Penrose tiling

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.

Farey sequence

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)

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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

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).