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.

Galton's Bayesian Machine

Chance magazine has an article by statistician and statistics historian Stephen Stigler on Galton's visualization of Bayes Theorem. He describes Galton's Bayesian machine, likely made up of beads, bins, and glass although the original device doesn't survive. These are, of course, familiar materials for Galton's statistical demonstrations.(This post was intended for some time ago. It got lost in the shuffle).

At the top level, beads are arrayed in deep vertical histogram bins representing the prior distribution, p(θ). A knob is turned and some fall into a bell-shaped horizontal room, representing the likelihood: f(x|θ). The room's back wall bulges away from us placing larger likelihood on central values of θ. If this room were moved more to the right, it would place larger likelihood on larger values of θ. At this second level, those beads falling to the front are retained, inside the bell curve wall, but some beads are rejected falling to the back, outside the wall. This performs the product: f(x|θ) p(θ). Of course at this level some bins are wide and deep, some are very shallow. The knob at this level is then turned and the beads are dropped to the bottom into vertical bins of equal width, rescaling them into a histogram proportional to the posterior distribution: f(θ|x)= f(x|θ) p(θ). Very clever and invented in 1877!

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!

Galton's Quincunx

A Qunicunx at the Instituto Butantan, São Paulo, Brazil.

The model was the invention of Sir Francis Galton, one of the English gentry scientists of the 19^th century. Galton was a cousin of Charles Darwin, and like Darwin, devoted himself to scientific explorations. He made significant contributions to meteorology, forensic science, and statistics. In the 1870s Galton developed a device to study dispersion of random events. His device consists of an array of pins that allows lead shot, encased behind glass, to cascade through. As a ball of shot falls it strikes a pin and falls randomly to the right or to the left, each equally likely. From there, the shot falls to the next level of pins where it repeats this random walk downward. The shot is collected in separate bins at the bottom of the device. The pattern of shot accumulated in the bins illustrates the variability associated with this simulation of a binomial experiment. He called his device a quincunx, due to the arrangement of pins like the pips on the number five side of a die. An illustration of Galton’s original quincunx can be found at the Galton Institute.

A Watery Histogram

A view of the side of an office building in Washington, DC after a rain shower.

What to look for:

Notice the water on the wall, leaking from the downspout.

Statistical Concept: A histogram showing the pattern of leaking water similar to the way a Galton board, also called Galton's Qunicunx or binomial board, is used to illustrate the binomial probability distribution, also a diffusion pattern demonstrating horizontal spread as the water seeps horizontally and down into the porous brick wall.