Sierpinski Triangle

* Sierpinski triangle is an example of Non-Euclidean shape.
* All man made objects belong to Euclidean geometry consisting of lines, planes, rectangular volumes, arcs, cylinders, spheres etc. Their dimensions are integral in nature i.e. 1,2,3 etc or 4 (including time)
* Increase in size S is given by the formula-- S= LD  where L is linear scaling & D is dimension or D = log (S) / log (L) ;
* In case of Euclidean objects  lying in a plane, when linear scaling is doubled, size increase 22 =4 times since D=2 and if in space, linear scaling is doubled, size increase 23 = 8 times  since D=3
* However, in case of Non-Euclidean shaped objects lying in a plane, if they are linearly scaled by a factor L ( for example 2 ), the area or size does not increase by L2 i.e. 22 =4 times but by an non-integral amount of D
* These types of geometries are called fractals and the physical and biological universe abounds with fractal shapes.
* Two identifying features of Fractal shapes : (1) They look similar to each other (self-similarity) but are not exactly same. Exa- Any two Human beings. (2) Their dimension is not integral, i.e. 2.45 or 3.973 etc. in stead of exactly 2 or 3.
* The whole universe is a fractal and each fractal is an universe unto itself since it is infinite.
* Whereas Euclidean objects are described by a simple algebraic formula, there is no such formula for fractals. These are described by iterative or recursive alogarithm.
* Wacław Sierpinski was a Polish mathematician (1882-1969) Who constructed the triangle which is a fractal image. Take an equilateral triangle. Join the mid-points of 3 sides to form a triangle. Remove the inner triangle and Continue joining the mid -points of the  generated 3 triangles. Again remove the middle triangle and continue  with a particular no. of iterations.
No. of Iterations No. of White / removed triangles
01 1
02 4
03 13
04 40
05 121
06 364
07 1093
08 3280
09 9841
10 29524
11 88573
12 265720
13 797161