mathrecreation

a fractal family

2012-02-15T12:50:00.000-08:00

These fractals were generated in a way similar to the Mandelbrot set. For the Mandelbrot set, you use the recursive formula z_{n+1} = z_n^2 +c, where z_0 is 0 and c is an element of C. As you input c values, and perform the recursion, if the magnitudes of the results get big they are not in the set, if they stay small, they are.

For these, the same recursive formula z_{n+1} = z_n^2 + c is used, except z_0 is your input value and c is a constant complex number. Different values for c yield different fractals. As with the Mandelbrot set, initial values (z_0 in this case) are in the set if the recursion stays bounded.

My favorite so far is this last one - when I look at it, there seems to be a slight optical illusion in play that makes the dark centers seem to grow slightly as you look at the picture. In all these images, points that stayed small for more iterations are darker.

fractal island

2012-02-10T16:25:00.000-08:00

This strange blob that looks a bit like an island was generated by the same process as the Mandelbrot set described here, except instead of using z_0 = 0 in the z_n = (z_{n-1})^2 +c recursive function, it uses z_0 = (-1,1/2).

Also, following Alexandre Muñiz's suggestion in his nice post Children of Julia Sets, I used 255 iterations, and assigned the grayscale colour to the points based on which iteration they failed to converge. This gives the wispy archipelagos in the south of the island.

off topic: event supporting alt.ed. in Ottawa

2012-02-06T08:50:00.000-08:00

If you know anyone in Ottawa who has an interest in education or parenting issues, please pass along word of this fundraiser for Churchill Alternative School.

An Evening With Dr. Gordon Neufeld, best-selling Author of "Hold On To Your Kids"

Churchill Alternative School Council is pleased to be hosting an evening with Dr. Gordon Neufeld, author of “Hold On To Your Kids”. Dr. Neufeld is a Vancouver-based psychologist known for groundbreaking work on peer orientation - peers replacing parents in the lives of our children.

Thursday, February 16, 2012 @ 7:30 p.m.
Nepean High School Auditorium, 574 Broadview Avenue, Ottawa

Tickets: $20 in advance - $25 at the door
Tickets can be purchased online through Dovercourt Recreation Association
(enter course # 76642), or call 613–798-8950

Dr. Neufeld ~ “What makes a child easy to parent”
http://www.youtube.com/watch?v=PcaMsZrElnE

Dr. Neufeld ~ “Why children need rest and how to provide it”
http://www.youtube.com/watch?v=nUHnMfa_aKE

What is an alternative school? It depends on who you ask, but a good place to learn about alternative education is at the Alternative Education Resource Organization's website. Basically it's progressive, student-centered, non-competitive, cooperative, and community-based education that seems to always be against the grain of what school boards, administrators, and corporations think schools should be like.

better late than never: Mandelbrot Set

2012-02-03T10:37:00.000-08:00

Although I have a math and computer science background, this (above) is my first attempt to draw the Mandelbrot set. It seems overdue since plotting the set is such a well established computer-math project that it's almost cliche (there is so much online about this, but this Math Munch article has some nice pointers). But if I am just finally getting around to this, then it's not too late for you too.

So, if you haven't written a little program to draw the set yet, I strongly recommend it. There are lots of nice things to think about as you explore this, and very little of it has much to do with fractals - although having that strange inkblot-like image appear at the end is the carrot, or cauliflower, that will motivate you along.

I'll share my little program and explanations below. You can find or create much more slick and efficient code if you try. For example, you don't really need to know the magnitude of the complex points (the square of the magnitude is enough), and I'm sure my point generation isn't the best.

what are you plotting?
I would guess that most students on the pre-calc train tend to think of plotting points almost exclusively in terms of single variable functions of real numbers. This is not one of those kinds of plots. Instead of (x,y) indicating y = f (x), we are plotting points belonging to C, and making them one colour if they belong to a particular set M, and a different colour if they don't.

