■ガウスの問題とデーンの定理(その10)
空間充填2(2^n−1)胞体の二胞角は
cosδ=−{(j+1)/(k+1)・(n−k)/(n−j)}^1/2
で与えられる.
n=6の場合
cosδ01=−{1/2・5/6}^1/2=−√(5/12)
cosδ02=−{1/3・4/6}^1/2=−√(2/9)
cosδ03=−{1/4・3/6}^1/2=−√(1/8)
cosδ04=−{1/5・2/6}^1/2=−√(1/15)
cosδ05=−{1/6・1/6}^1/2=−1/6
cosδ12=−{2/3・4/5}^1/2=−√(8/15)
cosδ13=−{2/4・3/5}^1/2=−√(3/10)
cosδ14=−{2/5・2/5}^1/2=−2/5
cosδ15=−{2/6・1/5}^1/2=−√(1/15)
cosδ23=−{3/4・3/4}^1/2=−3/4
cosδ24=−{3/5・2/4}^1/2=−√(3/10)
cosδ25=−{3/6・1/4}^1/2=−√(1/8)
cosδ34=−{4/5・2/3}^1/2=−√(8/15)
cosδ35=−{4/6・1/3}^1/2=−√(2/9)
cosδ45=−{5/6・1/2}^1/2=−√(5/12)
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cosδ01=−√(5/12),δ1=130.203
cosδ02=−√(2/9),δ2=118.125
cosδ03=−√(1/8),δ3=110.705
cosδ04=−√(1/15),δ4=104.963
cosδ05=−1/6,δ5=99.594
cosδ12=−√(8/15),δ6=136.911
cosδ13=−√(3/10),δ7=123.211
cosδ14=−−2/5,δ8=113.578
cosδ15=−√(1/15),δ4=104.963
cosδ23=−3/4,δ9=138.59
cosδ24=−√(3/10),δ7=123.211
cosδ25=−√(1/8),δ3=110.705
cosδ34=−√(8/15),δ6=136.911
cosδ35=−√(2/9),δ2=118.123
cosδ45=−√(5/12),δ1=130.203
2δ1+δ5=360°
2δ3+δ9=360°
2δ7+δ8=360°
δ2+δ4+δ6=360°
念のため,解析的にも検してみると,
arccos(√2/9)+arccos(√1/15)
=arccos(√2/9√1/15−√7/9・√14/15)
=arccos(−√8/15)=δ6
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