P^2+Σ2P

Penrose tiling is based on Golden Ratio (a=φ:b=1), but you can't use this rhombus because of aperiodicity. {a: Kite} {b: Dart} You connect these Kite and Dart randomly. Then you see tiling like these. This is like the Maurits Cornelis Escher's painting, but it is periodic .

F(1) 0/1 1/1 F(2) 0/1 1/2 1/1 F(3) 0/1 1/3 1/2 2/3 1/1 F(4) 0/1 1/4 1/3 1/2 2/3 3/4 1/1 F(5) 0/1 1/5 1/4 1/3 2/5 1/2 3/5 2/3 3/4 4/5 1/1 (φ(n)=Euler's totient function) φ(n) is the sum of numbers which are non divisors of n. You ignore 1. ex. φ(12)= 4 ⇒ (1,5,7,11) φ(13)=12 ⇒ (1,2,3,4,5,6,7,8,9,10,11,12) φ(14)= 6 ⇒ (1,3,5,9,11,13) Farey sequence is related to Ford circle . F(n+1)=(p+r)/(q+s) You see F(2). 1/2=(0/1)+(1/1) In F(3), 1/3=(0/1)+(1/2) and 2/3=(1/2)+(1/1) In F(4), 1/4=(0/1)+(1/3) and 3/4=(2/3)+(1/1) In F(5), φ(5)=4. 1/5=(0/1)+(1/4), 2/5=(1/3)+(1/2), 3/5=(1/2)+(2/3), 4/5=(3/4)+(1/1)

Ford circle is based on Golden Ratio. These circles are tangent. C(p/q), C(r/s), C((r-p)/(s-q))=C(1/0) ∴ ps-qr=±1 Then, you see that circles are the convergence . τ is the Golden Ratio. You also think about irrational numbers (ω), and this is zero (∞)=C(1/0).