Math with my brother, part 6

April 9th, 2018

Let’s review. We’re looking for a way to find a regular simplex — the simplest symmetric shape with flat boundaries — and we want our method to work no matter how many dimensions it has.

We already saw that to get a one dimensional simplex — a line — you go up to two dimensions and draw a line between the two points (1,0) and (0,1), keeping only the parts where all the coordinates are positive.

To get a two dimensional simplex — an equilateral triangle — you go up to three dimensions and draw a plane between the three points (1,0,0), (0,1,0) and (0,0,1), keeping only the parts where all the coordinates are positive.

It turns out this trick works in any number of dimensions. For example, suppose we want to get the three dimensional simplex — a regular tetrahedron.

You go up to four dimensions and draw a volume between the four points (1,0,0,0), (0,1,0,0), (0,0,1,0) and (0,0,0,1), keeping only the parts where all the coordinates are positive. In this picture, the x,y,z,w axes — all at right angles to each other — are in blue, and the resulting simplex is bounded by thick black lines:

Wait, what was that? Draw a volume???

Well, sure. Math doesn’t care how many dimensions you have. It doesn’t even care whether you can picture in your head the thing you’re talking about. It just does exactly what you tell it to.

And now, thanks to the statistically inspired math my brother showed me, I have a way to create a regular simplex in any number of dimensions — even a hundred dimensions, or a million. It may not make any visual sense to me, but I can describe it exactly.

And when you think about it, that’s pretty cool.

Math with my brother, part 5

April 8th, 2018

The logic of this way of looking at a regular simplex starts to become clear when we consider the regular simplex in two dimensions: the equilateral triangle.

Suppose there are three horses in the race, and the odds of each horse winning are, respectively, x, y and z. Since it is certain that one of the three horses will win, we know that:

      x + y + z = 1

But just like the race with two horses, we also know that the odds of any particular horse winning are at least zero. After all, there is no such thing as a negative probability. Well ok, sometimes there is, but not in horse races.

To my brother, the corresponding picture would look something like this:

In this three dimensional diagram, the diagonal plane indicated in red shows all the places where x+y+z=1. The equilateral triangle outlined in black shows all of the places within this plane where x, y and z are all at least zero.

Which means that triangle shape represents all the possible sets of odds for our horse race. It is also, not by coincidence, the shape of the regular simplex in two dimensions.

We’re starting to see a pattern here. Does this pattern continue to work for higher dimensions? We will look at that question tomorrow.

Math with my brother, part 4

April 7th, 2018

Because his work centers on statistics, my brother works a lot with probability. And the fundamental question of probability is: “Given several things that might happen, what are the odds that any particular one of them will happen?”

If two horses are racing (and if we don’t allow a tie), then we know for sure that one of those horses is going to win, and the other will lose. If the odds of the first horse winning are x, and the odds of the second horse winning are y, then:

      x + y = 1

where 1 is certainty (or 100%, as people often call it).

To my brother, this situation could be pictured something like this:

In the picture, x goes to the right, and y goes up. All of the possible outcomes of the horse race are along the diagonal black line that goes from bottom right to top left.

Further to the lower right (where x=1) it’s more likely that the first horse will win. Further to the upper left (where y=1) it’s more likely that the second horse will win.

That diagonal black line is exactly how my brother thinks of a regular simplex in one dimension — the shape that I drew as a horizontal line in yesterday’s post. For him, a one dimensional regular simplex isn’t just a line. It’s a long diagonal slice of a square.

The reason for this will become more clear when we go up one more dimension and talk about the three dimensional regular simplex — a triangle. Which we will do tomorrow.

Math with my brother, part 3

April 6th, 2018

In the world of computer graphics, it gets more and more difficult to place the vertices of a simplex as the number of dimensions increases. In one dimension, it’s easy. You can just draw a line from x = -1 to x = +1.

For two dimensions it’s a little more difficult, but still pretty easy. To create a regular triangle you go around a circle, marking off a vertex every third of the way around.

Starting with three dimensions it starts to get harder. To create a regular tetrahedron you need to resort to some fancy tricks. One trick is to first make a cube, and then use only half of its vertices. The edges of the tetrahedron are just diagonals of the cube’s faces.

simplex3

But once you get to four dimensions, those tricks don’t really work anymore. Which is very discouraging if you like doing things with simplex shapes.

