## Monday, 24 February 2014

### The making of Barry Cipra

On the second of October 2013 I got the chance to meet up with mathematical journalist Barry Cipra. He is a regular contributor to SIAM news as well as a correspondent for Science magazine. He writes the “What’s Happening in the Mathematical Sciences” series, and is the author of “Misteaks... and How to Find Them Before the Teacher Does: A Calculus Supplement”.

He was in Oxford to cover the opening of the new Mathematical Institute and the accompanying Clay conference which presented talks on the cutting edge of pure mathematics. However, my interest was piqued when he gave a talk called
“The benefits of not paying attention”.
Being a huge maths puzzle fan I turned the tables on Barry and so the reporter became the reported.

Over the next few weeks I will present Barry’s interview, in which we touch on: his history, his suggestions for people trying to break in to science journalism and, finally, some puzzles that he created when his attention should have been elsewhere.
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Since your job is to take mathematical ideas and make them understandable to a general audience, could you describe your Ph.D work on modular forms?
 Figure 1. Barry Cipra.
There is a famous sequence in a documentary of Fermat’s last theorem, in which a number of eminent mathematicians such as Peter Sarnak and John Conway were asked to describe modular forms… they all just laughed. However, I will try my best.

Modular forms are analytic functions that embody all sorts of number theoretic information. The specific thing I worked on was something called the Shimura lift [1], which allowed us to map modular forms of half integral “weight” to integral weight. Shimura’s original work was incredibly general, but only kicked in when the weight was 5/2, or larger. My work at Maryland was to reduce this to weights of 3/2.

As you can see, my work was very technical. Personally I am impressed at how much I can remember from over 30 years ago!

Where did your career take you after Ph.D?
During my first post-doc at MIT I began talking to a visitor of the chemistry department, who was really a mathematician at heart. We spoke about a problem he was having in ferromagnetism and its links to the Ising model, which is a problem in statistical physics. From our work together I wrote a beginners guide to the Ising model and its underlying mathematics. This led to me receiving a highly complementary letter through the mail, written in shaky handwriting, from a 90 year old Ernst Ising. I really should get the letter out some time to ensure that he really was saying nice things about me. At least I don’t remember him pointing out mistakes.

I then had a string of further post-docs and when I came to the end of my last one I looked around at academic positions and the alternatives of getting a “real job”. Luckily I had a number of contacts who pushed me in a different direction.

When was the decision to move towards journalism? Was it a conscious decision? Or was it a more gradual process?
It was pretty conscious as I’ve always had an interest in writing; ever since grade school. My only formal training was a journalism class I took at my high school and then worked on the school newspaper the following year.

Do you miss doing, rather than reporting, maths?
I still dabble in low level, recreational style problems. I try to come up with problems that may have some deeper connections. Quite a few of the problems I’ve generated are simple to state but defy simple explanations. However, if there is any true significance in my questions, I leave that up to the researcher trying to find the answer.

I did collaborate with some people from St Olaf’s college on a “billiards in polygons” problem, particularly in right angled triangles. We proved that if you started on one of the sides that wasn’t the hypotenuse and shot the billiard ball at a right angle from your chosen starting point then for almost all starting points, the trajectory is periodic.

My main role in all of this was to say,
“I don’t really understand what you just said, could you explain it a bit more?” and then, hopefully, “ah yes, I see we can now make that mathematically rigorous”.
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Next time Barry will be giving us a few tips on how best to go about getting into science journalism. See you then.

[1] A a pertinent paper can be found here. Although it is not for the faint hearted, or those without a degree in modular forms, you can clearly see from the first few lines that they use Cipra’s theorem. Sadly, being an applied mathematician I will never have theory named after me.

## Friday, 14 February 2014

### Be my mathematical Valentine.

Looking for an unusual gift to give your love this Valentine's day? Is the significant figure in your life a mathematician? Realise that you haven't bought anything and quickly need to knock something up? Then look no further as what could be more romantic than mathematics?

The simplest thing you could do is text the one you love with a simple I <3 U'. However, if your partner really is a mathematician, then I would suggest sending I/3 < U' instead and let them work out the message.

For those who are willing to go that extra mile (and have Matlab) here are a few suggestions of how to produce a Valentine token with a uniquely numerical twist.

1) The easiest method way to produce a heart is to use parametrized functions as seen in Figure 1. In fact the code to produce the heart in Figure 1 is so simple that I decided to use Matlab's annotation facilities to add an arrow to the picture. The whole code to produce this image can be found below.
 Figure 1. Produce a number of values of t spanning $-\pi$ to $\pi$. Calculate the appropriate values of x and y and, finally, plot them.

