Pythagoras tree
Encyclopedia
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
.
. 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 > .
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.
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.
mathematics teacher, in 1942.
s with only minor tweaking. This assumption is based on its very high Hausdorff dimension
.
Fractal
A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
constructed from squares
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
. Invented by the Dutch
Netherlands
The Netherlands is a constituent country of the Kingdom of the Netherlands, located mainly in North-West Europe and with several islands in the Caribbean. Mainland Netherlands borders the North Sea to the north and west, Belgium to the south, and Germany to the east, and shares maritime borders...
mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
teacher Albert E. Bosman in 1942, it is named after the ancient Greek
Ancient Greece
Ancient Greece is a civilization belonging to a period of Greek history that lasted from the Archaic period of the 8th to 6th centuries BC to the end of antiquity. Immediately following this period was the beginning of the Early Middle Ages and the Byzantine era. Included in Ancient Greece is the...
mathematician Pythagoras
Pythagoras
Pythagoras of Samos was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him...
because each triple of touching squares encloses a right triangle
Right triangle
A right triangle or right-angled triangle is a triangle in which one angle is a right angle . The relation between the sides and angles of a right triangle is the basis for trigonometry.-Terminology:The side opposite the right angle is called the hypotenuse...
, in a configuration traditionally used to depict the Pythagorean theorem
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
.
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
Lévy C curve
In mathematics, the Lévy C curve is a self-similar fractal that was first described and whose differentiability properties were analysed by Ernesto Cesàro in 1906 and G...
.
Construction
The construction of the Pythagoras tree begins with a squareSquare (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
. 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
Recursion
Recursion is the process of repeating items in a self-similar way. For instance, when the surfaces of two mirrors are exactly parallel with each other the nested images that occur are a form of infinite recursion. The term has a variety of meanings specific to a variety of disciplines ranging from...
to the two smaller squares, ad infinitum. The illustration below shows the first few iteration
Iteration
Iteration means the act of repeating a process usually with the aim of approaching a desired goal or target or result. Each repetition of the process is also called an "iteration," and the results of one iteration are used as the starting point for the next iteration.-Mathematics:Iteration in...
s in the construction process > .
 |
 |
 |
 |
| 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 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 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.
History
Pythagoras tree was first constructed by Albert E. Bosman (1891–1961), DutchDutch people
The Dutch people are an ethnic group native to the Netherlands. They share a common culture and speak the Dutch language. Dutch people and their descendants are found in migrant communities worldwide, notably in Suriname, Chile, Brazil, Canada, Australia, South Africa, New Zealand, and the United...
mathematics teacher, in 1942.
Uses
It is possible that the Pythagoras tree would make very useful fractal antennaFractal antenna
A fractal antenna is an antenna that uses a fractal, self-similar design to maximize the length, or increase the perimeter , of material that can receive or transmit electromagnetic radiation within a given total surface area or volume.Such fractal antennas are also referred to as multilevel and...
s with only minor tweaking. This assumption is based on its very high Hausdorff dimension
Hausdorff dimension
thumb|450px|Estimating the Hausdorff dimension of the coast of Great BritainIn mathematics, the Hausdorff dimension is an extended non-negative real number associated with any metric space. The Hausdorff dimension generalizes the notion of the dimension of a real vector space...
.
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. WeissteinEric W. WeissteinEric W. Weisstein is an encyclopedist who created and maintains MathWorld and Eric Weisstein's World of Science . He currently works for Wolfram Research, Inc.-Education:...
, The Wolfram Demonstrations ProjectWolfram Demonstrations ProjectThe Wolfram Demonstrations Project is hosted by Wolfram Research, whose stated goal is to bring computational exploration to the widest possible audience. It consists of an organized, open-source collection of small interactive programs called Demonstrations, which are meant to visually and...
. - Three-dimensional Pythagoras tree
- MatLab script to generate Pythagoras Tree
- http://ieeexplore.ieee.org/search/freesrchabstract.jsp?tp=&arnumber=5766703&queryText%3Dpourahmadazar%26openedRefinements%3D*%26searchField%3DSearch+All