Category Archives: goodmath

Searching for Topics

As regular readers have no doubt noticed by now, posting on the blog
has been slow lately. I’ve been trying to come back up to speed, but so
far, that’s been mainly in the form of bad math posts. I’d like to get
back to the good stuff. Unfortunately, the chaos theory stuff that I was
posting just isn’t good for my schedule right now. Once you get past
the definitions of chaos, and understanding what it means, actually
analyzing chaotic systems is something that doesn’t come easily to me – which
means that it takes a lot of time to put together a post. And
my work schedule right now means that I just don’t have that amount of

So, dear readers, what mathematical topics would you be particularly interested in reading about? Since I’m a computer scientist, my background obviously runs towards the discrete math side of the world – so, for the most part, the easiest topics for me to write about are from that side. But don’t let that limit you: tell me what you want to know about, and I’ll take the suggestions into consideration, and figure out which one(s) I have the time to study and write about.

I don’t want to limit you by making suggestions. I’ve tried that in the past, and the requests inevitably end up circling around the things I suggested. But I really want to know just what you want to know more about. So – fire away!

The Surprises Never Eend: The Ulam Spiral of Primes

One of the things that’s endlessly fascinating to me about math and
science is the way that, no matter how much we know, we’re constantly
discovering more things that we don’t know. Even in simple, fundamental
areas, there’s always a surprise waiting just around the corner.

A great example of this is something called the Ulam spiral,
named after Stanislaw Ulam, who first noticed it. Take a sheet of graph paper.
Put “1” in some square. Then, spiral out from there, putting one number in
each square. Then circle each of the prime numbers. Like the following:


If you do that for a while – and zoom out, so that you can’t see the numbers,
but just dots for each circled number, what you’ll get will look something like


That’s the Ulam spiral filling a 200×200 grid. Look at how many diagonal
line segments you get! And look how many diagonal line segments occur along
the same lines! Why do the prime numbers tend to occur in clusters
along the diagonals of this spiral? I don’t have a clue. Nor, to my knowledge,
does anyone else!

And it gets even a bit more surprising: you don’t need to start
the spiral with one. You can start it with one hundred, or seventeen thousand. If
you draw the spiral, you’ll find primes along diagonals.

Intuitions about it are almost certainly wrong. For example, when I first
thought about it, I tried to find a numerical pattern around the diagonals.
There are lots of patterns. For example, one of the simplest ones is
that an awful lot of primes occur along the set of lines
f(n) = 4n2+n+c, for a variety of values of b and c. But what does
that tell you? Alas, not much. Why do so many primes occur along
those families of lines?

You can make the effect even more prominent by making the spiral
a bit more regular. Instead of graph paper, draw an archimedean spiral – that
is, the classic circular spiral path. Each revolution around the circle, evenly
distribute the numbers up to the next perfect square. So the first spiral will have 2, 3, 4;
the next will have 5, 6, 7, 8, 9. And so on. What you’ll wind up with is
called the Sack’s spiral, which looks like this:

Sacks spiral.png

This has been cited by some religious folks as being a proof of the
existence of God. Personally, I think that that’s silly; my personal
belief is that even a deity can’t change the way the numbers work: the
nature of the numbers and how they behave in inescapable. Even a deity who
could create the universe couldn’t make 4 a prime number.

Even just working with simple integers, and as simple a concept of
the prime numbers, there are still surprises waiting for us.

Code in the Cloud: My Book Beta is Available!

As I’ve mentioned before, I’ve been spending a lot of time working on a book.
Initially, I was working on a book made up of a collection of material from blog posts;
along the way, I got diverted, and ended up writing a book about cloud computing using
Google’s AppEngine tools. The book isn’t finished, but my publisher, the Pragmatic Programmers,
have a program that they call beta books. Once a book is roughly 60% done, you
can buy it at a discount, and download drafts electronically immediately. As more sections
get done, you can download each new version. And when the book is finally finished, you
get a final copy.

We released the first beta version of the book today. You can look at
excerpts, or buy a copy, by going to
the books page
at Pragmatic’s website.

If you’re interested in what cloud computing is, and how to build cloud applications – or if
you just feel like doing something to support you friendly local math-blogger – please take
a look, and consider getting a copy. I’m not going to harp about the book a lot on the blog; you’re
not going to see a ton of posts that are thinly veiled advertisements, or updates tracking
sales, or anything like that. If there’s something that I would have written about anyway,
and it’s appropriate to mention the book, then I’ll feel free to mention it, but I won’t
waste your time hyping it.