To figure out whether a point is in M, for every point that you consider, you need to construct a particular sequence - if that sequence stays bounded, then it is in M, but if the sequence is unbounded it isn't in M. The further you go in calculating your sequence, the more sure you are that your points are actually in the set.

some complex number stuff
You don't need to write much code to implement the complex number operations that you need for this, but I'm partial to encapsulating this sort of thing in a general purpose class like this Processing example:

class CPoint {
float xvalue;
float yvalue;
float pointsize = 0.2;
CPoint(float x, float y){
xvalue = x;
yvalue = y;
}
CPoint sum(CPoint p){
return new CPoint(xvalue +p.xvalue, yvalue+ p.yvalue);
}
CPoint prod(CPoint p){
float newx = xvalue*(p.xvalue) - yvalue*(p.yvalue);
float newy = xvalue*(p.yvalue) + yvalue*(p.xvalue);
return new CPoint(newx, newy);
}
CPoint squared(){
return this.prod(this);
}
float magnit(){
float part = pow(xvalue,2) + pow(yvalue,2);
return sqrt(part);
}
void display(float xshift, float yshift, float zoom) {
fill(255,255);
ellipse(xvalue*zoom + xshift, yvalue*zoom + yshift, pointsize, pointsize);
}
}

The key item is the complex multiplication - this is what distinguishes a point in C from one in R^2.

I have a like/dislike relationship with Processing - I like how quickly things can be created and that it makes nice pictures without much effort, but I dislike how I can't seem to help breaking fundamental programming rules when using it (I end up using global variables, and always break model-view separation) - likely a personal problem, rather than a problem with Processing itself.

determining if a point is in the set M
You are going to be finding a sequence of points based on a special rule - if applying the special rule repeatedly causes the points to get too big, then they are out of the set. In addition to knowing the special rule, you also need to know "how big is too big" and "how many times will I apply the rule."

The calculation is encapsulated in this other Processing class Map. Can you figure out what the rule is, and how you specify the "how big" and "how many" parts of the calculation?

class Map {
CPoint c;
CPoint first;
CPoint current;

Map(CPoint initial, CPoint cValue){
first = initial;
current = initial;
c = cValue;
}

void iterate(){
current = current.squared().sum(c);
}

boolean iterate(int iterations, int bound){
for(int i=0; i< iterations; i++){
this.iterate();
if(current.magnit()>bound) break;
}
return (current.magnit() < bound);
}

void display(float xshift, float yshift, float zoom){
c.display(xshift, yshift, zoom);
}
}

If you only calculate for a few iterations, some points that should not be in the set get included - after 5 iterations, you get something that looks like a skate.

Having a somewhat less precise plot gives you funkier looking pictures than the crisper image that you get with more iterations. For this reason, people often include points not quite in the set in their pictures of the Mandelbrot set, and often use colour to show at which iteration a given point failed the membership test. These fuzzy pictures are really good examples of fuzzy sets - where instead of set membership being true or false, it is a range of values indicating by what level the point failed to be in the actual set. The picture below was generated using 20 iterations.

plotting the set
The last thing you need to do is to set up your window, and generate your points. Here's the main Processing file that I used for this (no zoom or panning in this one - exercise left to the reader):

//various magic numbers
int windowX = 600;
int windowY = 500;
int iterations = 200;
int zoom = 250;
int disk = 2;
int numberPoints = 100000;
int randomRange = 100;
float centerX = -0.5;
float centerY = 0;

//init
void setup() {
size(windowX, windowY);
noStroke();
smooth();
background(0);
}

void draw()
{
loop();
CPoint zero = new CPoint(0,0);
float x;
float y;
int signx = -1;
int signy = -1;
Map newMap;
CPoint cPoint;
for( int i=0; i < numberPoints; i++){
cPoint = randomCPoint(1.5, randomRange);
newMap = new Map(zero,cPoint);
if(newMap.iterate(iterations, disk)){
newMap.display(windowX/2 - centerX*zoom, windowY/2 - centerY*zoom, zoom );
}
}
}

CPoint randomCPoint(float bound, float depth){
float x;
float y;
int signx = -1;
int signy = -1;
x = bound*random(depth)/depth;
y = bound*random(depth)/depth;
if(random(100)<50) signx *= (-1);
if(random(100)<50) signy *= (-1);
return new CPoint(signx*x,signy*y);
}

If you implement it like this, the set will slowly emerge as more and more points are tested. This crisper version below used 200 iterations (maybe a bit excessive).

primes on a log spiral

2012-01-11T11:29:00.000-08:00

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

2012-01-10T14:45:00.000-08:00

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

2011-11-20T18:40:00.001-08:00

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

2011-11-11T20:30:00.001-08:00

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

2011-11-11T10:38:00.001-08:00

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.

some gsp fractal sketches

2011-11-05T19:32:00.000-07:00

I found an old GSP file with a bunch geometric fractals in them - I thought that some of them looked nice, so I've posted them here. If you'd like to try them out, you can get the GSP file here - for the most part, they involve pretty standard use of the "iterate" feature.

