■DE群多面体の面数公式(その375)
[1]221
221の頂点間距離が2のとき,半径は√(8/3)
R^2=1+1/3+1/6+1/10+1/15+a6^2=8/3
=1+1/3+1/6+1/10+2/5+b6^2
1+1/3+1/6+1/10=(30+10+5+3)/30=8/5
R^2=8/5+2/5+b6^2=8/5+1/15+a6^2=8/3
a6^2=(40−24−1)/15=5/3
b6^2=(40−24−6)/15=2/3
===================================
[2]321
頂点間距離が2のとき,半径は√3
R^2=1+1/3+1/6+1/10+1/15+1/21+a7^2=3
=1+1/3+1/6+1/10+1/15+2/6+b7^2
1+1/3+1/6+1/10+1/15=(30+10+5+3+2)/=5/3
R^2=5/3+1/3+b7^2=5/3+1/21+a7^2=3
a7^2=(63−35−1)/21=9/7
b7^2=(9−5−1)/3=1
===================================
[3]421
頂点間距離が2のとき,半径は2
R^2=1+1/3+1/6+1/10+1/15+1/21+1/28+a8^2=4
=1+1/3+1/6+1/10+1/15+1/21+2/7+b8^2
R^2=12/7+2/7+b8^2=12/7+1/28+a8^2=4
a8^2=(112−48−1)/28=9/4≧2
b8^2=(28−12−2)/7=2≧2
R^2=7/4+a8^2=2+b8^2
===================================
もしこれ以上を考えるのであれば,
R^2=1+1/3+1/6+1/10+1/15+1/21+1/28+1/36+a9^2
=1+1/3+1/6+1/10+1/15+1/21+1/28+2/8+b9^2
1+1/3+1/6+1/10+1/15+1/21+1/28
=Σ2/j(j+1)
=2(1−1/8)=7/4
1+1/3+1/6+1/10+1/15+1/21+1/28+1/36+=2(1−1/9)=16/9
1+1/3+1/6+1/10+1/15+1/21+1/28+2/8=2=7/4+1/4=2≧2
R^2=16/9+a9^2=2+b9^2
===================================