Fractals With Stars

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Fractals with stars
[[Image:Picture 11.png|center|400px]]
Fields: Graph Theory, Fractals, and Geometry
Image Created By: MWillis

Fractals with stars

The final star creation.

Basic Description

Start by trying to make a star tool. Plot point A and plot point B far away from it. Mark the first point as center and rotate B around it 4 times at 72º until five points are created around a center point. Then construct a segment between points A and B and create a midpoint of that line. Rotate the midpoint around point A at 36 degrees to create point C and then rotate C at 72º to create point D and then rotate point D by 72º 4 times to create the outline points of a star. Then hide C and connect all the points to form an outline of a star. Then fill in the star and highlighted everything and create a tool for it. Then select the tool and utilize it, marking point D as the center and an adjacent point as the end point. The adjacent point should be as equidistant from the center point as point B. The second star should be 2.2676 times smaller than the original star. Keep doing this in succession on each leg of the star until you have 5 stars on each leg (26 total). Then locate the conversion point, which is point D or the point equidistant from point A as point D, and plot that point as the origin and map a polar grid around it and then try to find a parametric function that intersects the centers of the stars.

Screen shot 2013-06-18 at 12.29.17 PM.png The final product of my design and the polar equation that runs along the center points of it. As you can see, the conversion point of the star fractals is the origin of the grid and the starting point of the function.

A More Mathematical Explanation


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Formula for the spiral that matches the system of points is:

r=(.192/θ), θ=(.1005*(.53/r)^5

Picture 10.png

The lengths of the distances between the center points of the stars is shown above. The lengths vary depending on how big the stars are but they all are 1.50588 times longer than the next smallest one. In addition, Each star is 2.2676 larger than the next smallest. Everything stays consistent.

Why It's Interesting

I find this interesting because of all the assignments and projects, specifically on GSP, that we have done in class, we have never done anything involving stars, which for some reason are my favorite shape. It is also the first time that i have ever played with a polar grid and plotted polar equations.

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