Animating them in random ways creates some strange looking forms - the same sketch that produces the pentagon fractal above also gives the one below.

The same sketch that gives the snowflake-like pattern at the top of the post gives this odd looking sponge:

butterflies, bus transfers, cotangents

2011-11-02T19:16:00.000-07:00

Because it is my habit to do paper-folding while using public
transportation, people sometimes turn their heads and cast pitying
eyes upon me; but because of my strong concentration
at these times, their looks do not bother me.

- Kazuo Haga, Origamics

The origami model that I fold most frequently is Nick Robinson's A4 butterfly. You can find this model in Nick's book The Origami Bible (unfortunately I don't think that the instructions are posted on his website). Being in North America, A4 paper is not so easily obtained, but luckily I get handed a little piece of almost-the-same-ratio-as-A4-paper every workday morning in the form of a bus transfer.

Rectangles that have the same proportions as A4 paper have nice geometric properties - they are silver rectangles (named in contrast to golden rectangles), and the niceness of these silver rectangles is due the fact that the ratio of the long side to the short side is sqrt(2). If you don't have an appropriately proportioned bus transfer, or you want to make your own A4-style silver rectangle, Nick Robson provides some helpful instructions here.

Really, you don't need a perfect silver rectangle for the butterfly model - it is pretty forgiving, and tends to work well for bus transfers, ticket stubs, and magazine-subscription inserts (golden rectangles, for example, work too). However, if you look at the simplified crease pattern you can see that the model completely breaks down if the ratio of long to short side is too small or too large. To make things precise, things don't work at all if the ratio of long-side to short-side, r, approaches r = cotan(pi/4) = 1 on the low end, or r = cotan(pi/8) ~= 2.414 on the high end.

The reason that these ratios are as they are is that the fold that creates the outer edge of the wing has an angle of pi/4 with the midline of the paper, while the fold that creates the inner edge of the wing has an angle of pi/8. If either of these lines hit the corner of the rectangle, the model no longer works. That is why the ratio of long-side to short-side (or in trig-ratio speak, adjacent to opposite) is bounded by the cotan of these angles.

Butterflies attempted with almost-square paper have large bodies and almost no wings, while the long paper produces butterflies that have too-long wings and undersized bodies. Although it seems that the model enforces sharply defined boundaries on the range of paper can be used, finding the size of paper that produces the optimal butterfly is another problem. Are silver-rectangle butterflies the best, golden ones, or maybe ones with r = cotan(3pi/16)? This might be a question of personal origami-aesthetics rather than mathematics.

tintin - the calculus affair

2011-10-20T14:29:00.000-07:00

There is a larger version for printing here - I suggest that you just print the first 4 pages.

grade one functions

2011-10-05T18:48:00.000-07:00

The function concept is sometimes said to be pervasive in school mathematics - although it is not until secondary school that it is formally introduced, many elementary school math math activities can be thought of as preparing the way for understanding and working with functions.

However, few students, or teachers, ever make a connection between the functions that they eventually learn about (or teach) in high school and those implicitly encountered in early primary grade activities.

Take for example, the symbol patterns that grade one (and kindergarten) students are often asked to look at, like this one:

Given such a pattern, students are often asked to describe the pattern, continue it, or identify its "core-pattern" or kernel - the repeating unit within the pattern.

How are patterns like this related to functions? Or how do we model such a pattern using functions?

Any sequence of symbols can be thought of as a function from the natural numbers N onto a set of symbols S. You can show this mapping by simply labeling the pattern with N.

