■高次元タイルの二面角(その4)

【1】空間充填2^n+2n胞体

  cosδ=−1/√n

  cosδ=−(n−2)/n

  n=4のとき両者は一致し,δ=2π/3

 n=5のとき

  cosδ1=−1/√5,δ1=116.565°

  cosδ2=−3/5,δ2=126.87°

  2δ1+δ2=360°

 念のため,解析的にも検してみると,

  2arccos(−1/√5)=2π−arccos(−3/5)

  2arccos(−1/√5)+arccos(−3/5)=2π

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【2】空間充填2(2^n−1)胞体

 n=4の場合

  cosδ01=−√(3/8),δ1=127.761

  cosδ02=−√(1/6),δ2=114.761

  cosδ03=−1/4,δ3=104.477

  cosδ12=−2/3,δ4=131.81

  cosδ13=−√(1/6),δ2=114.761

  cosδ23=−√(3/8),δ1=127.761

  2δ1+δ3=360°

  2δ2+δ4=360°

 念のため,解析的にも検してみると,

  2arccos(−√3/8)=2π−arccos(−1/4)

  2arccos(−√3/8)+arccos(−1/4)=2π

  2arccos(−√1/6)=2π−arccos(−2/3)

  2arccos(−√1/6)+arccos(−2/3)=2π

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