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).

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 NumberSpiral.com, 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).

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." 

why stop at four?

The Ontario Social Studies Curriculum says:

By the end of Grade 2, students will:
– recognize and use pictorial symbols
(e.g., for homes, roads), colour (e.g., blue
line/river), legends, and cardinal directions
(i.e., N, S, E,W) on maps of Canada and
other countries;

I was helping out with some Grade 2 homework the other day, and when it came time to mark the cardinal directions I reached for an old trigonometry text book to show a much more detailed compass rose. Young students can figure out how many points you have each time you sub-divide the compass (4, 8, 16, 32, ...), and the naming conventions for the points also make sense to kids (what's between North and East? North East! What's between North and North East? North North East!).

calculated thought experiment

I always feel that I come away with something new whenever I read Ludwig Wittgenstein's Remarks on the Foundations of Mathematics - likely because I understood so little on each previous read. In the book, one thing  he tries to get at is what we mean by words "mathematics" and "calculation," and in doing so he asks questions that are so basic that they call into question our implicit assumptions about what these words mean. One of these sets of questions ask about whether our mental state and attitude in any way influences whether or not we are actually "doing mathematics."

For example,

"Imagine the geometry of four-dimensional space done with a view to learning about the living conditions of spirits. Does that mean that it is not mathematics?... Could people be imagined, who in their ordinary lives only calculated up to 1000 and kept calculations with higher numbers for mathematical investigations about the world of spirits?" 

Does it matter what we think we are doing when we are doing math? As long as we are moving the symbols around correctly does it still count as mathematics?

Elsewhere he asks "What would happen, if we rather often had this: we do a calculation and find it correct; then we do it again and find it isn't right; we believe we overlooked something before - then we go over it again and our second calculation doesn't seem right, and so on. Now should I call this calculating, or not?"

Does calculation require a social convention - if one person performed something once, could it be considered an algorithm? "What about this consensus - doesn't it mean that one human being by himself could not calculate? Well, one human being could at any rate not calculate just once in his life."

Some of the most fascinating thought experiments that Wittgenstein proposed (way back in 1942-1944) were about (what we would now call) computers or "mobile devices":

"Does a calculating machine calculate? Imagine that a calculating machine had come into existence by accident; now someone accidentally presses its knobs (or an animal walks over it) and it calculates the product 25 x 20."

Has any calculation happened in this case? Later he suggests a scenario that now seems quite familiar:

"Imagine that calculating machines occurred in nature, but that people could not pierce their cases. And now suppose that these people use these appliances, say as we use calculation, though of that they know nothing. Thus e.g. they make predictions with the aid of calculating machines, but for them manipulating these queer objects is experimenting. These people lack the concepts which we have, but what takes their place?"

Very (unintentionally) prescient Ludwig! We are actually now living in a reality which closely resembles this thought experiment - and an environment that sounds like the classroom of the future as imagined by Computer Based Math. What will replace current concepts of number once our experience with calculation is mediated entirely by machines whose cases cannot be pierced? And will we even notice that they have been replaced once they are gone?