So, the pattern can be thought of as a rule f that associates a symbol with each natural number. That's nice, but how can you represent the "rule" that determines which symbols appear where? One way is to return to the core-pattern idea, which reminds us that our pattern is cyclical. In this example, it is a 4-cycle, which you might visualize as a sort of loop among four vertices, or maybe as a clock with 4 positions:

A nice function, sometimes introduced in school math, that maps the natural numbers onto this kind of structure is the modulo k operation. In this case "modulo 4" is what we need: for any natural n, n modulo 4 returns the remainder of n divided by 4, which is always one of the numbers 0, 1, 2, or 3. Modular arithmetic is sometimes called "clock arithmetic" because it is cyclical, and the function "modulo 4" can be visualized as wrapping the number line around a clock with 4 positions on it. So let's say that h is the function that maps N onto the set K={0,1,2,3} following the rule that h(n) = n modulo 4. It's graph looks like this:

Now consider another function g that maps the set K onto the symbols in our set S, it's co-graph looks like this:

The overall pattern can be described as the composition f = hg, and perhaps visualized as a clock labeled with the symbols from the pattern. This shows both the cycle that defines the pattern (the function h) and the arbitrary association of this core-pattern onto particular symbols (the function g), so I think this is a pretty nice way to represent the pattern in terms of functions, but there are other possible descriptions.

So what's the point?

Well, it's fun to do this sort of thing isn't it?

I think it is worthwhile thinking about why we don't often look back and apply the language of functions to early (very early!) math. This may be in part because, as described in the Common Core Standards, "In school mathematics, functions usually have numerical inputs and outputs and are often defined by an algebraic expression." Here we have a non-algebraic function that has a non-numeric output. So, although we do eventually learn about and teach functions in school, they don't tend to be the kinds of functions that lend themselves to talking about the patterns, sortings, colourings, etc. that are encountered in elementary school.

Although this whole thing may seem weird from a school math perspective, it might seem more natural to think of the pattern this way if you are used to thinking like a programmer, and were trying to find a way to generate patterns like this. The pictures in this post, for example, were made in Fathom using the idea of the two functions g and h:

Another thing you might ask: in doing this activity, did we "uncover" the functions that lie behind this simple elementary school patterns, or did we "model" the pattern using functions? I would probably say that we used the language of functions to make some of the underlying structure more apparent.

simple origami and math: traditional envelope

2011-09-27T10:22:00.000-07:00

Another very simple origami project that can help spark mathematical conversations is the traditional envelope. Like the other simple origami projects that I've mentioned in previous posts (the jumping frog and the paper cup), the instructions for the envelope are available from the Origami USA diagrams page.

This project is made with letter-sized paper (8.5 by 11 works great, I haven't tried A4), and is an accessible and appealing "practical" project (who doesn't like the idea of folding up a note so that it is its own envelope?).

The crease pattern (at the top of the post) is a great potential source of math-themed conversation. Identifying the types of shapes (the envelope itself is hexagonal - a rectangle with two corners cropped) and finding their area (what is the area of the final envelope compared to the size of the original note paper?) provide some things to explore. Parallel and perpendicular lines, and a few 45 degree angles, make talking about lines and angles in the pattern accessible for younger students.

The pattern is obviously symmetrical, but what kind of symmetry does it have? Many crease patterns that you might look at, like the paper cup (pattern below), have reflective symmetry. The envelope, on the other hand, has rotational symmetry (albiet a simple 180 degree rotational symmetry).

Something else to take note of is the "handedness" of the finished envelope. If you are careful when you follow the instructions, you will end up with an envelope with a front that has its top left and bottom right corners cropped (which is best if you want to affix a stamp to the top right corner).

However, if you are folding the envelope by watching someone else or folding from memory, you are just as likely to end up with its mirror image, an envelope that has its bottom left and top right corners cropped. The envelope, like a many modular origami units (like Sonobe units) has a right-handed and a left-handed version - if you fold a certain way you get one orientation, if you fold another way, you get the mirror-image. Which fold in constructing the envelope determines the orientation of the final model?

punctured knight's tours

2011-09-23T13:09:00.000-07:00

It's pretty clear that a chess knight cannot travel to every square on a 3x3 board. If the knight starts in the center square, it cannot move at all, while if it starts on any other square, it cannot reach the center. If you puncture the board by removing the central square, your dissapointment with the simplicity of the remaining problem might be somewhat relieved by the niceness of the solution: there is only one possible knight tour on a punctured 3 by 3 board (up to rotation, reflection, and change of direction), and it is closed with a nice star-shaped path.

