■置換多面体の空間充填性(その160)
H3,H4,F4に対応する頂点回りのファセット数公式ができたので,f0,f1公式についても考えてみたい.まずはf0公式から.
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【1】H3,H4
{3,5}のj次元面の中にあるk次元面の数を<j,k>で表すと,
<3,2>=4,<3,1>=6,<3,0>=4
<2,1>=3,<2,0>=3
<1,0>=2
これらの組み合わせと面数の積を考える.
{3,5}(1,0,0)=12
{3,5}(0,1,0)=30
{3,5}(0,0,1)=20
{3,5}(1,1,0)=<1,0>30=60
{3,5}(1,0,1)=<2,0>20=60
{3,5}(0,1,1)=<2,1>20=60
{3,5}(1,1,1)=<2,1><1,0>20=120
{3,3,5}(1,0,0,0)=120
{3,3,5}(0,1,0,0)=720
{3,3,5}(0,0,1,0)=1200
{3,3,5}(0,0,0,1)=600
{3,3,5}(1,1,0,0)=<1,0>720=1440
{3,3,5}(1,0,1,0)=<2,0>1200=3600
{3,3,5}(1,0,0,1)=<3,0>600=2400
{3,3,5}(0,1,1,0)=<2,1>1200=3600
{3,3,5}(0,1,0,1)=<3,1>600=3600
{3,3,5}(0,0,1,1)=<3,2>600=2400
{3,3,5}(1,1,1,0)=<2,1><1,0>1200=7200
{3,3,5}(1,1,0,1)=<3,1><1,0>600=7200
{3,3,5}(1,0,1,1)=<3,2><2,0>600=7200
{3,3,5}(0,1,1,1)=<3,2><2,1>600=7200
{3,3,5}(1,1,1,1)=<3,2><2,1><1,0>600=14400
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【2】F4
{3,4}のj次元面の中にあるk次元面の数を<j,k>で表すと,
<3,2>=8,<3,1>=12,<3,0>=6
<2,1>=3,<2,0>=3
<1,0>=2
これらの組み合わせと面数の積を考える.
{3,4,3}(1,0,0,0)=24
{3,4,3}(0,1,0,0)=96
{3,4,3}(0,0,1,0)=96
{3,4,3}(0,0,0,1)=24
{3,4,3}(1,1,0,0)=<1,0>96=192
{3,4,3}(1,0,1,0)=<2,0>96=288
{3,4,3}(1,0,0,1)=<3,0>24=144
{3,4,3}(0,1,1,0)=<2,1>96=288
{3,4,3}(0,1,0,1)=<3,1>24=288
{3,4,3}(0,0,1,1)=<3,2>24=192
{3,4,3}(1,1,1,0)=<2,1><1,0>96=576
{3,4,3}(1,1,0,1)=<3,1><1,0>24=576
{3,4,3}(1,0,1,1)=<3,2><2,0>24=576
{3,4,3}(0,1,1,1)=<3,2><2,1>24=576
{3,4,3}(1,1,1,1)=<3,2><2,1><1,0>24=1152
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