■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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