Pascal's Video

Here is a cute video about the wonders of Pascal's Triangle, although as they note he was very late to the party. Via Flowing Data.

 

MOMATH

I'm just back from a trip to NYC that included a visit to MOMATH, the Museum of Mathematics (thank you Nick and Katie). The museum is full of hands-on and feet-on exhibits similar to those found at the Exploratorium in San Francisco. Some examples:

Here kids are walking over a lighted floor that forms Voronoi regions of all points closer to one child than to any others.

Here are two visitors riding square-wheeled tricycles in circles on a catenary floor. A catenary is the curve of a hanging chain. Inverted, this curve forms the bumps on the floor that smoothly match with the rotating square wheels. It is quite a ride!

 Even the restroom sinks have math connections.

 

Alas, the sole probability and statistics exhibit was a Galton board or quincunx, that was out of service.

This model has a lever to change the positioning of the pegs that direct the travel of the balls to the bottom. The lever is shifted to allow for non-fair (non-50-50) directions of the ball drop. The left-shifted pile of balls at the bottom suggest, that before it stopped working, the lever was set for the balls to fall to the left with a greater probability than to the right. This is much like a device that Karl Pearson, Galton's protégé, illustrated and wrote about in 1895 showing many individually sliding rows of pegs to vary the probabilities of fall to the left and right.

It was cold and windy in the city. We all bundled up.

Quincunx: a designer's view

A very interesting qunicunx or Galton board illustration by Bob O'Keefe and Springer publishers from a few years back. Most obvious is the segregation of the colors. What could cause this? Perhaps a magnet? May the force be with you!
More subtle is the resulting distribution. It should look more bell-shaped rather than this triangular shape. You could get a triangular distribution from the sum of two uniform random variables, but it would require not pins to jostle the balls, no central hole for them to fall through, and a more symmetric supply of black and white balls. Something like this..



Imagine someone has loaded the balls in the equal-sided diamond-shape shown. The balls are held, waiting to fall above the the v-shaped retainer. Suppose the the entire retainer is removed, all at once. The balls fall straight down, through the slots, into the waiting bins below. The retainer is then replaced and refilled with balls. This is the image we see. This would achieve the resulting triangular distribution.

This, of course, still doesn't explain the segregation of the colors!