2) A slightly different heart shape can be produced using an implicitly defined set of points given by the equation shown in Figure 2. This equation is slightly harder to plot because given a value of x you have to solve a cubic equation to work out the corresponding value of y. Thankfully, we're able to let Matlab bare the brute force work and simply plot the delightful result.
 Figure 2. Unlike the previous equation were we were able to calculate x and y explicitly, this equation has no such nice closed form solution, or parametrization.
3) For those of you with a love that cannot be contained within two dimensions there is also a three-dimensional heart shape shown in Figure 3. Once again, this is slightly more difficult than the previous example. Not only do we have to contend with an implicit equation, but it is in three variables, instead of two! In fact I was unable to plot the equation easily using the basic functionality of Matlab, so I resorted to using one of the files from the file exchange, Ezimplot3.
 Figure 3. A fully rendered three-dimensional heart produced using Ezimplot3.
4) Finally, the last one has a special place in my heart for a number of reasons. Each year I get my wife n roses for the n years that we have been together. Often I try to vary the type I get. For example, one year I got roses made of wood, another year I got roses made of feathers. Last year I decided to make her a rose using my coding abilities. This is shown in Figure 4.
 Figure 4. A rose by any other name would smell like a Turing pattern.
Regular readers will no doubt know that Alan Turing's theory of Morphogenesis is my favourite piece of mathematics. So, I used an image of a rose to form the initial condition of a Turing pattern and you can see the whole evolution of the pattern in Figure 4.

As I say, this animation is very special to me, so unlike the other images, I won't be giving the code out for it. However, for the interested party, the coding behind it is not too difficult. All you need to do it to load a grey scale image into Matlab and use the image values as a two-dimensional network on which you run the reaction-diffusion equations.
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CODES
CODE 1
%% Clear variables
clear
close
clc

%% Set up figure size
screen_size = get(0,'screensize');
figure_position =[0 0 500 screen_size(4)/1.2];
h=figure('outerposition',figure_position);

%% Variables
fs=50; %Fontsize
t=linspace(-pi,pi,1000);
x=16*sin(t).^3;
y=13*cos(t)-5*cos(2*t)-2*cos(3*t)-cos(4*t);

%% Plot
area(x,y,'facecolor','r','edgecolor','none')

text(0,15,'I','fontsize',fs,'HorizontalAlignment','center','color','b')
text(0,0,{'x=16sin(t)^3';'y=13cos(t)-5cos(2t)-2cos(3t)-cos(4t)'},'fontsize',12,'HorizontalAlignment','center','color',[1,1,1])
text(0,-20,'You','fontsize',fs,'HorizontalAlignment','center','color','b')
text(20,-24,'By Thomas E. Woolley','fontsize',fs/7,'HorizontalAlignment','right','color','k')

annotation('line',[.3 .4],[.4 .4-0.6667*(.3-.4)],'linewidth',5,'color','k')

%% Tidy up the plot
axis equal
axis([-20 20 -25 20])
set(gcf,'PaperPositionMode','auto')

%% Save
print(gcf, '-r300',['./1D.png'], '-dpng');
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CODE 2
%% Clear variables
clear
close
clc

%% Set up figure size
screen_size = get(0,'screensize');
figure_position =[0 0 500 screen_size(4)/1.2];
h=figure('outerposition',figure_position);

%% Variables
fs=30; %Fontsize
[x,y] = meshgrid(-3:.01:3);
f = (x.^2+y.^2-1).^3-x.^2.*y.^3;

%% Plot
contourf(x,y,f,[0 0],'r','linewidth',3)

text(0,0.25,{'I am';'implicitly';'yours'},'fontsize',fs,'HorizontalAlignment','center','color','b')
text(0,1.5,'(x^2+y^2-1)^3-x^2y^3=0','fontsize',fs/2,'HorizontalAlignment','center','color','b')
text(1,-1,'By Thomas E. Woolley','fontsize',fs/5,'HorizontalAlignment','right','color','k')

%% Tidy up the plot
axis equal
axis([-2 2 -2 2])
set(gcf,'PaperPositionMode','auto')

%% Save
print(gcf, '-r300',['./Implicit_heart.png'], '-dpng');
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CODE 3
%% Clear variables
clear
close
clc

%% Set up figure size
screen_size = get(0,'screensize');
figure_position =[0 0 500 screen_size(4)/1.2];
h=figure('outerposition',figure_position);

%% Variables
fs=20; %Fontsize

%% Variables
f = '(2*x^2+y^2+z^2-1)^3-x^2*z^3/10-y^2*z^3';

%% Plot
ezimplot3(f,[-3 3],200)

text(1,-.5,.6,{'You fill all the';' dimensions';'of my life'},'fontsize',fs,'HorizontalAlignment','center','color','y')
text(.7,0,-.8,{'(2x^2+y^2+z^2-1)^3-x^2z^3/10-y^2z^3=0'},'fontsize',fs/2,'HorizontalAlignment','center','color','k', 'rotation', 2)
text(0.5,-.9,-.9,'By Thomas E. Woolley','fontsize',fs/3,'HorizontalAlignment','right','color','k', 'rotation', -75)

%% Tidy up the plot
grid off
axis equal
view([85 20])
set(gcf,'PaperPositionMode','auto')

%% Save
print(gcf, '-r300',['./3D.png'], '-dpng');
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This post is dedicated to my wife and best friend. Happy Valentine's day.