More as I think of them. Feel free to post math pics you find or make, too.
Author  Comment  

Philosophaster 
Math pictures 
Lead  
Squares represent numbers from 2 to 4551, going from left to right and top to bottom. Darker numbers have fewer factors, and primes are in dark blue. Made it today:
More as I think of them. Feel free to post math pics you find or make, too.
May all beings be free from enmity, affliction, and anxiety, and live happily.
Last Edited By: Philosophaster Jun 30 11 9:06 AM.
Edited 2 times.




xpsyuvz  
Nice, Philo.
I'm seeing staircases and pillars. Trying not to be too obtuse, but how did you do that? (Did you use some kind of computer program?  It would take me a year or a decade to paint each square in such a methodical way...)


Philosophaster  
I programmed it, using a language called Processing. You can download it from the site, if you'd like to try it out.
Here is the program itself: http://philosophaster.com/c/factors.txt
May all beings be free from enmity, affliction, and anxiety, and live happily.


wannabealot  
That is incredibly cool.


Naturyl  
Looks like there is a vertical and semihorizontal "grid." Interesting.
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Philosophaster  
Naturyl wrote:Yeah, the vertical lines are multiples of 10. Diagonals I'm not so sure about, but maybe I'll write a program to print the same thing as a list and check it out.
May all beings be free from enmity, affliction, and anxiety, and live happily.




Philosophaster  
...and out to infinity. Gives you the famous "golden ratio," also called Phi. 

wannabealot  
Cool looking. What is Phi used for?


Naturyl  
Philadelphia cream cheese.
And also architecture, famous paintings and art, etc. It is considered a particularly beautiful ratio. The Parthenon, for example, was based on Phi in the form of a "golden rectangle:"
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Last Edited By: Naturyl
Mar 29 12 12:06 PM.
Edited 1 times.


Philosophaster  
Yep, it is mostly used in artistic stuff. It also shows up in the Fibonacci numbers:
Fibonacci numbers are usually defined like this: fib(0) = 0fib(1) = fib(2) = 1 fib(n) = fib(n1) + fib(n2) You start with the list {0, 1, 1}, and to get a new Fibonacci number you add the two numbers before it. So fib(3) = 1+1 = 2, fib(4) = 1+2 = 3, fib(5) = 3+2 = 5, and so on. And you end up with {0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89...}. As you get further and further into the sequence, the ratio of a Fibonacci number to the number before gets closer and closer to Phi, which is about 1.61803399. You can see the progression: 1 / 1 = 12 / 1 = 2 3 / 2 = 1.5 5 / 3 = 1.666... 8 / 5 = 1.6 13 / 8 = 1.625 21 / 13 = 1.615384... You might be curious if we could calculate some arbitrary Fibonacci number without first calculating every Fibonacci number before it (or you might not be, but I will tell you anyway). How would we calculate fib(50) or something, if we had not already figured out fib(49) and fib(48), which would (using the formula above) require knowing fib(47) and fib(46), etc. etc.? It turns out there is a way, and it also has to do with Phi. The Fibonacci formula above is called a "recursion relation," which means that it is kind of defined in terms of itself. But there is a way of turning certain kinds of recursion relations into formulas that are not defined recursively. Basically we look at the formula for fib(n) fib(n) = fib(n1) = fib(n2) ...and notice that it looks sort of like this one: x^{n} = x^{n1}  x^{n2}If we divide that equation by x^{n2}, we get: x^{2} = x  1, or (rearranged) x^{2}  x + 1 = 0. This is just a quadratic equation with two solutions for x, which we can find using the quadratic formula. The solutions (also called "roots") turn out to be...: Phi and 1  Phi. The number (1  Phi) is also known as "Psi." After some more algebra, which would be kind of tedious to lay out here, we end up with the final formula (using the Greek symbols for Phi and Psi): So you can use that to calculate any Fibonacci number you want, without calculating any of the ones before it. The exact values for Phi and Psi are:
Last Edited By: Philosophaster
Mar 29 12 1:18 PM.
Edited 2 times.


Sherezada  
Phi is the beauty ratio. The closer your facial proportions (distances) are to phi, the more beautiful you are, and vice versa. A man devoted about two decades to designing a beauty mask, which is available on the internet.
I should point out that some (many) people disagree that beauty can be quantified so coldly. 