- Pythagoras tree
The Pythagoras tree is a plane
fractal constructed from squares. It is named afterPythagoras because each triple of touching squares encloses aright triangle , in a configuration traditionally used to depict thePythagorean theorem .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 theLévy 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
iteration s in the construction process.Area
Iteration "n" in the construction adds 2"n" 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".
External links
* [http://mathpaint.blogspot.com/2007/03/pythagoras-tree.html Gallery of Pythagoras trees]
* [http://www.phidelity.com/blog/fractal/pythagoras-tree/ Pythagoras tree with different geometries as well as in 3D]
*" [http://demonstrations.wolfram.com/PythagorasTree/ Pythagoras Tree] " by Enrique Zeleny based on a program byEric W. Weisstein ,The Wolfram Demonstrations Project .
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