■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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