Scientific problems often lead to a system of equations to solve. One of us (Tim Davis) is an expert in methods for solving them, and he also collects equations from scientists with real problems to solve.

The pictures you see here are diagrams of some of those equations, created by Davis' colleague (Yifan Hu) who is an expert in visualizing them. Each dot, or intersection of two or more lines, is an equation for a particular unknown. This is called a "node." Suppose the equation for x includes y, and the equation for y includes x. Then we draw a line between the nodes for x and y. This line is called an "edge".

For example: (x): x+y=3, (y): x-y+z=0, (z): y+z=5 could be drawn as this picture:

x--y--z.

Not every equation involves every unknown. If it did, you would see this picture instead:

x---y \ / z

You've seen these kinds of figures before -- anytime you've looked an airline route map at the back of the magazine on an airplane. They're called graphs, as in "graph theory," with nodes (the dots) and edges (the lines between the dots).

To draw the dots and lines, we need to put the dots in place. But where?

Sometimes equations have a natural place. Take a helicopter and color it with dots, draw lines between the dots, and then remove the helicopter. You're left with just lines and dots, and the equations you've drawn might represent how the helicopter flies, or how it reacts to forces applied to it, or how it vibrates. Whatever they represent, the dots have a natural place in 3D space, so we can draw them there.

What if you forgot it was a helicopter, and all you had were the equations? Or what if your equations did not represent a physical object, but something intangible such as a financial portfolio optimization problem, in which the unknowns and equations don't have a place? If you drew a diagram of investment decisions, "where" is an asset? "Where" is a decision? "Where" is an equation? Anywhere, and nowhere, but to draw it we need to put it somewhere.

We use physics to solve this problem. Make each line a spring, and give each dot an electrical charge. Then let the diagram "wobble" around so that the energy of this physical system dissipates, and the system reaches a stable minimal energy position. The final positions of the nodes now gives us the "where". This gives a good way to draw the graph. The lines between the nodes are given a color based on their length: hot (red) colors for short ones, green for middle ones, and blue (cool) for long ones.

You can see an animated demo of the dots and lines moving into place here: http://www.cise.ufl.edu/research/sparse/matrices, or below:

(click reload on your browser to see a replay of the animation).

If you do this to the equations for the helicopter, you'll find that the graph looks like a warped version of the helicopter, below. The five main rotors are to the top left, the nose to the bottom left, and the tail rotor to the right.

If you do this for a more abstract system of equations, you'll get something that shows how the equations relate to each other. For the engineer or scientist who created the system of the equations, or for someone (like me) who figures out new ways to solve them, these pictures help us to understand how these equations "work". They give us insight into methods for solving them.

And quite often, the pictures are simply beautiful.

For more background, see:

- [1] T. A. Davis and Y. F. Hu (2011). The UF Sparse Matrix Collection ACM Transactions on Mathematical Software, 38(1).
- [2] Y. F. Hu (2005). Efficient and high-quality force-directed graph drawing. The Mathematica Journal 10, pp. 37-71.

These web pages have selected some interesting images:

- Pretty Math Problem, IEEE Spectrum, Oct. 2010.
- FastCoDesign.com: Geeky Science Problems Double as Works of Art
- But does it float.com,
*The most intense moments the universe has ever known are the next 15 seconds*, catalogued under generative art, by Folkert and Atley.

This image is a blow-up of the thumbnails on this page: www.cise.ufl.edu/research/sparse/matrices. On that page, you can click on each of the figures to get a higher-resolution one (with the gray text, however). For high-resolution versions of the above image: