Friday, September 30, 2016

Shams Square Ripple

Shams Square ripple:

The central yellow square is divided into 4 triangles, is flipped outward along their hypotenuse. Similar division and flipping is repeated on the resultant triangles. From the 4th phase onwards ‘island’ squares are formed, reason for which is that similar triangles trying to flip from the adjacent side block and forbid such an operation. All the squares thus  formed are joined along their vertices forming a chain. The total area of this chain equals the initial square. The formation of island squares is in the sequence :{1, 0, 0, 0, 4, 8, 20, 32, 60, 88, 148, 208, 332, ... }

So the recurrence relation is :

a(n+2) = 2 a(n)+4 (n-2) (for all n>=1)

Generating function:
G_n(a_n)(z) = (-2 z^4-4 z^3+z^2+2 z-1)/((z-1)^2 (2 z^2-1))
The initial single square during the 100th phase spawns 4503599627370096 squares.
During the 200th phase it will have
5070602400912917605986812820704..During the 528 th phase it will exceed the number of atoms in the visible universe,  ie  ≈1.1857 × 10^80, ,which is 1.2 x the total number of atoms in the visible universe..!

Formula for finding the total number of island squares at any phase:
For odd numbered phase :  
 (-2 + 2^(1 + (n - 3)/2) - (n - 3)/2) 8 - (-1 + 2^((n - 3)/2)) 4
For even numbered phase: 
 (-2 + 2^(1 + (n-4)/2) - (n-4)/2) 8
,where n is the number of the phase.

see above figure:
The boundary of this progression will never (ever) cross the red square, whose side is 3 times the size  of the initial square !
Here the shape of the initial square is preserved in each of the square islands.The total area (of the initial square) is also preserved. Though the area remains constant ( finite) the perimeter or the length of the curve grows to infinity !
As the flipping continues the shape of the initial square turns to an octagon .The peripheral triangles on the outermost area create more squares as the progression continues.
Among all the other regular polygons only the square generate self-similar polygons in a homogeneous and fractal progression..!!

The green arrows show the path and direction of the flipping triangles and the red arrows show how the flipping gets forbidden and so generate square islands