This nice pattern inspired me to look at knight tours on other punctured square boards - boards with odd dimensions that had the central square removed. On boards that are 5 by 5 I could only find two distinct solutions (other non-distinct tours can be found by rotation, reflection, and reversing direction), but it is likely that there are more. Neither of the ones that I found are closed - the first follows a spiral path and always travelling in the same direction (like the 3 by 3 case), while the second starts out as a spiral in one direction and then changes direction after the ninth move.

However, I found that a nice closed tour can be created on a 7 by 7 punctured board by "gluing" together rotated copies of an open 3 by 4 tour. The technique of building up knight tours from smaller ones by gluing them together in a way that the knight can move from one to the next is a common one, and is particularly helpful when you want to create symmetric or semi-magic tours (this is described in Martin Gardner's essay "Knights of the Square Table" from Mathematical Magic Show).

Although this technique does not give you a spiral pattern like the 3 by 3 case or the first 5 by 5 example, the copy and rotate technique gives the path another nice pattern. You can see this symmetry in the 7 by 7 punctured board can be seen if you look at the cell values modulo 12 (doing this tracks where the values in the original board are rotated to).

If you connect the values that are equal to each other mod 12, you get a nice pattern of rotated nested squares - this pattern is completely independent of the tour on the initial 4 by 3 board: all it shows is the rotation that was applied to make the larger board. The image below has done this for some values (not all) - for example 5, 17, 29, and 41 form a square, as do 8, 20, 32, and 44.

This pattern is reminiscent of a more ideal version of the same pattern, which can be made using iterations in Geometer's Sketchpad (the gsp file used to create this is here). It seems that one way or another, we end up finding spirals.

You can use the same "rotate and glue" process to create a closed 9 by 9 punctured tour, made up of copies of a specially constructed open 4 by 5 tour. There are several open 4 by 5 tours that can be glued together to make a closed 9 by 9 punctured tour - here's one below:

Here are some previous posts about knight tours:
knight moves
closing time
kixote, or knight tour puzzles
more kixote

spirals

2011-09-10T18:54:00.000-07:00

More than any other book that I know of, Theodore Andrea Cook's The Curves of Life shows the extent of our fascination with spirals. First published in 1914, it is an odd blend of 19th-century natural history, amateur mathematics, and art history. On the mathematics of spirals, it is not the best source, Conway and Guy's The Book of Numbers has a better overview on spirals in plants, but it is unmatched as a compendium of all things spiral.

I was thinking about the allure of spirals while I finally got around to attempting some better renderings of spirals from earlier posts. The older pictures in this blog were made with Fathom, which worked well, but these drawn using Processing look a bit nicer I think, and the code is easier to play with.

The spiral below is a quadratic spiral displaying the triangular and hexagonal numbers, originally from this post.

This other spiral is a phyllotaxis spiral like the ones described here. The picture at the top of the post is based on the one below - with edges between points shown instead of the points themselves.

For what it's worth, the Processing code for these and other similar spirals is here. If Processing isn't your thing, you can find Mathematica and Python versions of polygonal-numbers-on-quadratic-spirals at Walking Randomly.

tesseracts and factor lattices

2011-08-26T14:32:00.000-07:00

I shall just sit down for a moment and pop on my boots and then I'll be on my way.
Speaking of ways, pet, by the way, there is such a thing as a tesseract.
Mrs. Murry went very white and with one hand reached backward and clutched
her chair for support. Her voice trembled. "What did you say?"
- A Wrinkle In Time, Madeleine L'Engle

We're likely not as troubled by tesseracts as Mrs Murry, but it is a nice little surprise when you come across them when they're unlooked for. You'll encounter them (or at least their 1-skeletons, the vertices and edges) while drawing factor lattices.

A factor lattice for a number n has as its nodes all factors of n. Two nodes a and b have an arrow (directed edge) between them if a divides b. Usually we don't draw all the arrows, just the ones where b/a is a prime factor of n - all the other arrows are found by composition, or are the trivial arrows that are the loops on each node (a divides a). (In this way, factor lattices are nice examples of very simple and special categories). Some notes on drawing factor lattices are here.

The factor lattice for 1 is just a single node.

