Pythagoras tree (fractal)
The Pythagoras tree is a plane fractal constructed from squares. Invented by the Dutch mathematics teacher Albert E. Bosman in 1942, it is named after the ancient Greek mathematician Pythagoras because each triple of touching squares encloses a right triangle, in a configuration traditionally used to depict the Pythagorean theorem. If the largest square has a size of "L" × "L", the entire Pythagoras tree fits snugly inside a box of size 6"L" × 4"L". 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 /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. This is the simplest symmetric triangle. Alternatively, the sides of the triangle are recursively equal proportions, leading to the sides being proportional to the square root of the inverse golden ratio, and the areas of the squares being in golden ratio proportion. Area. Iteration "n" in the construction adds  2 squares of area formula_1, for a total area of 1. Thus the area of the tree might seem to grow without bound in the limit as "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