Monthly Archives: June 2010

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!

Saturday Recipe: Ginger Scallion Sauce

Today’s recipe is something I made this week for the first time, and trying it was like a revelation. It’s simple to make, it’s got an absolutely spectacularly wonderful flavor – light and fresh – and it’s incredibly versatile. It’s damned near perfect. It’s scallion ginger sauce, and once you try it, it will become a staple. To quote David Chang, whose cookbook I learned this from: if you’ve got ginger scallion sauce in the fridge, you’ll never be hungry.

There are two main variations of this: there’s a cooked version, and a raw version. Mine is the raw version. I love the freshness of flavor, and while cooking it will intensify some of the flavors, it will also detract from that delightful freshness.


  • Fresh ginger – roughly one inch, peeled.
  • A bunch of fresh scallions.
  • A teaspoon, give or take, of coarse salt.
  • 1 tablespoon of soy sauce.
  • 1 tablespoon rice vinegar.
  • 1/4 cup oil – peanut oil, canola oil, or something
    other neutral oil.

  • A dash of sesame oil.


  • Mince the ginger. Toss the minced ginger into a food processor.
  • Cut the roots off of the scallions, cut them coarsely, and add them to the food processor.
  • Add the rest of the ingredients to the food processor.
  • Run the food processor until everything is finely ground into a smooth sauce.

That’s it. Ginger scallion sauce. Taste it – make sure it’s got enough salt. Don’t add any soy sauce – just use plain salt if it needs any.

So what can you do with it? Just about anything. A few
great ideas:

  1. Ramen noodles. Just cook up a batch of ramen, and toss it with a tablespoon of the sauce. You can also add some stir fried meat and veggies to make it a bit more filling.
  2. Grilled meats. Use a bit of the sauce as a marinade, then grill it, and dress it with a bit of the sauce when it’s done.
  3. Use it instead of mayo on a sandwich.
  4. Add a bit more vinegar, and use it as a vinaigrette over a salad.
  5. Saute some shrimp, and toss some ginger-scallion sauce in just before they’re done.
  6. Get a nice whole fish, steam it cantonese style with just a bit of salt, soy, and sake. Spoon a bit of the sauce over it when it’s done.

If you wanted to try to cooked version, you take the ginger, scallions, and salt, and puree them in the food processor. Then put them into a large pot. In a different pot, heat the oil up until it just starts to smoke, and then pour it over the ginger/scallion/salt mixture. When it cools, whisk in the rest of the ingredients.

But like I said – I think it’s best to just stick with it raw.

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.

Metaphorical Crankery: a bad metaphor is like a steaming pile of …

So, another bit of Cantor stuff. This time, it really isn’t Cantor
crankery, so much as it is just Cantor muddling. The post
that provoked this
is not, I think, crankery of any kind – but it
demonstrates a common problem that drives me crazy; to steal a nifty phrase
from, people who can’t count to meta-three really shouldn’t try
to use metaphors.

The problem is: You use a metaphor to describe some concept. The metaphor
isn’t the thing you describe – it’s just a tool that you use. But
someone takes the metaphor, and runs with it, making arguments that are built
entirely on metaphor, but which bear no relation to the real underlying
concept. And they believe that whatever conclusions they draw from the
metaphor must, therefore, apply to the original concept.

In the context of Cantor, I’ve seen this a lot of times. The post that
inspired me to write this isn’t, I think, really making this mistake. I think
that the author is actually trying to argue that this is a lousy metaphor to
use for Cantor, and proposing an alternative. But I’ve seen exactly this
reasoning used, many times, by Cantor cranks as a purported disproof. The
cranky claim is: Cantor’s proof is wrong, because it cheats.

Of course, if you look at Cantor’s proof as a mathematical construct, it’s
a perfectly valid, logical, and even beautiful proof by contradiction. There’s
no cheating. So where do the “cheat” claims come from?

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The Unfalsifiable Theory Of Everything from viXra

Today is another bit of rubbish from viXra! In the comment thread from the last post, someone (I presume the author of this paper) challenged me to address this. And it’s such a perfect example of one of my mantras that I can’t resist.

What’s the first rule of GM/BM? The worst math is no math.

And what a whopping example of that we have here. It’s titled “Spacetime Deformation Theory”, by one Jacek Safuta. I’ll quote the abstract in its entirety, to give you the flavor.

The spacetime deformations theory unifies general relativity with quantum mechanics i.e. unifies all interactions, answers the questions: why particles have mass and what they are, answers the question: what is energy, unifies force fields and matter, implies new theories like spacetime deformations evolution.

This is a theory of principle (universal theory delivering description of nature) and not constructive theory (describing particular phenomenon using specific equations).

The theory is falsifiable, background independent (space has no fixed geometry), not generating singularities or boundaries.

This is hard to believe but a belief has nothing to with it. The real intellectual challenge is to falsify the theory.

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Gravity, Shmavity. It's the heat, dammit!

Sorry for the ridiculously slow pace around here lately; I’ve been ridiculously busy. I’m changing projects at work; it’s the end of the school year for my kids; and I’m getting close to the end-game for my book. Between all of those, I just haven’t had much time for blogging lately.

Anyway… I came across this lovely gem, and I couldn’t resist commenting on it. (Before I get to it, I have to point out that it’s on “”. viXra is “ is an e-print archive set up as an alternative to the popular service owned by Cornell University. It has been founded by scientists who find they are unable to submit their articles to because of Cornell University’s policy of endorsements and moderation designed to filter out e-prints that they consider inappropriate.”. In other words, it’s a site for cranks who can’t even post their stuff on arXiv. Considering some of the dreck that’s been posted an arXiv, that’s pretty damned sad.)

In my experience, when crackpots look at physics, they go after one of two things. Either they pick some piece of modern physics that makes them uncomfortable – like relativity or quantum mechanics – and they try to force some argument that their discomfort with it must mean that it’s wrong. The other big one is free energy – whether it’s perpetual motion, or vacuum energy, or browns gas – the crackpots claim that they’ve found some wonderful magical process that defies the laws of thermodynamics in order to make limitless free energy. The cranks rarely (not never, but rarely) go after the kinds of physics that we experience every day.

Well, this is something different. This guy basically wants to claim that gravity doesn’t really exist. And along the way, he claims to have solved the problems of dark matter and dark energy. See, we’ve all got it totally wrong about gravity! Gravity isn’t a force where matter attracts other matter. It’s a force where warm things attract other warm things! Gravity is actually a force created when things radiate heat.

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