The factor lattice for any prime p consists of two nodes, and the only non trivial arrow is one from 1 to p. Sometimes people ask why 1 isn't a prime - this doesn't quite answer the question, but provides an example of how 1 is different than numbers that are prime.

A number like 6, that is the product of two primes gives a factor lattice that looks like a square.

A number like 30 that is the product of three distinct primes produces a factor lattice that looks like a cube.

And 210 is, you guessed it, the smallest number that produces a factor-lattice that looks like a (squished frame of a) tesseract (the 4-dimensional hypercube).

You can go further (of course) but it gets harder to see what's going on in the diagrams. 2310 is the smallest number that gives us a 5-cube.

more kixote: knight's tour puzzles

2011-08-17T09:05:00.000-07:00

Here are some new kixote or knight's tour puzzles. The pdf file "kixote set 2" has three new puzzles and solutions, while "kixote set 1" includes the two puzzles and solutions that I posted earlier. These are not too difficult - I haven't figured out how to make difficult puzzles yet...

Kixote set 1: 2 knight's tour puzzles and solutions

Kixote set 2: 3 knight's tour puzzles and solutions

Any feedback would be welcome. :)

perspective

2011-07-25T12:12:00.000-07:00

Increasingly, it seems that many of the things I read or otherwise stumble across are connected. I have two competing theories for this apparent synchronicity. The first hypothesis is that I'm a perceptive fellow who is increasingly attuned to relationships in the world around me. The competing theory is that as I get older I am becoming more closed-minded to the point I automatically interpret everything using narrow categories of thought that are becoming ever-more rigid. I like the first theory, but the second is pretty compelling.

So, whether by perceptiveness or closed-mindedness, a number of things I've seen over the last month or so have fallen under the rubric of "art in mathematics and mathematics in art."

A few weeks back I visited the art gallery near to me (NGoC) and found that it had a few works by Albrecht Druer, who is often mentioned when mathematical techniques to achieve hyper-realism and realistic perspective drawing are discussed.

The same mathematical techniques that produce realistic perspective drawings can also produce surprising anamorphic images. For example, an anamorphic garden was created recently at Paris city hall by François Abelanet.

According to an essay from insidehighered.com, the role of mathematics producing perspective in art can motivate students to take a closer look at art-enabling mathematics (see also this textbook and this video).

In addition to sometimes using mathematical techniques, art can be inspired directly by mathematics itself - obviously mathematical art is often inspired by geometric forms such as polyhedra and tessellations. Take a look at the Bridges organization (see in particular the archives of the art presented at past Bridges confrerences), and the sculpture that hangs in the Fields Institute, recently captured in a post by Ivars Peterson. Math inspired art is not new - Jean-Pierre Luminet writes of the long history of our desire to see the beauty of pure geometric forms reflected in our arts and in the actual world in which we live.

Mathematics-based inspiration is most obvious when it takes the form of visual arts but it is not unknown to inspire performance and theater. Tony Orrico has posted videos of his circle-generating performance art on youtube, which reminded me of similar, but less artistic, compass-less circle drawing at the first (and last?) world freehand circle drawing championship of a couple of years back.

Jump math's John Mighton is a an example of the infrequently encountered mathematician-playwright. One of his plays, The Little Years, is being staged as part of this year's Stratford Festival (in Stratford Ontario). A study guide that was produced a few years ago for this play when it was performed in Ottawa features an overview of some famous women mathematicians.

Mathematics itself, and not just geometry, is often said to be beautiful, sometimes in laments that its beauty is not generally appreciated, or included in mathematics education. N Sinclair has an interesting paper on the role of mathematical aesthetics in mathematical learning and inquiry.

There's lots out there devoted to math-in-art and art-in-math - feel free to share anything that you've come across.

math blog aggregators

2011-07-18T07:58:00.000-07:00

Over the last few days I've been seeing a lot of math blog aggregators.

First, was the Math Teachers at Play #40 blog carnival over at Math Mama Writes.

Second, was Mathblogging.org - "your one stop shop for mathematical blogs."

Third, was the Alltop math topic page - "the most topular stories."

