## 2.11.12

### Factor Visualization

Stephen Von Worley has a lot of great data visualizations on his site, but this one struck me as especially cool: http://www.datapointed.net/visualizations/math/factorization/animated-diagrams/

Dots are added one at a time and they all reconfigure to make patterns which display all of the prime factors of the new number. 70 for example, is shown as 7 groups of dots in a regular heptagon arrangement. Each of the groups is 5 groups in a regular pentagon arrangement. And each of these 5 groups is a set of 3 dots in an equilateral triangle. 105 = 7 x 5 x 3. The 7, 5 and 3 are clearly visible. The animation is mezmorizing.

I did a series of screenshots and made this composite image of the integers 1-49.

1. Is there a poster available or can I make my own?

2. Good job, Mike. Thanks a lot for sharing this idea with everyone.
I notice a pattern here. The prime numbers are represented by dots arranged in a circle while composites are arranged in compositions. But look at 4, the dots are arranged in circle there too. As if 4 is a prime. Shouldn't it be 2X2 ? But depending on how 4 is represented, all the numbers divisible by 4 also could need redrawing. I would love to hear your thoughts on it. Cheers.

1. Sharp eye Raiyan! I agree. The artist who made the animation decided it looked better as a square, when a rectangle would be better. I tried both ways when experimenting with the designs and agree that the square looks better.

So it's not mathematically correct but aesthetically better. Such are the tradeoffs we make sometimes in the name of Art!

3. Fair enough.

I wonder if 49 was a deliberate choice so as to arrange the numbers such that there will be one diagonal consisting of primes except for the one in centre which is 25. This also leads me to think whether it is possible to find a prime P such that kP-k+1 is prime for values of k that result in positive integer kP-k+1. If one finds that prime, the numbers could be arranged in a square and there would be an entire diagonal of primes. A quick mental check reveals it holds for P=3 (the numbers on the diagonal are 3,5,7).

1. No it wasn't deliberate, just the right size for fitting on a poster. I made a whole series of cards with these patterns on them, with all variations, so that 15 for example was three groups of pentagons and also 5 groups of triangles. They're really quite lovely, and were good fun to make!

I blogged about them on my Norwegian blogg... it's a little tricky to read but you can see the pictures here:

http://matematikksenteret.no/content/2116/Faktorvisualisering-og-faktorspill

4. Wow! This is particularly interesting to me because I develop iOS applications for teaching math to kids.

The first app I ever released was a little matching game based on a similar technique. Check it out here: https://itunes.apple.com/us/app/fortynine/id909512755?mt=8&ign-mpt=uo%3D4

5. At line: 70 for example, is shown as 7 groups of...
What 70?