| * 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. |
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