Any other good aggregators out there?

knight's path puzzle solutions

2011-07-02T16:48:00.000-07:00

Here are solutions to the "kixote" puzzles from the previous post.

kixote, or knight's path puzzles

2011-06-30T11:27:00.000-07:00

If you can make puzzles out of king's tours, why not try to do the same with knight's tours? Here is a first attempt. Remember, a knight moves this way:

So, the goal of these puzzles is to complete a "knight's tour" of the board, honoring the numbers that have already been filled in. The types of boards that you can use for a knight's tour are much more limited than those you can use for a king's tour. The two puzzles in this post are based on a 5x6 board (as in "quick chess" if you have ever played that).

First draft of "kixote" instructions: Fill in the missing numbers so that when followed in order they form a "knight's path" around the grid, where each square is visited exactly once. A knight is a chess piece that follows an L-shaped path, moving in one of these ways: (a) two steps vertically followed by one step horizontally, or (b) two steps horizontally followed by one step vertically.

Kixote Puzzle A

Kixote Puzzle B

OK, but what about the silly name "kixote"? Well, it's clear that any number puzzle has to have a three syllable name that sounds vaguely Japanese, and this one should have something to do with knights that roam around.

It seems that one way to make a puzzle is to start with a problem and its solution, and then hide part of the solution. Finding the solution to the problem is now just a puzzle, since some of it is right there in front of you. In this case, the original problem is finding a knight's tour - the puzzle is generated by taking some particular solution and hiding some of it. To be a well-formed puzzle, you should only be able to obtain the original solution, not several other solutions that just match on their exposed parts. Making sure that the puzzle is well-formed either takes good puzzle-making skills, or good puzzle-checking algorithms. Right now, I have neither - please let me know if these puzzles work or not.

Update: the solutions to the puzzles in this post are here.

Hidato, or a king's tour

2011-06-28T20:02:00.000-07:00

I picked up a logic puzzle book on the weekend and learned of Hidato puzzles - a type of number puzzle that was developed (recently, I think) by Gyora Benedek. This appealing genre of puzzle was, according to Benedek, inspired by trying to follow the path taken by small fish as they darted through a coral reef.

In a Hidato puzzle you are given an n-celled grid where some cells contain numbers while others are left blank. Your task is to fill in the blanks so that the numbers form a the consecutive series from 1 to n. Cells are considered adjacent if they are touching edges or corners. Another way to think of the puzzle is to imagine moving along the grid while trying to fill in the numbers 1 to n in order - you are allowed to move one square at a time either horizontally, vertically, or diagonally.

Just as Sudoku are latin squares with missing values, so the Hidato are really king's tours in disguise. A chess king can move one cell in any direction, and a king's tour is a complete covering of the chessboard where each square is visited exactly once (a hamiltonian path on the king's tour graph). As with knight tours, a king's tour can be open (if the final square is not adjacent to the start) or closed (if you are one move away from the beginning when you get to the end). Below is an example of an open king's tour on standard 64 cell chess board.

A solved Hidato is a completed king's tour since it is played on a chess-like grid and the Hidato moves are precisely those which a king is allowed. Whether by accident or design, the Hidato puzzles I've seen so far are not 8x8 boards like in standard chess - I've seen 6x6, 7x7 and a variety of rectangular and odd-shaped boards.

Here are two not-too-hard Hidato-inspired king's tour puzzles (note that I am not calling them Hidato - that's a trade mark... they are "king's tour puzzles") on a 5x5 board (the solutions at the bottom of this post). These puzzles were created by first creating a king's tour and then only labeling some of the cells - hopefully enough to guarantee the uniqueness of the solution.

King's Tour Puzzle A

King's Tour Puzzle B

You can generate king's tours using the same algorithm as used for knight's tours. You can also use the same knight-tour techniques to transform an open king's tour into a closed king's tour. The "king's graph" is much more connected than the knight's graph, so tours are much easier to generate (and close). The interior cells have 8 neighbours, with cells at the edges and corners with 5 and 3, respectively.

If you picture the board as a graph, kings have a lot more edges to travel along than knights do.Interestingly, the sequence, 6, 20, 42, 72, 110,... (A002943) gives the number of edges in a king's graph for an n+1 by n+1 board, and this sequence shows up as a diagonal in the Ulam number spiral of a few posts back.

Here are solutions to the small king-tour puzzles above.

