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