In other news, here’s the main reason that things have been dead on this blog since
the weekend:


That’s the view from my driveway as of monday morning. Over the weekend,
we had one of the worst windstorms to hit New York in about thirty years. That
mess is two oak trees, each close to 2 meters in diameter, which came down on
our street on saturday. (If you look closely towards the right hand side, you
can see the remains of my neighbors car.) The telephone pole in the picture
was snapped not by getting hit by a tree, but simply by the wind. Since that
pole had our electrical transformer, and those trees took out the wiring that
fed that transformer, we are (obviously) without electricity, internet, or
(most importantly) heat.

Con-ed is promising to restore our electricity by friday. I’m not holding my

Anyway, back to the happy stuff. The book exists in electronic form! Buy
a copy for yourself, your friends, your neighbors, and your dog! We’ve got lots
of wonderful new expenses to deal with recovering from that storm! 🙂

What is math?


I’ve got a bunch of stuff queued up to be posted over the next couple of days. It’s
been the sort of week where I’ve gotten lots of interesting links from
readers, but I haven’t had time to finish anything!

I thought I’d start off with something short but positive. A reader sent
me a link to a post on Reddit, with the following question:

Throughout elementary and high school, I got awful marks in math. I always
assumed I was just stupid in that way, which is perfectly possible. I also
hated my teacher, so that didn’t help. A friend of mine got his PhD in math
from Harvard before he was 25 (he is in his 40’s now) I was surprised the
other week when I learned he isn’t particularly good at basic arithmetic etc.
He said that’s not really what math is about. So my question is really for
math fans/pros. What is math, really? I hear people throwing around phrases
like “elegant” and “artistic” regarding math. I don’t understand how this can
be. To me, math is add, subtract, etc. It is purely functional. Is there
something you can compare it to so that I can understand?

This hits on one of my personal pet peeves. Math really is a beautiful
thing, but the way that math is taught turns it into something
mechanistic, difficult, and boring. The person who posted this question
is a typical example of a victim of lousy math education.

So what is math? It’s really a great question, and not particularly
an easy one to answer.

Continue reading

The Balance of Screening Tests

As you’ve no doubt heard by now, there’s been a new recommendation issues
which proposes changing the breast-cancer screening protocol for women under
50, by eliminating mammograms for women who don’t have significant risk
factos. While Orac has done a terrific job of covering this here and
here, I wanted to throw
in a couple of notes and a personal perspective.

Continue reading

Chaos: Bifurcation and Predictable Unpredictability


Let’s look at one of the classic chaos examples, which demonstrates just how simple a chaotic system can be. It really doesn’t take much at all to push a system from being nice and smoothly predictable to being completely crazy.

This example comes from mathematical biology, and it generates a graph commonly known as the logistical map. The question behind the graph is, how can I predict what the stable population of a particular species will be over time?

If there was an unlimited amount of food, and there were no predators, then it would be pretty easy. You’d have a pretty straightforward exponential growth curve. You’d have a constant, R, which is the growth rate. R would be determined by two factors: the rate of reproduction, and the rate of death from old age. With that number, you could put together a simple exponential curve – and presto, you’d have an accurate description of the population over time.

But reality isn’t that simple. There’s a finite amount of resources – that is, a finite amount of food for for your population to consume. So there’s a maximum number of individuals that could possibly survive – if you get more than that, some will die until the population shrinks below that maximum threshold. Plus, there are factors like predators and disease, which reduce the available population of reproducing individuals. The growth rate only considers “How many children will be generated per member of the population?”; predators cull the population, which effectively reduces the growth rate. But it’s not a straightforward relationship: the number of individuals that will be consumed by predators and disease is related to the size of the population!

Modeling this reasonably well turns out to be really simple. You take the maximum population based on resources, Pmax. You then describe the population at any given point in time as a population ratio: a fraction of Pmax. So if your environment could sustain one million individuals, and the population is really 500,000, then you’d describe the population ratio as 1/2.

Now, you can describe the population at time T with a recurrence relation:

P(t+1)= R × P(t) × (1-P(t))

That simple equation isn’t perfect, but it’s results are impressively close to accurate. It’s good enough to be very useful for studying population growth.

So, what happens when you look at the behavior of that function as you vary R? You find that below a certain threshold value, it falls to zero. Cross that threshold, and you get a nice increasing curve, which is roughly what you’d expect. Up until you hit R=3. Then it splits, and you get an oscillation between two different values. If you keep increasing R, it will split again – your population will oscillate between 4 different values. A bit farther, and it will split again, to eight values. And then things start getting really wacky – because the curves converge on one another, and even start to overlap: you’ve reached chaos territory. On a graph of the function, at that point, the graph becomes a black blur, and things become almost completely unpredictable. It looks like the beautiful diagram at the top of this post that I copied from wikipedia (it’s much more detailed then anything I could create on my own).

But here’s where it gets really amazing.