Puzzle A Solution

Puzzle B Solution

graph paper collection 1

2011-06-24T13:29:00.000-07:00

Neat graph-paper variants are easy to come by, but I decided to pull together some collections that lend themselves to certain geometrically-inspired math doodling. The first collection includes isometric dot paper, tumbling block paper, and some detached square paper - all generated at incompetech.com (what a name!).

Isometric dot paper is now a staple of middle-school math (I only first saw it a few years ago though).

Tumbling block paper is a quasi-regular rhombic tiling that's really just the isometric paper with the dots connected in a certain way.

One of the things that the "detached square" paper helps you draw are overlapping columns that look like the Penrose impossible staircase.

The graph paper and other stuff from this blog can be found here.

jump math

2011-06-07T07:52:00.000-07:00

I attended a JUMP-math presentation last week - here are some of my notes. JUMP is a Canadian charity that produces mathematics teaching and learning resources for grades 1-8. It was founded by John Mighton, a very interesting guy.

Some things that I've learned about JUMP:

1. JUMP provides resources for the classroom teacher that are grounded in a very positive and research-based philosophy for mathematics education. JUMP 1-8 resources line up with the Ontario Curriculum (and the Western Curriculum, and to a lesser extent, the Atlantic Curriculum). I can't really do the JUMP approach justice in this note, but I'll try to highlight a few reasons why I am impressed with it below.

2. JUMP has been by adopted in various schools throughout Canada and around the world. JUMP should be more widely adopted in Ontario public schools, but currently it is not. I think there are three key reasons for this: a) its history; b) its charity status; c) some misunderstandings about its methods.

3. The various JUMP resources that are available include: (1) JUMP teacher-resources are available online for free at their website. You have to register to get these because JUMP has to track who is using its materials as part of its charity designation. (2) You can order the JUMP workbooks from here also (these will cost you) - these are intended to be used along with the teacher resources. (3) There are also 'JUMP at home' books which are available from book stores and are based on materials that have been licensed from JUMP.

4. John Mighton, JUMP's founder strongly believes that no one has an innate inability that prevents them from learning and enjoying mathematics. I agree, and I think sharing this assumption is a necessary starting point for anyone involved in teaching math to young people.

A bit more context:

JUMP, having started as a tutoring program founded by an education-industry outsider is not always understood to be a complete classroom program, which it now is. Its charity status means that it threatens the big industry text-book publishers who, while promoting their own (for-profit) texts and resources, likely do their best to block JUMP from gaining any ground. JUMP's charity status has led it to adopt a business model where it offers its teacher-resources for free, and recoups costs through printing and selling workbooks (which they claim are not even necessary - the teacher resources are the main part of the program). As a charity, JUMP has not joined the text-book publishing business, and because it does not publish a bound textbook it is not fully sanctioned by Ontario's Trillium List. In addition to these barriers, JUMP has often been misunderstood, and sometimes misrepresented by its proponents, as a "back to basics" approach that is somehow at odds with current approaches in math education.

Seeing JUMP as "back to basics / skill and drill" approach to mathematics misunderstands its approach at a fundamental level. This misunderstanding has developed in part because of the workbooks (which at a quick glance look like a bunch of worksheets), and because JUMP is very task-oriented. Folks from JUMP take pains to point out that it does not emphasize drill, but instead "guided exploration" and "micro-inquiry" - the workbooks are supplementary, and shouldn't be looked at out of context from the teacher resources. Although the teacher is called up on to explicitly teach concepts, the lessons are set up so that students are continuously working on tasks, providing many opportunities for formative evaluation and feedback.

In this approach, numeracy is developed through careful and precise attention to details - nothing is glossed over or assumed. This shows an appreciation both for the underlying mathematics and for difficulties learners can face. Nothing is ever presented as "this is how it is done, just memorize the algorithm." For example, the development of the standard long-division algorithm in grade 5 is extremely well done, with great use of base-10 blocks, and constant references to the way "sharing" concepts were modeled in earlier lessons. Going through this helped me understand long-division better. :)

There should not be a trade-off between skill-mastery and creativity in mathematics. According to the JUMP philosophy, strong skills, factual knowledge, and confidence open up possibilities for creativity and risk-taking. They a pre-requisite for problem solving - you can't do higher-order work if you are mired in calculation difficulties and lack the confidence to experiment.

There are lots of informational/promotional videos about JUMP on their youtube channel, and also an extended interview with John Mighton on the TVO site.