Pythagorean Tree

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Pythagorean Tree, in 2 Dimensions
Le pytho.jpg
Fields: Algebra and Fractals
Image Created By: Enri Kina and John Wallison

Pythagorean Tree, in 2 Dimensions

A Pythagorean Tree is a fractal that is created out of squares. The space between the squares in each iteration creates a right triangle. The top line of the square becomes the hypotenuse of the triangle above it.

Basic Description

Output ANIihE.gif

This animation shows how the angles of the triangle affect the shape of the tree.

A More Mathematical Explanation

Note: understanding of this explanation requires: *Basic Algebra

\frac{1}{x} Pythagoreantree.png

In this image, the original square ha [...]

\frac{1}{x} Pythagoreantree.png

In this image, the original square has an area of 36, meaning its side length s = 6. This can be put into the Pythagorean theorem, a^2 + b^2 = c^2, to get a^2 + b^2 = 6^2. This means the sum of the areas of the two branched-off squares will always be equal to the original square. These areas are dependent on the side lengths of the right triangle in the middle, which in turn are dependent the angles. With the hypotenuse 6 and angle \theta, the length of side b can be found with sin\theta = \frac{b}{6}. If \theta = 60º, then b would equal, about, 5.196.

If \theta = 60º, then sin\theta = \frac{b}{6} = 5.196...

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