Pythagoras tree (fractal)

The Pythagoras tree.

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 6L × 4L. 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 0	Order 1	Order 2	Order 3
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 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.

Varying the angle

An interesting set of variations can be constructed by maintaining an isosceles triangle but changing the base angle (90 degrees for the standard Pythagoras tree). In particular, when the base half-angle is set to 30° = arcsin(0.5), it is easily seen that the size of the squares remains constant. The first overlap occurs at the fourth iteration. The general pattern produced is the rhombitrihexagonal tiling, an array of hexagons bordered by the constructing squares.

Order 4	Order 10

In the limit where the half-angle is 90 degrees, there is obviously no overlap, and the total area is twice the area of the base square. It would be interesting to know if there's an algorithmic relationship between the value of the base half-angle and the iteration at which the squares first overlap each other.

Modified and Unmodifed Pythagoras tree (fractal) for antenna engineering application

Unmodified Pythagoras Tree Fractal (UPTF) was invented by the Dutch mathematician, Albert E.Bosman, in 1942 [3]. The Pythagoras Tree is a 2D fractal constructed from squares [3]. It is named after the ancient Greek mathematician Pythagoras because each triple of touching squares encloses a right triangle (∠45 deg) based on configuration traditionally used to depict the Pythagorean theorem [3]. If the largest square has a size of L × L, the entire Pythagoras tree fits snuggles inside a box of size 6L × 4L [3]. The construction of the Pythagoras tree begins with a square. Upon this square are constructed two other squares, each scaled down by a linear factor of ½√2, such that the corners of the squares coincide pair wise [10]-[13]. The same procedure is then applied recursively to the two smaller squares, ad infinitum. Fig.1 shows illustration of the first five iterations in the construction process. Iteration n in the construction adds 2^n squares of size〖(½√2)〗^n, for a total area of 1 [11]. Thus the area of the tree fractal might seem to grow without boundary (n→∞) [3]. However, starting at the 5th iteration some of the squares overlap, and the tree fractal actually has a finite area because it snuggles in to 6×4 box. For this reason, to delay the overlap of left and right hand fingers of UPTF in 4th iteration (Figure. 1 in reference [3]), we design MPT fractal by eliminating 1st iteration large side square and change isosceles right-angled triangle to isosceles triangle with steep angles (α=10 deg) to reduce the fractal height to design compact antennas. This triangle change is our fractal freedom degree that helps antenna designer to make a novel fractal shape. Our purpose in designing a MPTF is to use this fractal to control impedance bandwidth and resonances. Figure. 2 in reference [3] shows illustration of the first five iterations for Modified Pythagorean Tree Fractal (MPTF) with different colors (Odd iterations with Black and Even iterations with White colors). Note that all the triangles are isosceles triangles with steep angles equal α=10 deg and other angles values of triangles and squares can be calculated by geometrical theories. In this research authors from NorthWest Antenna and Microwave Research Labaroury (NAMRL), Urmia University presents the design of a novel Modified Pythagorean-Tree-Fractal (MPTF) based antenna using multi fractal technique for Ultra wide band application. Based on simulation results, the Modified Pythagoras Tree Fractal exhibited very good miniaturization ability due to its self-similar properties, without reducing significantly the bandwidth and the efficiency of the antenna.

History

Pythagoras tree was first constructed by Albert E. Bosman (1891–1961), Dutch mathematics teacher, in 1942.

See also

• Bram Oome

External links

[1] Gallery of Pythagoras trees [2]Pythagoras tree with different geometries as well as in 3D [3]Pythagoras Tree by Enrique Zeleny based on a program by Eric W. Weisstein, The Wolfram Demonstrations Project. [4] Weisstein, Eric W. "Pythagoras Tree". MathWorld. [5] Three-dimensional Pythagoras tree [6] MatLab script to generate Pythagoras Tree [7] Pourahmadazar, J.; Ghobadi, C.; Nourinia, J.; (2011). Novel Modified Pythagorean Tree Fractal Monopole Antennas for UWB Applications. New York: IEEE. doi:10.1109/LAWP.2011.2154354.{{cite book}}: CS1 maint: extra punctuation (link) CS1 maint: multiple names: authors list (link)