Take a look at that graph. You can see that it looks fractal. With a graph like that, we can look for something called a self-similarity scaling factor. The idea of a SS-scaling factor is that we’ve got a system with strong self-similarity. If we scale the graph up or down, what’s the scaling factor where a scaled version of the graph will exactly overlap with the un-scaled graph/

For this population curve, the SSSF turns out to about 4.669.

What’s the SSSF for the Mandelbrot set? 4.669.

In fact, the SSSF for nearly all bifurcating systems that we see, and their related fractals, is virtually always exactly 4.669. There’s a basic structure which underlies all systems of this sort.

What’s this sort? Basically, it’s a dynamical system with a quadratic maximum. In other words, if you look at the recurrence relation for the dynamical system, it’s got a quadratic factor, and it’s got a maximum value. The equation for our population system can be written: P(t+1) = R×P(t)-P(t)2, which is obviously quadratic, and it will always produce a value between zero and one, so it’s got a fixed maximum value, and Pick any chaotic dynamical system with a quadratic maximum, and you’ll find this constant in it. Any dynamical system with those properties will have a recurrence structure with a scaling factor of 4.669.

That number, 4.669 is called the Feigenbaum constant, after Mitchell Fiegenbaum, who first discovered it. Most people believe that it’s a transcendental number, but no one is sure! We’re not really sure of quite where the number comes from, which makes it difficult to determine whether or not it’s really transcendental!

But it’s damned useful. By knowing that a system is subject to recurrence at a rate determined by Feigenbaum’s constant, we know exactly when that system will become chaotic. We don’t need to continue to observe it as it scales up to see when the system will go chaotic – we can predict exactly when it will happen just by virtue of the structure of the system. Feigenbaum’s constant predictably tell us when a system will become unpredictable.

Why Math?


So, why math?

The short version of the answer is remarkably simple: math provides
a tool where you can, without ambiguity, prove that something is true or false.

I’ll get back to that – but first, I’m going to make a quick diversion, to help you understand my basic viewpoint on things.

This blog actually started in response to something specific. I was reading
Orac’s blog “Respectful Insolence”, and
he was fisking a study published by the Geiers, purporting to show a change in the trend in autism diagnoses. Orac was attacking it on multiple bases, but it struck me
that the most obvious problem with it was that it was, basically, a mathematical argument, but the math was blatantly wrong. It was making a classic statistical analysis mistake which is covered in first-year statistics courses. (And I mean
that very literally: when I was in college, I lazily satisfied some course requirements by taking a statistics course given by the Poly Sci department, and
in statistics for political scientists, they covered exactly the error made by the Geiers in November of the fall semester.) It struck me that while there were a lot of really great science bloggers – people like Orac, PZ Myers, Tara Smith, and so on – that I didn’t know of anyone doing the same thing with math.

So I started this blog on Blogger. And my goals for the blog have never changed. What
I’ve wanted to do all along is:

  1. To show people the beauty of math. Math is really wonderful. It’s
    fun, it’s beautiful, it’s useful. But people are taught from
    an early age that it’s useless, hard, and miserable. I want to show
    otherwise, by describing the beauty of math in ways that are approachable
    and understandable by non-mathematicians.
  2. To help people recognize when someone is trying to put something past
    them by abusing math – what I call obfuscatory mathematics. Because so many people don’t know math, hate it,
    think it’s incomprehensible, that makes it easy for dishonest people
    to fool them. People throw together garbage in the context of a mathematical
    argument, and use it to lend credibility to their arguments. By pointing
    out the basic errors in these things, I try to help show people how to
    recognize when someone is try to use math to confuse them or trick them.
  3. To show people that they use and rely on math far more than they think.
    This relates back to the first point, but it’s important enough to
    justify its own discussion. Lots of people believe that they can’t
    understand math, and avoid it like the plague. But at the same time, they’re
    using it every day – they just don’t know it. My favorite example
    of this is from my own family. My older brother had a string of truly horrible
    math teachers, and was convinced that he was horrible at math, couldn’t
    understand it, couldn’t do it. You couldn’t even try to teach it to him,
    because he was so sure that he couldn’t do it that he’d psych himself out
    before he even started. But he’s a really smart guy. When he went to college,
    he studied music. I visited him at one point, and was watching him do an
    assignment for his music theory course, where they were studying something
    called serial composition. He was analyzing a musical score – and what
    he was doing to analyze it was taking determinants of matrices in mod-12
    arithmetic! Of course, he didn’t know that that was what he was
    doing; instead of the numbers 0 through 11, he was using the notes of the
    musical scale. But it was taking a determinant, just using a different
    symbol set. He had no trouble doing that; but try to teach him to compute
    a percentage, and he’ll insist not just that he can’t do it, but that
    he’s incapable of learning to do it. That kind of thing is
    all too common – people do math every day, without knowing it. If they
    understood whata they were doing, they might be open to learning more,
    to being able to do more themselves – but because they’ve been taught
    that they can’t do it, they don’t see that they do.