But then my brother told me how they make simplex shapes in his field, and it is completely awesome. More tomorrow.

Math with my brother, part 2

April 5th, 2018

It turns out that my brother and I, in our work, both deal with a mathematical object called a simplex. A simplex is just the simplest thing with straight sides in any particular number of dimensions.

For example, in one dimension a simplex is a line. In two dimensions it is a triangle, and in three dimensions it is a tetrahedron (a three sided pyramid).

In each case, the number of vertices (or corner points) is one more than the number of dimensions. For example, a line has two vertices (one on each end), a triangle has three, and a tetrahedron has four.

In my work in computer graphics, I have often needed to create a simplex shape. But once things go beyond two dimensions, it can get hard to figure out where to put the vertices.

Or at least I thought so, until I talked about simplex shapes with my brother. Because he comes from the world of statistics, he had a whole different way of looking at the shape of a simplex. More tomorrow.

Math with my brother

April 4th, 2018

The other day I spent time with my brother, and as usual we ended up discussing mathematics. We both use math in our work, but because our work is very different, he and I end up using it differently.

My brother develops powerful techniques for analyzing and identifying DNA. For example, after the attack on the World Trade Center, our federal government turned to him to help them properly identify what remained of the victims.

Meanwhile, I just do computer graphics. So you can see why I think of my brother as the Mycroft in the family.

Because he uses math for statistical analysis, and I use it to make pictures, it would seem that we would find little common ground. Yet it turns out that we are often thinking about the same mathematical objects. We just think about them very differently.

Tomorrow I will give an example.

The ending of Casablanca

April 3rd, 2018

My colleagues and I just got a research proposal rejected by the National Science Foundation. The reviewers were very nice about it. All of the reviews were excellent, and the reviewers were all extremely positive about our proposal.

They loved our mission statement, our discussion of broader impact, and our explanation of scientific merit. They thought our scientific paradigm was right on target, and that our research protocol was both properly focused and well thought out.

They clearly loved our proposal. They just didn’t fund it.

That may have had less to do with the relative merits of our submission than with a fairly recent phenomenon — the rapidly dwindling federal budget for evidence based research. I won’t go into the reasons for that, but feel free to follow the evidence. :-)

I was talking with my colleagues about this odd outcome, and we were wondering what we should do next. Resubmitting an improved version of the proposal next year would be one option, but there didn’t seem to be anything to improve.

I told my colleagues that the situation reminded me of the ending of Casablanca. In the end, Ilsa knows that she will never see Rick again. But it’s ok, because she will spend the rest of her life knowing that he really loves her.

Answers to the minions quiz

April 2nd, 2018

As promised, I am not putting on this page the answers to the other day’s “Can you name their minions?” quiz. So anybody who wants to continue to work on it is free from the curse of OIISICNBU (you can probably figure out what that stands for).

For those who want to know, HERE are the answers to the quiz. If anybody spots an error in my answers, please speak up, so I can make suitable corrections!

A Perilous Rampage Is Launched

April 1st, 2018

I am postponing the continuation of yesterday’s topic till tomorrow. It seems I’ve experienced a bit of a production emergency.

Out of curiosity, this afternoon I tried out Google’s new Bad Joke Detector App. As usual, their level of engineering is very impressive, and the software performs as advertised.

Unfortunately, when I launched the App, I didn’t realize it was set to run a “clean-up” process in the background. The program efficiently searched through all of my directories, and promptly deleted most of my text files.

The good news is that I now have an enormous amount of free disk space, and hopefully by tomorrow I will have figured out how to turn the damn thing off.

I suppose I should make some sort of bad joke about the whole episode. But if I do, this post will probably disappear.

Eclectic trivia contest

March 31st, 2018

I realize, looking back on the challenge I issued in yesterday’s post, that it would be nearly impossible for any one person to get all ten answers right. For any given item, there are people who could rattle off the answer to that item without even thinking. But for other items on the list, those same people might not even recognize the name that I provided as a clue.

It would be interesting to do some demographics, using a challenge like this one as a litmus test. What might be the profile of a person who could get all of the correct answers to such a challenge, without needing to look anything up?

Does such a person even exist? And if they do, what must they be like?

Tomorrow I will provide a link to all of the correct answers. I will do this in the form of a link so as not to spoil it for people who want to keep working on it.