■プラトン立体の二面角(その2)

 正多面体の二面角δは

  {3,3}→cosδ=1/3

  {3,4}→cosδ=−1/3

  {3,5}→cosδ=−√5/3

  {4,3}→cosδ=0

  {5,3}→cosδ=−√5/5

と計算される.

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メッサーの式

sin(δ/2)=cos(π/q)/sin(π/p)

{(1-cosδ)/2}^1/2=cos(π/q)/sin(π/p)

と比較してみたい.

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  {3,3}→1/3=(1-cosδ)/2→cosδ=1/3

  {3,4}→2/3=(1-cosδ)/2→cosδ=−1/3

  {3,5}→τ^2/3=(1-cosδ)/2→cosδ=−√5/3

  {4,3}→1/2=(1-cosδ)/2→cosδ=0

  {5,3}→4/(10-2√5)=(1-cosδ)/2→cosδ=−√5/5

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半稜線に対する中心角φ

cosφ=cos(π/p)/sin(π/q)

ε=2φ

ε+双対多面体の二面角=πが成り立つ

  {3,3}→2arccos1/√3=arccos(-1/3)

  {3,4}→2arccos1/√2=arccos0

  {3,5}→2arccos√(5+√5)/10)=arccos(√5/5)=arctan2

  {4,3}→arccos(1/3)

  {5,3}→2arccos(τ^2/√3)=arcsin(2/3)

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