Fractal Project Mariellen Hemmerling

Fractals “A fractal is "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole.” – Wikipediageometric shape “An irregular geometric object that is self-similar to its substructure at any level of refinement. The fractal dimension is a measure of the irregularity of the boundary of the object.” - amsglossary.allenpress.com/glossary/browseamsglossary.allenpress.com/glossary/browse –Meaning the fractal zoomed in is smaller size of the whole picture, no matter how many times you zoom in you will always get the same picture.

Sierpinski’s Triangle Discovered by Waclaw Sierpinski, in This fractal is a bunch of tiny triangles in one big triangle, which is how it makes it a fractal.

The Golden Ratio This is divided so that the smaller part fits in the slightly larger part etc. This is found in many famous paintings, and many buildings such as the Parthenon.

Pascal’s Triangle “A geometric representation of the binomial coefficients in a triangle” The top numbers added together to make the bottom numbers.

Pythagoras Tree It’s a fractal constructed from squares. It was named after Pythagoras because each triple of touching squares makes a right triangle.

Lorenz Fractal It’s named after Edward N. Lorenz. It’s a 3-D system that creates chaotic flow/ Unpredictable.

Van Koch Founded by Helge Von Koch. This is basically a snowflake, but the repeat is the star points. This fractal is a curve.

IFS Fractal A method of constructing fractals, that make a whole fractal self similar. In the picture below, the leaf has tiny leaves making the big leaf.

Cesaro’s Sweep This fractal has a “base” and a “motif”, when together it makes an upright peano Curve. So if you place the motif in different spots it will create a sweep.

Levy Curve This fractal is a self similar fractal, meaning the smaller shapes make the whole fractal. It makes a sort of boat like figure, with shapes that are similar to a loop, that repeats itself to make a curve.

Dragon Fractal This Fractal was discovered by William G. Harter. It’s a pretty cool fractal, and uses many cool colors. It resembles the Julia set, and in the Mandelbrot set zoomed in it appeared.

Star fractal This is made by a five cornered star, and connecting smaller stars at each corner. This fractal can be made of stars, triangles, or squares.

Polya’s Sweep This is very much alike with the Cesaro’s Sweep, although it makes a different type of triangle. It has the base and motif.

Circle fractal It closely resembles the Cantor set. The Circle Fractal is basically one big circle, with a bunch of smaller circles inside.

Mandelbrot Set Named after Benoit Mandelbrot in 1978 It is similar to the Julia Set when modified. It has a similar process of iteration (reapeating itself when zoomed in)

Julia Set When modified, it is found on the exterior points of the Mandelbrot. The math controls the foundation of the map.

Nature

Software This software supports animation, and makes fractals.

Common Terms Iteration: A repeating pattern. Trees: A fractal tree, make branches grow off of branches and repeat themselves. Base/Motif: Any shape that is composed as a line. Base: A line that is just straight Motif: Lines that switch directions that starts with a base.

Thoughts. I think fractals are really cool, I never thought art could be a mathematical thing. I love the colors, and the shapes and the fact that they repeat when zoomed in. How this can help people This can help people mathematically by learning Trig, Algebra, and more mathematical terms. It can also help artists.

Research Pythagoras Tree: Sierpinski's Triangle: Pascal’s Triangle: The Golden Ratio: mark/homepage/goldenratio/goldenratio.html Star Fractal: Levy Curve: Cesaro’s Sweep: Dragon Fractal: uct.html Polya’s sweep: Julia Set: Lambda Fractal: ml Circle Fractal: =517&start=0 ml Lorenz Fractal: IFS fractal: m orange.fr/en/art/ifs.html Van Koch Fractal: