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Pascal’s Triangle, the Sierpenski Triangle, and the Mandelbrot Set

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.

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StudyCorgi. (2022, May 9). Pascal’s Triangle, the Sierpenski Triangle, and the Mandelbrot Set. Retrieved from https://studycorgi.com/pascals-triangle-the-sierpenski-triangle-and-the-mandelbrot-set/

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StudyCorgi. (2022, May 9). Pascal’s Triangle, the Sierpenski Triangle, and the Mandelbrot Set. https://studycorgi.com/pascals-triangle-the-sierpenski-triangle-and-the-mandelbrot-set/

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"Pascal’s Triangle, the Sierpenski Triangle, and the Mandelbrot Set." StudyCorgi, 9 May 2022, studycorgi.com/pascals-triangle-the-sierpenski-triangle-and-the-mandelbrot-set/.

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StudyCorgi. "Pascal’s Triangle, the Sierpenski Triangle, and the Mandelbrot Set." May 9, 2022. https://studycorgi.com/pascals-triangle-the-sierpenski-triangle-and-the-mandelbrot-set/.

References

StudyCorgi. 2022. "Pascal’s Triangle, the Sierpenski Triangle, and the Mandelbrot Set." May 9, 2022. https://studycorgi.com/pascals-triangle-the-sierpenski-triangle-and-the-mandelbrot-set/.

References

StudyCorgi. (2022) 'Pascal’s Triangle, the Sierpenski Triangle, and the Mandelbrot Set'. 9 May.

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