The definition extracted from Wikipedia: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
If the Wikipedia one was a bit hard to understand here's how I think of it:
Starting with a 3,4,5 right triangle, if you think of the sides of the triangle as each being one of the sides of a square, the areas of the two smaller squares add up to the area of the larger square.
Now thinking of the two smaller sides (in this case, 3 and 4) as being the bases of two other 3, 4, 5 right triangles. As long as the ratios are equivalent, a right triangle with sides 9/5,12/5,3 has the exact same angle and side ratios as a right triangle of sides 3,4,5, a right triangle with sides 12/5,16/5,4The Pythagoras tree emerges by continuing this process many, many times (as needed).
Each triangle is geometrically similar to the first as the angle-angle and side-side ratios are the same for each
The beauty of the fractal is that if you look at simply a single triangle and the three squares involved with said triangle, the "tree" (trunk as largest square and branches as two smaller squares) is basically the same as any other triangle-and-three-square figure in the whole fractal.Thinking of Pythagorean Theorem in a physical sense was easy to understand by me
The area can be determined as followsBy Pythagorean theorem
S₁² + S₁² = L²
2S₁² = L²
S₁² = L²/2
S₁ = L/√2
S₂ = S₁/√2 = (L/√2)/√2 = L/(√2)²
S₃ = S₂/√2 = (L/(√2)²)/√2 = L/(√2)³
S₄ = S₃/√2 = (L/(√2)³)/√2 = L/(√2)⁴
. . .
Sn = L/(√2)ⁿ
We start with 1 square
In each iteration, we add twice the number of square added in previous iteration
In iteration 1, we add 2 squares
In iteration 2, we add 2*2 = 2² squares
In iteration 3, we add 2*2² = 2³ squares
In iteration 3, we add 2*2³ = 2⁴ squares
. . .
In iteration n, we add 2ⁿ squares
So in each iteration, we add 2ⁿ squares, each with side length of L/(√2)ⁿ
Area of all squares in iteration n
= number of squares * area of each square
= number of squares * (side length )²
= 2ⁿ * (L/(√2)ⁿ)²
= 2ⁿ * L²/2ⁿ
= Area of original square
So why does Wikipedia get total area = 1?
Because, they have obviously assumed that dimension of original square = 1 x 1, which means L = 1, therefore L² = 1. However, this is NOT explicitly stated in the link, which it should have been, since the last mention of original (i.e. largest) square assumes square of size L x L, not 1 x 1.More designs using Pythagoras tree
: http://www.redbubble.com/groups/apophysis-tutorial-fun/forums/14903/topics/332497-volume-64-pythagoras-treesDraw on your own