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Avinash Kumar
410 followers -
Programmer & Digital Artist
Programmer & Digital Artist

410 followers
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Nice one!
Only Linux users will understand this wisdom ;) 
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The Bluebird of happiness......
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DataCollection.js: http://bit.ly/1BOn58N - handy library for working with (filtering + manipulation) semi-structured data in the browser.
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A Pythagoras tree is constructed by starting with a square, then attaching two squares on top of it with a size of root 2 over 2. 

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,[1] 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.[2][3] 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 12/5,16/5,4

The 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 follows

By Pythagorean theorem: 

S₁² + S₁² = L² 
2S₁² = L² 
S₁² = L²/2 
S₁ = L/√2 

Similarly: 
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 
= L² 
= 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-trees

Draw on your own:

http://www.wolframalpha.com/input/?i=pythagoras+tree

http://math.mercyhurst.edu/~credmond/computer_art/artwork/pythagoras.php

#maths   #mathematics   #amazing   #gif   #animation   #geometric   #geometry   #science  
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It's time I need to stop buying American products (excluding Apple). Most of the American products here are cheap Made in China stuff. American companies keep getting richer selling Chinese crap in the name of their brands. #quality   #product   #america

I received a brand new Canon EF 50mm F1.8 II lens today with dust particles on the back side of the lens.

Not only does the build quality seem MUCH cheaper than Sony standard lenses, the overall quality is equally crap (Except photo quality. The photos that it captures are amazing for the price of the lens). #canon   #lens  

All the stuff currently going on between Russia and the USA doesn't sound good. It'll be very sad if it turns into some kind of nuclear war between the two.

Why does America always have to step into someone else's affairs? I bet they are the cause of more killings in the world than all other countries combined.

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Meet Dopamine, the bus driver!
It is a common misconception that we "drive our bus", or in other words have conscious control over our daily actions. Such a quaint notion, long fully laid to rest by neuroscience. 

Instead of "us" driving our buses, we have many "bus drivers" - various brain mechanisms, hormones and neurochemicals well below our conscious awareness and control that drive our behaviour. 

Here I introduce "Dopamine the Bus Driver". 

#dopamine   #neuroscience   #subconscious  

http://bradleyesau.blogspot.ca/2013/12/dopamine-bus-driver.html
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