Welcome to this week’s Math Munch!
This beautiful picture comes to you from Brent Yorgey and Stephen Von Worley. If you counted the dots, you probably didn’t count them one at a time. (And, if you did, can you think of another way to count them?) If you counted them like I did, you noticed that the dots are arranged in rings of five. Then maybe you noticed that the rings of five are themselves arranged in rings of five. And then, finally, you may have noticed that those rings are also arranged in rings of five! How many dots is that? 5x5x5 = 125!
In this blog post, Brent describes how he wrote the computer program that creates these pictures. The program factors numbers into primes. Then, starting with the smallest prime factor, the program arranges dots into regular polygons of the appropriate size with dots (or polygons of dots) at the vertices of the polygon.
Here’s how that works for 90. 90’s prime factorization is 2x3x3x5:
As Brent writes in his post, this counting gets much harder to do with numbers that have large prime factors. For example, here is 183:
When Stephen saw Brent’s creation, he decided the diagrams would be even more awesome if they danced. And so he created what he calls the Factor Conga. If you only click on one link today, click that one. The Factor Conga is beautiful and totally mesmerizing.
Next up, a few months ago we posted about the puzzles of Sam Loyd – one of which was a puzzle called Get Off the Earth. In this puzzle, the Earth spins and – impossibly – one of the men seems to vanish. This puzzle is a type of illusion called a geometrical vanish. In a geometrical vanish, an image is chopped into pieces and the pieces are rearranged to make a new image that takes up the same amount of space as the original, but is missing something.
Here’s a video of another geometrical vanish:
No matter the picture, these illusions are baffling for the same reason. Rearranging the pieces of an image shouldn’t change the image’s area. And, yet, in these illusions, that’s exactly what seems to happen.
Check out some of these other links to geometrical vanishes. Print out your own here. And think about this: Are these illusions math – and, if it so, how? I came across geometrical vanishes because a friend asked if I thought the Get Off the Earth puzzle was mathematical. He isn’t convinced. If you have any ideas that you think can convince him either way, leave them in the comments section!
Finally, the Math Munch team’s home, New York City, (and this writer’s other home, New Jersey) was hit by a hurricane this week. The city and surrounding areas are still recovering from the storm. Sandy left millions of people without power and many without homes. One way people have tried to communicate the magnitude of what happened is to make infographics of the data. Making a good infographic requires a blend of mathematics, art, and persuasion. Here some of the most interesting infographics about the storm that I’ve found. Check out how they use size, placement, and color to communicate information and make comparisons.
To those in places affected by Hurricane Sandy, be safe. To all our readers, bon appetit!