Fractal Dimension

This video introduces the idea that fractals generally have a dimension that is not a whole number. For instance, the line has dimension 1, a square has dimension 2, and a cube has dimension 3, but fractals often have dimensions that are between these whole numbers (though they can be greater than 3 or less than 1 as well).

The dimension of fractals can be found by noticing a pattern while looking at simple self similar shapes (a line, a square, and a cube). Each shape is broken up into smaller shapes by a scaling factor which results in a greater number of smaller shapes. For example, breaking up the sides of a square into pieces that are 1/5 the size of the original side lengths results in 25 smaller squares. A relationship can be found between the scale factor, the number of pieces created, and the dimension of the shape.

The equation generated from the above process can be used to find the dimension of fractals.

[thedeathofbirth]

in the square each piece is 1/9th and for cube each piece is 1/27th of the original .