primes on a log spiral

Since looking again at Theodore Andrea Cook's The Curves of Life a few posts back I've been planning on playing with logarithmic spirals, which are identified in that book as the type of spiral that you often encounter in nature and in architecture. I was inspired to finally spend some time with them after reading a recent post on Math Hombre.

For fun I treated the curve like a number line and plotted prime numbers on it  using Processing. It seems to me that a nice thing about curling up the number line is that it allows you to take in more of the line at a glance. You can notice both the (seemingly) increasing gaps that occur between primes, as well as the (apparent) persistent occurrence of twin primes.

the best of 2011

Once in a while I get sent books to review and recommend - this is very nice, but unfortunately I haven't had the chance to post many reviews. It is not only in the book review department that I'm failing - I seem to be having a more general problem finding time to do any recreational mathematics (and then to post about it here).

So it was a treat to receive a copy of The Best Writing on Mathematics, 2011 (Mircea Pitici, ed.), a book that solves both problems: it is a book that I really have to recommend, and it is also certain to inspire me in more mathematical recreations.

   

The anthology gets off to a good start: In the forward, eminent physicist Freeman Dyson proclaims that "Recreational mathematics is a splendid hobby which young and old can equally enjoy... To enjoy recreational mathematics you do not need to be an expert." A great statement that I should probably take as the motto for this blog.

This anthology offers a lot for recreational mathematicians, mathematics educators, professional math practitioners, and hopefully others as well. A couple of the articles in the collection were "old favorites" that inspired posts on this blog when they appeared in their original contexts. Doris Schattschneider's article on Escher and Coxeter prompted this post, and Dana Mackenzie's article on Apollonian gaskets motivated this one and another. These articles remain among my favorites in the collection, but there are many others that make interesting reading, including others like these that focus on aesthetic aspects of mathematics (in ribbed sculptures, in bronze and stone, and in strange-attractors).

Some of the articles are against the grain of our prevailing zeitgeist - Melvyn B. Nathanson strikes a somewhat contrarian tone against the promises of polymath, and Martin Campbell-Kelly wistfully recalls the now obsolete numerical table. I particularly liked how Underwood Dudley asks "What is Mathematics For" and takes aim at an assumption that is now almost sacrosanct: that we teach mathematics because it is useful.

Dudley's thesis, that mathematics (particularly school algebra) may not be used very often but helps us learn to think and reason, although not currently popular, is actually one of the oldest arguments in favor of learning algebra. The very first English-language algebra textbook (published by Robert Recorde in 1557) was titled "The Whetstone of Whitte" precisely because algebra was considered by its author to be like a knife-sharpener for the brain. Of algebra, it said:

Its use is great, and more than one. 

Here if you lift your wits to wet, 

Much sharpness thereby shall you get. 

Dull wits hereby do greatly mend, 

Sharp wits are fined to their full end.

   

I think that many who appreciate the appeal to the aesthetics and cultural significance of mathematics in Lockhart's Lament will agree with Dudley's call for a more subtle (and accurate) understanding of what mathematics education gives us beyond the merely utilitarian.

Although there is a broad appeal to these articles, I'm guessing that the audience that will most appreciate this collection are those involved in mathematics education. Of particular interest to teachers are two career retrospectives by eminent math-education- theorists Alan Schoenfeld and John Mason, the previously mentioned paper by Underwood Dudley, two other papers specifically about mathematics education, as well as a paper on the cognitive aspects of perceiving numbers.

 Thinking about these kinds of articles, I was reminded that when Martin Gardner died in 2010, many wrote about how his columns inspired them to take up mathematics as a hobby and as a profession. With Gardner as an example, it is clear that the authors of these and other popular mathematics articles are doing something worthwhile.

more window patterns in gsp

You'd be right in saying 'hey, these are just a bunch of overlapping squares.' Yes. The only redeeming thing that I can point to is that they are made by following a rule, and the rule is one that is easy to reproduce without using any external measuring device (like a ruler or protractor), only the squares themselves. Think origami: you find midpoints by folding, etc. In this case, GSP is used, but only simple constructions like mid-point finding and segment creating.

The trick is to find a rule that allows you to start with a square and then construct two points that you can base another square on, and then repeat.

These were made from the same window-pattern instructions mentioned here.

A4 window patterns and special triangles

A short while ago I mentioned that A4 paper has nice proportions - it's a silver rectangle, which means that the ratio of its long side to its short side is sqrt(2). Because of their nice proportions, silver rectangles can be used to construct special triangles that we know and love from trigonometry.

One nice way to note the angles in these triangles is to form window patterns based on them - these are shapes made from overlapping pieces of paper that have been rotated according to a rule. The term window pattern comes from William Gibbs - so named because if you put them up in a window, the light shining through the different layers of paper reveals additional patterns and shapes.

Here's one example of the special-triangle-window-pattern process. Start with an A4 or similarly proportioned rectangle, and find the midpoint of one of the shorter sides (by folding the paper, for example).

Now take a second rectangle  the same size, and place it so that one vertex lines up with the midpoint drawn, and the other vertex along the same short side of the second rectangle touches the long side of the first. It's easier to see this in a picture:

By doing this, you've constructed the tricky length of sqrt(3)/2 and built the 30-60-90 (pi/6, pi/3, pi/2) triangle. You can confirm that the angle that you've formed a 60 degree triangle by repeating the process and finding that you come "full circle" after 6 pieces of paper (360/6 = 60).

If you change the first placement a bit so that the second rectangle lies mostly across the interior of the first, you get the pattern at the top of the post.

These are nice patterns, but they don't actually use the special properties of A4 (you could do a similar thing with square or letter paper). A little more complicated placing of one rectangle over the other can allow you to create a right triangle with one leg equal to 1 and the other equal to sqrt(2)-1. This is not one of your "standard" special triangles, but it is special in that it allows you to calculate exact values of certain angles (which angles, we'll find out when we complete our pattern).

Here's what the placement looked like that constructed this triangle. I'm afraid that text instructions for the placement would be just too much for this post - maybe you can figure out how it is done from the diagram :).

If you continue placing the rectangles, you will find that it takes 16 of them to come back to the start, which tells us that our triangle contains an angle of pi/8 or 22.5 degrees - the others are pi/2 (90) and 3pi/8 (67.5).

So.. our new special triangle tells us, for example, that tan(pi/8) is equal to sqrt(2)-1 (what other exact values do we get?).

animated iterations - too much fun

Not much to this post - just playing with the GSP sketch that I pointed to earlier. These are just iterations, plus animation, plus tracing, with what I think are some nice results.

The 'colored Pythagoras tree' fractal below is a classic that I learned in a GSP workshop years ago, and it's based on one of the projects in the free booklet 101 Project Ideas for GSP. I'm sure there are some instructions for the whole thing floating around somewhere. [Update: See the Nov 15th blog post at sine of the times for some instructions on the basic tree.]

The image below is a later stage of the one at the top of the post - an iteration made up of pentagons and curves  - the bottom image shows what the first generation of this iteration looks like.