A geometric proof of the Pythagorean Theorem

 

Ulam spiral on triangle on hexagonal grid

Ulam spiral on triangle on hexagonal grid transformed by 12 symmetries and added. Black is no primes, white is 8 primes. Note the many stars embedded in the hexagon.

Ulam spiral on triangle on hexagonal grid

Ulam spiral on triangle on hexagonal grid transformed through 12 symmetries and added, and a color scheme chosen to represent the number of primes. White is 9 primes, yellow 8, purple 7...

Ulam spiral on a triangle on hexagonal grid is transformed through 6 symmetries and added. Black is no primes, white is 5 primes. When the remaining 6 symmetries are added, the curl should disappear.

Ulam spiral on triangle. I think this is the largest one on the net, with 7503 primes. It is interesting that the spiral brings out the symmetries of triangle, as it does in the square case.

Ulam spiral transformed through the 8 symmetries of the square and added. Black is no primes, white is 8 primes. It is interesting that the primes are highly correlated wrt to the symmetries of the square. Note the resemblance to Buddhist sand paintings.

Universe in a box.

24 ASPECTS OF THE UNIVERSE

A basic classification of the universe can be derived from the Ideal Gas equation;

P*V = k*N*T

where k is the Boltzmann constant. It is one of the most fundamental constants of physics. The 4 thermodynamic variables can be mapped to the most universal categories of physics;

P: (Energy,Moomentum)
V: (Time, Spaace)
N: (Mass,Charge)
T: (Gravity,Boson)

Where;

Moomentum = (Momentum,Angular-Momentum,Spin)
Spaace = (Distance,Area,Volume)
Charge = (Electric,Weak,Strong)
Boson = (Light,Weak,Gluon)

The overall 4ness is reflected within each subcategory giving 4*4 = 16 subcategories. It is a mixture of (1,3) and (2,2) symmetries, inside and outside.

Where are the Higgs Boson, Dark Matter and Dark Energy? I think they are in k. They are the fields that the rest float on. They may even make a quartet, predicting another field;

(Higgs,?,DE,DM)

to make 5 categories and 20 subcategories in all.

Where are the Black Hole and the rest? I think they make another quartet;

(Atom,Planet,Star,Black-Hole)

Thus making 6 categories and 24 subcategories. I had a yarn before about the specialness of the number 6. 24 is also a very special number. It is one of John Baez's favorite numbers.

http://en.wikipedia.org/wiki/6_(number)
http://en.wikipedia.org/wiki/24_(number)
math.ucr.edu/home/baez/numbers/24.pdf

If all listed together in a certain order, the symmetry simplifies somehow;

(Higgs,?,DE,DM)
(Time, Distance,Area,Volume)
(Energy,Momentum,Angular-Momentum,Spin)
(Gravity,Light,Weak,Gluon)
(Mass,Electric,Weak,Strong)
(Atom,Planet,Star,Black-Hole)

There is deep symmetry to it with many aspects, each reflecting a mathematical property of 24, starting with the simple arithmetic ones like;

4*6 = 2*2*2*3 = 2*12 = 3*8