View source for Pythagoras tree (fractal)

{{Short description|A plane fractal constructed from squares}}
[[File:Animated_self-resembling_Pythagoras_tree_(fractal).webm|thumb|Animation of an imperfectly self-resembling Pythagoras tree]]
[[Image:Pythagoras Tree Colored.png|thumb|The Pythagoras tree with an angle of 25 degrees and smooth coloring]]
The '''Pythagoras tree''' is a [[Plane (geometry)|plane]] [[fractal]] constructed from [[Square (geometry)|squares]]. Invented by the [[Netherlands|Dutch]] [[mathematics]] teacher Albert E. Bosman in 1942,<ref>{{cite web |url=http://www.arsetmathesis.nl/bruno0402.htm |title=Bruno's column (0402) Boom van Pythagoras |access-date=2012-03-10 |url-status=dead |archive-url=https://web.archive.org/web/20090118100209/http://www.arsetmathesis.nl/bruno0402.htm |archive-date=2009-01-18 }}.</ref> it is named after the [[Ancient Greece|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''&nbsp;&times;&nbsp;''L'', the entire Pythagoras tree fits snugly inside a box of size 6''L''&nbsp;&times;&nbsp;4''L''.<ref name="Wisfaq.nl">[http://www.wisfaq.nl/showrecord3.asp?id=32367 Wisfaq.nl].</ref><ref name="Pourahmadazar">{{cite journal |author1=Pourahmadazar, J. |author2=Ghobadi, C. |author3=Nourinia, J. |title=Novel Modified Pythagorean Tree Fractal Monopole Antennas for UWB Applications|journal=IEEE Antennas and Wireless Propagation Letters |volume=10 |publisher=IEEE |location=New York |year=2011 |pages=484–487|doi=10.1109/LAWP.2011.2154354 |bibcode=2011IAWPL..10..484P }}</ref> The finer details of the tree resemble the [[Lévy C curve]].

==Construction==
The construction of the Pythagoras tree begins with a [[square (geometry)|square]]. Upon this square are constructed two squares, each scaled down by a linear factor of {{radic|2}}/2, such that the corners of the squares coincide pairwise. The same procedure is then applied [[recursion|recursively]] to the two smaller squares, ''ad infinitum''. The illustration below shows the first few [[iteration]]s in the construction process.<ref name="Wisfaq.nl"/><ref name="Pourahmadazar"/>

{|align="center"
|-
| [[image:PythagorasTree1.png|none|Construction of the Pythagoras tree, order 0]]
| [[image:PythagorasTree2.png|none|Order 1]]
| [[image:PythagorasTree3.png|none|Order 2]]
| [[image:PythagorasTree4.png|none|Order 3]]
|-
|align="center"| Order 0
|align="center"| Order 1
|align="center"| Order 2
|align="center"| Order 3
|}

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<sup>''n''</sup> squares of area <math>\tfrac{1}{2^n}</math>, for a total area of 1. Thus the area of the tree might seem to grow without bound in the limit as ''n''&nbsp;→&nbsp;∞. 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&times;4 box.<ref name="Wisfaq.nl"/>

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

{|align="center"
|-
| [[File:Ptang4th.svg|220px|fourth-order tree, one overlap visible]]
| [[File:Ptang10th.svg|220px|tenth-order tree]]
|-
|align="center"| Order 4
|align="center"| 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.


The Pythagoras tree was first constructed by Albert E. Bosman (1891–1961), a [[Dutch people|Dutch]] mathematics teacher, in 1942.<ref name="Wisfaq.nl"/><ref>[http://www.arsetmathesis.nl/bruno0402.htm Arsetmathesis.nl] {{webarchive|url=https://web.archive.org/web/20090118100209/http://www.arsetmathesis.nl/bruno0402.htm |date=2009-01-18 }}</ref>

==See also==
* [[Lévy C curve]]

==References==
{{Reflist|2}}

==External links==
{{Commons|Pythagoras tree}}
* [https://web.archive.org/web/20190215125315/http://mathpaint.blogspot.com/2007/03/pythagoras-tree.html Gallery of Pythagoras trees]
* [https://github.com/BoleeMan/PythagorasTree Filled Pythagoras Tree using VB6 by Edward Bole (Boleeman)] 
* [https://samu.github.io/pythagoras-tree/ Interactive generator with code]
* {{cite web | url=http://www.phidelity.com/blog/fractal/pythagoras-tree/ | title=Pythagoras tree with different geometries as well as in 3D | url-status=dead | archive-url=https://web.archive.org/web/20080115080140/http://www.phidelity.com/blog/fractal/pythagoras-tree/ | archive-date=2008-01-15 }}
* [http://demonstrations.wolfram.com/PythagorasTree/ ''Pythagoras Tree''] by Enrique Zeleny based on a program by [[Eric W. Weisstein]], The [[Wolfram Demonstrations Project]].
* {{MathWorld |title=Pythagoras Tree |urlname=PythagorasTree}}
* [http://wonderfl.kayac.com/code/1ee53bd0cff107d44c831a36b92502e90734b0ec Three-dimensional Pythagoras tree]
* [http://www.mathworks.com/matlabcentral/fileexchange/26816-pythagoras-tree MatLab script to generate Pythagoras Tree]
* [http://www2.ual.es/neotrie/arbol-de-pitagoras/ Construction step by step in the virtual reality software Neotrie VR]
* {{cite journal |author1=Pourahmadazar, J. |author2=Ghobadi, C. |author3=Nourinia, J. |title=Novel Modified Pythagorean Tree Fractal Monopole Antennas for UWB Applications|journal=IEEE Antennas and Wireless Propagation Letters |volume=10 |publisher=IEEE |location=New York |year=2011 |pages=484–487|doi=10.1109/LAWP.2011.2154354 |bibcode=2011IAWPL..10..484P }}

{{Fractals}}

{{DEFAULTSORT:Pythagoras Tree}}
[[Category:Fractals]]