The Pythagoras tree is a plane fractal constructed from squares. It is named after Pythagoras because each triple of touching squares encloses a right triangle, in a configuration traditionally used to depict the Pythagorean theorem.

centre

If the largest square has a size of 1×1, the entire Pythagoras tree fits snugly inside a box of size 6×4. The finer details of the tree resemble the Lévy C curve.

Construction

The construction of the Pythagoras tree begins with a square. Upon this square are constructed two squares, each scaled down by a linear factor of ½√2, such that the corners of the squares coincide pairwise. The same procedure is then applied recursively to the two smaller squares, ad infinitum. The illustration below shows the first few iterations in the construction process.

Construction of the Pythagoras tree, order 1	Order 2	Order 3	Order 4
Order 0	Order 1	Order 2	Order 3

Area

Iteration n in the construction adds 2ⁿ squares of size (½√2)ⁿ, for a total area of 1. Thus the area of the tree might seem to grow without bound in the limit n→∞. However, some of the squares overlap starting at the order 5 iteration, and the tree actually has a finite area because it fits inside a 6×4 box.

It can be shown easily that the area A of the Pythagoras tree must be in the range 5 < A < 18, which can be narrowed down further with extra effort. Little seems to be known about the actual value of A.

Uses

It is possible that the Pythagoras tree would make very useful fractal antennas with only minor tweaking. This assumption is based on its very high Hausdorff dimension

External links

• Gallery of Pythagoras trees
• Pythagoras tree with different geometries as well as in 3D
• Pythagoras Tree by Enrique Zeleny based on a program by Eric W. Weisstein, The Wolfram Demonstrations Project.

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