Category Archives: Data Visualization

Mathematics Visualization

First, I should make long-overdue mention of a visualization done by Rachel Binx using some of the data I had for tagging of papers as a way of figuring out the relationships between different mathematics fields. I’ve never met her, but she is apparently “a feisty young woman operating out of the bay area.”  She was kind enough to let me know about it last year.  So now, I’m letting you guys know about it (all three of you).  The visualization is interactive and you can check it out by clicking here.

It can be hard to keep up with a blog especially when you have a lot of other things going on … for instance, being the Editor-in-chief on a second blog. So, I will try not to neglect this one. I will also, probably, do something cheap on occasion like link to posts on the other blog.

A Map of Math?

There has been some recent discussion at Reddit of an attempt I made to turn some data from arXiv.0rg into a visualization of the relationships between research areas in mathematics.  I was never completely pleased with the dataset. 

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Planarity

I recently discovered the game, Planarity. The goal of the game is to take a graph and put it in a form where none of the edges cross. That is to say, to prove that it is planar. If you don’t know what I’m saying; or you think it sounds boring; or you think it sounds complicated; go play it anyway. It’s lots of fun.

The Monty Hall Problem

When I was younger, I would often play a game with myself: He knows.  I know he knows.  He knows I know he knows. I pondered these sentences, turning them over slowly in my mind, thinking that eventually, I would have a feel for them.  Unsurprisingly, for the mind of a little boy, the examples that came to mind were adversial.  He tried to out know me, as I tried to out know him.  So, I could see this as a game of countering strategies.  When I felt I had an intuition built up, I tried for more,

I know he knows I know he knows.

He knows I know he knows I know he knows!

I know he knows I know he knows I know he knows!!

He knows I know he knows I know he knows I know he knows!!!

I know he knows I know he knows I know he knows I know he knows!!!!

There was always a point at which the sentences attained true meaninglessness.  It was at that point, I gave things a rest until next time.

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An Attempt at Mapping Mathematics

Update: A new version of this diagram can be found at the bottom of the article.

Click for the full version

Click for full version

On October 25th, Terry Tao wrote about the idea of creating a display visualizing the applications of mathematics.  Soon after, there was a lot of activity on MathOverflow, collecting possible uses of different fields of mathematics.  The original idea came from a visualization which illustrated the uses of the elements of the Periodic table.

I started writing this blog because of an overlap which I perceived between Tufte’s view of information display and mathematical thought. Tufte’s insight was mainly that effective information display requires an exacting form of clarity of expression where what is extraneous is identified as deleterious, and is therefore stripped away. In this view there is little room for ornamentation for the sake of ornamentation. Whether in computer code or in good writing, mathematical or otherwise, there is always something to be said for concise and clear expression.

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Information and Display

There was a point not too long ago when people were really excited about the display of information. This excitement was epitomized for me by Edward Tufte’s work in information display. It brought home to me a unity in the several ways of transcribing information from the way we understand it to the page. When I write, read, program, draw, manipulate equations or design graphical displays, there is some unity in what needs to be done. I should have clarity, of thought and of expression; and this requires both a clarity of understanding and a facility with the tools at hand.

I am being abstract so I’ll just say what brought on this discussion: good displays of information don’t always have to be flashy, sometimes a list will do. Terry Tao has a really cool break down of several numbers that are relevant to the current discussion of the economic climate. I think it would have taken deliberate effort on my part to have come away without understanding the relative proportions of the US budget and the numbers concerning the housing crisis better.

The list succeeds by bridging the key difficulty in understanding the financial numbers — that they are simply too large. It surmounts this difficulty by analogizing what we understand poorly with what we understand well. This all seems very straightforward and basic and yet, we have probably each seen literally millions of examples of graphics, advertisements, textbook illustrations, or journal article figures, where the designer did not accomplish this. He or she was not sure what the key difficult was and failed to make the information as understandable as it could have been. I am sure I am guilty of this myself. This is because displaying information is not a straightforward task and there is often room for improvement.

Matrix multiplication

matrix.jpg

Have you ever wondered what you are doing when you multiply a vector by a matrix?  There is a lot more to this idea than I will discuss here.  However, I wanted to share this image which gives quite a bit of geometric intuition.  The blue area represents a set of randomly generated points which lie within a circle of radius 1 which is centered at the origin.   I randomly generated a 2 by 2 matrix and multiplied the coordinates of all of the points shown in blue by that matrix.  The result was the coordinates of all the red points shown in the top left of the illustration.  If I again multiply all the red points in the previous image by the same matrix, I get the red points in the top center image.  If I repeat this process a few times I get the above illustration.  The orientation of the ellipse and the change in scale are related to the eigenvectors and eigenvalues of the matrix I generated.

When you multiply the points in a unit circle by a 2 by 2 matrix, the result is always an ellipse (possibly a circle) where the length of the major and minor axes are the eigenvalues of the matrix.  This idea generalizes to any m by n matrix which maps an m dimensional unit sphere to an n dimensional ellipsoid where the length of the axes of the ellipsoid are the square root of the eigenvalues of the matrix created by multiplying the m by n matrix by its transpose.

Which Root?

I thought it might be interesting to see the previous image coloured according to the root to which the iteration eventually converges. Here for simplicity, I concentrate on a single root whose location is circled. All points which converge to this root are colored red with the intensity of the color representing the same information as in previous diagrams. Points which converge to all other roots are colored blue. I also have a version of this image where the root is not identified which I find more aesthetically appealing.

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Another Chaotic search

Newton’s Method

 Newton’s method is a more simple way of finding a root.   The formula is

x_{n+1}=x_n - \frac{f^{'}(x)}{f(x)}

 However, the behavior of the algorithm in finding roots is anything but simple.  Above, I display the behavior of this algorithm on the complex interval where the real part of and the imaginary part of each complex number if between -1 and 1.  The coloring scheme is identical to the previous post.

A Chaotic search

As consumers of mathematical and computational intellectual captial, we often find ourselves implementing algorithms which we do not understand. With massive computer resources, the number of methods discovered which seem to work but whose mechanisms remain shrouded in mystery are steadily growing. The behavior of some algorithms can astound us with their beautiful perplexity. Below, I display the number of steps (dark for small numbers and light for large numbers) to convergence of Halley’s method being used to solve

\displaystyle x^7+x^2-1=0

which has 7 solutions. The areas of nonuniform color are clearly chaotic. For each point in the image, I implemented the following iteration

{x_{n+1} = x_n-\frac{F(x_n)}{F^{'}(x_n)-\frac{F^{''}(x_n)F(x_n)}{2F^{'}(x_n)}}}

which terminates when the solution is within machine precision of a root of the equation. The largest value of n is then stored. The numbers on which I executed the algorithm are in the interval where the real part and imaginary part of the number are between -1 and 1.

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