Relation between Pascal’s triangle and the Sierpenski triangle
Both Pascal’s and Sierpenski are triangles. The Sierpenski triangle is obtained from Pascal’s triangle by marking or coloring the odd numbers and leaving the even numbers without color.
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Properties of the Sierpenski triangle and the Mandelbrot set
Both the Sierpenski triangle and Mandelbrot set a deal with non-integer dimensions (fractal dimensions) and can be described in an algorithm. They are self-similar and are recursive.
Comparison of Pascal’s triangle, Sierpenski triangle, and Mandelbrot set
|Pascal’s triangle||Sierpenski||Mandelbrot set|
|Beauty||Triangular in shape.||Triangular in shape. |
Its ever-repeating pattern of triangles makes it beautiful
|It is kidney-shaped and bordered by cardioids. |
When colored, the pattern is beautiful.
|Complexity||It is simple to generate since the numbers of the coefficients of the previous row are added to obtain the current row coefficients.||Simple to generate since the triangles are obtained by shrinking the initial triangle over and over. |
3D construction is possible.
|Its generation is slightly complex since points on the complex plane have to be tested in the equation z=z2+c.|
|Mathematics||Binomial expansion is used to obtain its coefficients and it is used for calculating combinations or discrete values of data.||Total area of triangle tends to ∞ as triangle sizes decrease. |
Is used for determining logarithms, symmetry, transformations and geometric construction.
|Deals with complex numbers. |
Plotted on the complex plane by iterating z=z2+c.
Points tested must converge i.e. be <2 for them to be within the set.