This will come around back to my basic point; keep reading below the fold.

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Nobel Prize Blogging: Symmetry Breaking

Today the 2008 Nobel Prize winners were announced for physics. It was given to three physicists who described something called symmetry breaking. Since most people don’t know what symmetry breaking is, but people remember me writing about group theory and symmetry, I’ve been getting questions about what it means.

I don’t pretend to completely understand it; or even to mostly understand it. But I mostly understand the very basic idea behind it, and I’ll try to pass that understanding on to you.

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ScienceBlogs DonorsChoose Drive 2008

Every year at ScienceBlogs, we do a charity drive for If you haven’t heard of them, DonorsChoose is a charity that takes proposals from schoolteachers, and lets people pick specific proposals to donate money to. We run our charity challenge through the month of October.

For personal reasons, I couldn’t participate last year. The year before that, Good Math/Bad Math readers donated just over two thousand dollars to support math education in impoverished New York area schools.

This year, I’m still focusing on the NYC area, because with where I live and work, I get to directly see how these schools and students are treated, and how desperately they need help. I’ve selected a number of projects that fit into two main categories:

  1. Basic Supplies: These are proposals to buy very
    basic classroom essentials, like pencils and paper. It’s
    pathetic that teachers need to come to a charity asking for this
    kind of stuff. But the unfortunate fact is, there are schools in
    NYC where the school can’t even provide paper and pencils for their
    students. How can you possibly expect a student to learn math, when
    they don’t even have blank paper to work problems on? I guess we all too busy trying to figure out
    cool toys for two year old boys to care about the future.
  2. Math Manipulatives: manipulatives are much more
    interesting. They’re things like shaped pattern blocks blocks,
    which can be used to provide a direct, tactile tool for
    understanding math. I live in an excellent school district, and my
    children’s classrooms have enormous supplies of manipulatives of
    several different kinds. I’ve seen very directly how these simple
    things make basic mathematical concepts concrete and understandable
    for elementary school children. Manipulatives can be used to help
    develop the intuitions behind shapes, numbers, addition,
    multiplication, patterns, areas, fractions, angles, and basic
    geometry. They’re an amazing tool for elementary education, and I
    think every school should have boxes of manipulatives in every
    elementary classroom. They’re not terribly expensive, and they’re

To me, there’s a bit of a personal element to this stuff. The work that I’m most proud of in my life is several summers that I spent working for something called the New Jersey Discovery program. Discovery took children from some of the worst schools in the state of New Jersey, and set out to help them. The kids spent their weekends all year in tutoring. Then during the summer, they came to Rutgers, and spend the summer at the university. Three days a week, they took classes; the other two days, they worked for the university. If the kids did this for the two full years, then they got scholarships to get their bachelors degree at Rutgers.

I got to teach the Discovery kids computers and math. They were great kids: smart, hard-working, and motivated. I got to hear from them what their schools were like: places where they didn’t have basic essentials. In 1990, they were using math textbooks from 1960 – books that were falling apart, with missing pages. And that was one of the better equipped classes. They had english classes where they had no books at all. They didn’t have paper. They had classrooms where the blackboard were cracked and broken, and couldn’t be written on.

Before that experience, I had no idea that poor schools
were anything remotely close to that bad. It astonished me, and
opened my eyes. It’s easy to not realize, to not notice
how bad things can be for people who are less fortunate that you.

It’s a bad time in the economy, so I understand that a lot of us have a lot less money available to donate to things like this. But at the same time, it’s important to realize that inner city
impoverished school systems will generally be hurt much worse by economic troubles than the schools of those of us are better off.

So please: go to Donors Choose, and find some good proposals, and help
fund them!

Of course, I’ll throw in some goodies. If you throw in over $100, you can pick a topic for me to write an article about. I’ll probably also set up something at Cafepress to make some Good Math/Bad Math goodies for donors. Don’t wait though – I’ll still send you the goodies even if you donate before I decide what they are.

Inflation Conversions – What's 1972£10,000 worth today?

I’ve been getting a ton of questions about an article from the Independent about a guy named Bertie Smalls. Bertie was a british thief who died quite recently, who was famous for
testifying against his organized crime employers back in the 1970s. The question concerns one
claim in the article. Bertie was paid £10,000 for his part in a robbery in 1972. The article alleges that £10,000 in 1972 is equivalent to £200,000 today.

Lots of people think that that looks fishy, and have been sending me mail asking
if that makes any sense.

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