Thursday, April 25, 2013

patterns abound

Since noticing this graphic, I've spent a little time contemplating the patterns you can make by connecting 12 dots that are evenly distributed around a circle (above: no lines, and all lines). Twelve dots around a circle has some cultural resonance - we still see them up on many walls keeping time. Feeling the pull of patternicity and agenticity, other people, more mystically minded than I, have spent a lot of time making patterns on zodiac circles that look a lot like the ones here. What really makes 12 so useful and open to this circle pattern making are all those divisors (1, 2, 3, 4, 6, 12), which contribute towards making 12 the smallest abundant number (6 is perfect because it is equal to the sum of its proper factors, 12 is abundant as the sum of its proper factors is greater than itself).

Tuesday, April 16, 2013

good ole prime spiral

For a while I have been putting up images from this blog on tumblr (if you like the pictures but not the words, that's the version of this blog for you). Far and away the most popular picture is this one, which reminded at least one fellow tumblrite of the Death Star. To be honest, I don't really know why this picture stands out from the others, or why you might choose to re-blog it rather than a picture of a kitten (I guess no one blogged this over a kitten... most choose this and a kitten).

I do occasionally throw some things up there that don't have a blog post associated with them, and I think this might be one of them (until now).

This is a picture of a quadratic number spiral with only the primes showing. The site that introduced this type of number display to me is the aptly named, which has a great overview that I won't bother summarizing here - best that you read it there.  There is lots of playing around that you can do with these spirals (you can plot polygonal numbers on them, and look for other interesting sequences, for example).

Wednesday, April 3, 2013

frames and octahedrals

Looking at a polka-dotted shower curtain the other day, I started to play a game of connect the dots. I was looking at "frames" of dots like the ones below, and counting the squares that could be made using only the dots on each frame as vertices.

So, consider an n by n square grid with points at each vertex and points one unit apart around the perimeter  For an such a grid, how many squares can be drawn by connecting 4 points on the perimeter of the grid, and what is the total area of all the squares drawn? Read no further if you want to do this yourself.

Counting the squares is pretty straight forward - for an grid with sides length n there will be n squares. There is the full n by n square formed by joining the vertices of the grid, and then a series of smaller rotated squares, the base of each formed by joining the ith point along the bottom with the (n-i)th point along the left side (consider the lower left corner to be the origin (0,0) and count to the right and up).

The areas of these squares are also easy to calculate, thanks to the right triangle that is made by the square and the frame that it is tilted in.

If you add up the areas for a given n, and look at the sequence that you get - you'll find that these give the octahedral numbers (OEIS A005900). A nice surprise (for me at least). Octahedrals are figurate numbers, like like polygonal numbers but in this case three-dimensional: two square-pyramidal numbers (OES A000330) stuck together to form an octahedron.

Figurate numbers have long been a favorite topic in recreational mathematics (there are several posts about them on this blog - like this one), and sometimes they show up when you are not expecting them.

The geometric aspect of figurate numbers sometimes allow you to express numerical relationships nicely using pictures. I don't think the picture below (which shows the n = 3 case) quite qualifies as "a proof without words", but I think it helps to show why the octahedral numbers pop out when you "draw squares on frames."