−c^2/2)(1−b^2/2)(1−a^2/2)−(1−c^2/2)^2(1−b^2/2)^2−(1−c^2/2)^2(1−a^2/2)^2+(1−c^2/2)^3(1−a^2/2)(1−b^2/2)−(1−a^2/2)^2(1−b^2/2)^2+(1−a^2/2)(1−b^2/2)^3(1−c^2/2)+(1−a^2/2)^3(1−b^2/2)(1−c^2/2)−(1−a^2/2)^2(1−b^2/2)^2(1−c^2/2)^2
[右辺]一方,
(1−c^2/2)^2c^2{1−(c/2)^2}+(1−b^2/2)^2b^2{1−(b/2)^2}+(1−a^2/2)^2a^2{1−(a/2)^2}+2(1−c^2/2)(1−a^2/2)(1−b^2/2){c^2{1−(c/2)^2}+b^2{1−(b/2)^2}+a^2{1−(a/2)^2}}−(1−a^2/2)^2(1−b^2/2)^2c^2{1−(c/2)^2}−(1−c^2/2)^2(1−a^2/2)^2b^2{1−(b/2)^2}−(1−b^2/2)^2(1−c^2/2)^2a^2{1−(a/2)^2}
=Σc^2{1−(c/2)^2}{(1−c^2/2)^2+2(1−c^2/2)(1−a^2/2)(1−b^2/2)−(1−a^2/2)^2(1−b^2/2)^2}
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うまく整理されないので,a^2=b^2=c^2=8/3とおいてみる.
[右辺]8{1−8/12}{(1−8/6)^2+2(1−8/6)^3−(1−8/6)^4}=−8/3{(1/3)^2−2(1/3)^3+(1/3)^4}=−8/3・(9−6+1)/3^4=−32/3^5
[左辺](8/3)^3(1−8/12)^3+{(1−8/6)−(1−8/6)^2}^3=(8/9)^3+(−1/3)^3(2/3)^3=(8^3−2^3)/9^3
どこかに誤りがあるようだが,これにて打ち切り.
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