The Art of Geometry

Representing the brain is a matter of geometry. Just as the equation c=2[pi]r describes the circumference of a circle, various functions in the mathematician’s toolbox can be used to construct a picture of the brain.

It all starts with engineering principles in one-dimensional space. Before computers, industrial designers used flexible wooden sticks called splines to help estimate what the curves of cars and airplanes should look like. The splines had pegs driven through them that could be pinned to a surface. Move one peg, and you changed the shape of the curve.

In the world of geometry, a spline is a curve defined by points along its length. Each point corresponds to the pegs of the spline’s physical counterpart. The simplest “curve” is a straight line. Focus on, say, five of its infinite number of points, and you have dissected the line into four segments. Each segment could be described by a straight-line function like f(x) = x.

Change one section from a straight line to a u-shaped parabola, like f(x)=x squared, and you change the shape of the entire curve. Develop a different equation for each segment, piece those segments together, and you can build a complex curve out of four convoluted lines (below, right). The four separate equations are the splines which combine to describe a complex form.

three illustrations demonstrating the different curves associated with mathematical functions

Similarly, mathematicians trying to model certain objects begin with a known set of points. But real-world geometry is a lot more complex than a one-dimensional curve. To deal with the complexity, designers break the problem into pieces.

Picture the two-dimensional weave of a wire mesh fence. The weave could be made of squares or diamonds, it doesn’t matter. What’s important is that the weave is flat.

Now take that fence and lay it on the ground over a grassy hill. The underlying points of the fence’s grid haven’t changed, but now the straight lines between them have become curves. The set of equations that describes those curves also describes a surface. Two dimensions have become three.

illustration showing how a flat mesh covers a surface

The more points a designer has to work with, the more accurate the estimates become. For example, the pentagonal sides of a soccer ball approximate the shape of a sphere, but the multitude of glittering facets on a spinning disco globe do it better. Designers of cars and airplanes can piece together any number of different patches to reach their goals.

Brain modelers have a few more constraints. They aren’t trying to construct a shape from scratch; they are trying to describe the shape of a very complex object that already exists. They usually must develop their own equations to accomplish the task.

“We create the right kind of math to do the geometry of the brain,” says Robert Barnhill, professor of computer science and head of the Computer Aided Geometric Design group at Arizona State University.

CAGD group members combine mathematical expertise with the power of computers and graphics to better model the human brain for medical researchers. It’s a challenging task that requires methods other than simple curves or the engineer’s spline.

The trouble with creating math is that it can be unpredictable. A mathematical equation doesn’t know what it’s supposed to look like. The unsuspecting mathematician can run into what Barnhill calls a “catastrophic” function, one that is inherently unstable.

Even if they’re only one part of a geometric description, such functions can drastically change the shape of the oject being modeled. If used, they could cause the whole description to “explode.”

“A cliff is a good example of a catastrophic surface,” Barnhill explains. “A little change in one place introduces a big change in another. We can deal with cliffs (if they already exist in the data), but we don’t want to introduce them.”

That mathematics is constantly being developed is a concept Barnhill says is difficult for some people to understand. He says most people think of math as some magical, ready-to-use solution that simply can be taken off a shelf.

But without people to develop the underlying principles, math wouldn’t exist. Take the problem we began with, the formula that describes calculating the circumference of a circle. Most people would think of that formula as simple and basic. But the Greeks couldn’t do it.

“They didn’t have pi,” Barnhill says. “Without that concept, you’re sort of stuck.”

Such unique mathematical challenges make the solutions—the modeled product—all the more beautiful, says ASU computer scientist Alyn Rockwood, a specialist in geometric design. He looks at his work as a creative experience, much like that of Renaissance art masters who wove elaborate Greek math into their paintings.

picture of Alyn Rockwood
“It’s similar to what you do when you fit on a pair of shoes,” says Alyn Rockwood explaining how ASU researchers create three-dimensional images of the brain. “You move your foot around in the shoe to find the most comfortable position. But you don’t actually change the shape of your foot.”

“They would look at art and say, ‘Look how this artist has interwoven the Fibernacci sequence with some Golden means,’” Rockwood says. “Of course, they thought there was something mystical in the numbers. Maybe there is. I don’t know. But they were appreciating a great deal more than the lay person—(because) they could see the intricacies of the mathematics.”—Alana Mikkelsen