Interactive Computations
Use the free Wolfram Mathematica Player to interact with the math behind NUMB3RS.
```                     BRETT
Hey, Spielberg, you might wanna
take the lens cap off.
```

Designing Optical Systems »

This demonstrates an interactive lens system. Download the linked Demonstration, then move any of the blue dots to control the location, thickness, and curvature of the lenses, or the starting point or angle of the rays.
```                     CHARLIE
Think of a gum ball machine,
packed with hundreds of gum balls.
And imagine striking it with a
sledgehammer...
```

Animated Projectile Motion »

In the linked Demonstration, use the trigger to animate the simulated projectile. Change the initial conditions by varying the sliders.
```                     CHARLIE
It's the same with this image.
It's being scattered by the forces
of refraction... But using Snell's
Law, a dash of Goos-Haenchen shifts
and a motion-tracking algorithm...
```

Snell's Law of Refraction (Wavefronts) »

Light travels at different speeds through different materials, causing it to bend according to Snell's law. Here light is represented by traveling wavefronts coming from a light source above the boundary between two materials.
```                     CHARLIE
From the picture, we can see that
the final water level in the tub
```

How Many Gallons in My Pool? »

Filling a pool requires a lot of water. This Demonstration gives the volume of water contained within many pool types.
```                     LARRY
I did think some of the math was
Apollonian Networks in particular.
```

At each step, inscribe new tangent disks into each open curvilinear triangle. These open regions fill up quickly, leaving a fractal of infinitely many points that are never in any circle. The resulting figure is known as the Apollonian gasket.
```                     CHARLIE
Part of Game Theory is Risk and
Response Analysis.
```

Bilateral Accident Model »

Often the probability of an accident is determined by both the behavior of someone who is not injured by the accident in the legal sense and the behavior of someone who is injured.
```                    CHARLIE
The model becomes a predictor.
Same way economists predict grain
futures or currency markets.
```

The Minimal Model of the Complexity of Financial Security Prices »

A representative investor trades a single asset using the simplest possible rule and still generates complexity. The investor looks back a fixed number of days to decide whether to buy or sell.