Upper half-plane

H is the upper half-plane which contains the set of complex numbers with positive imaginary part.

This is modular group.

ad − bc = 1

This includes point at infinity.

Modular Group

a,b,c,d are integers.

ad-bc=1

This is modular group which is also isomorphic of PSL(2, Z).

a/b, a/c, c/d and b/d are irreducible.

(a,b,c,d)=(7,5,4,3),(4,11,17,47),(4,7,13,23),(2,5,17,43),(6,19,23,73),(8,13,19,31),(48,13,11,3)

You see 3 prime numbers and 1 even number in it.

Τ=3P+N

(P:prime number, N:even number)
2 is prime number, but it is also even number. This has special meaning in my blog.

This is also irreducible .

You see (7,5,4,3).

7p+5q

4p+3q

(p=2, q=3)

14+15=29

8+9=17

29/17=1.705882353

PSL(V)=SL(V)/SZ(V)

PSL is projective linear group, and V is vector space. Moreover, SZ(V) is the center, and SL(V) is factor group.

This is order.

d is the center.

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).

Sphere Packing

You need to fill spheres in packing infinitely. This is more complicated than packing oranges.

Two dimensions are easy to image, but spheres are three dimensions and you pile them infinitely.

This is n-dimensional Pythagorean theorem, the distance between (a,b,c,d) and (w,x,y,z) in R^n.

There are always spaces between spheres. You need to put small ones. Fraction of boundary is less than 0.99. You can find 99% spaces for spheres anytime.

https://prime2011.tumblr.com/post/179661361213

∴

0.99^n

0.99^300≒0.049
0.99^500≒0.00657
0.99^800≒0.00032

Therefore, spheres are almost zero in higher dimensions.

Four Color Theorem

You need just four colors to fill any realms in your flat spaces. This is called Four color theorem.

You prove it by induction.

At first, there are 3 countries in your map. You can add the one color.

There is the rule which must be the different color when they face each other. Therefore, yellow is acceptable.

Then, you have 4 countries in your map. Do you need 5 colors to fill your space?

This is complicated, and you need to choose the color to adjust the Four color theorem.

You don't need 5 colors.

Is this induction satisfied with any situation?

This is beyond our ability. The Four color theorem seems to be correct, but we rely on the computer.