■サマーヴィルの等面四面体(その434)

  1−cp/1−cq/1−cr/1−cs

では

=1−cp/1−cq/(1−cr/(1−cs))

=1−cp/1−cq(1−cs)/(1−cs−cr)

=1−cp(1−cs−cr)/{(1−cs−cr)−cq(1−cs)}

={{(1−cs−cr)−cq(1−cs)}−cp(1−cs−cr)}/{(1−cs−cr)−cq(1−cs)}

 cp=1/4,cq=1/4,cr=1/4とおくと

={{16−16cs−4−(4−4cs))}−(4−4cs−1)}/{(16−16cs−4)−(4−4cs)}

=(5−8cs)/(8−12cs0

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 cs=cos^2π/s=5/8のとき,

  1−cp/1−cq/1−cr/1−cs=0

  π/s=arccos(√(5/8))

  s=π/arccos(√(5/8))=4.76679

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