## Tai Melcher / Mathematics

### “Math in Place”

To anyone who has driven the bypass through Charlottesville often enough, the sight of whales’ tails disappearing into the hillside has become familiar. But my children still get excited every time they see them, and anticipate their passing with cries of “Here come the tails!”. I confess I look forward to seeing them myself, every time. The tails are but one of the Art In Place installation exhibits around the city, a program sponsored by the city council with the purpose of making art accessible to the general public and based on the idea that art has the power to show us our familiar world in new ways.

My idea is that carefully designed pieces of art can create a similar awareness of and excitement for fundamental mathematical concepts. Working with a small group of UVa undergraduates, I would like to create a series of installation-style mathematical exhibits, which are themselves beautiful while simultaneously demonstrating beautiful mathematics.

**INTRODUCTION**

This dream idea primarily comes from two motivations.

First: I have experienced the mathematician’s common frustration that generally people feel mathematics is not relevant. Too often mathematics in the classroom comes off as artificial, as a construction imposed upon the world. However, the intention of mathematics is to simply describe observed phenomena, to quantify the rules that govern the physical world.

Second: I enjoy building things. Working on projects with my hands provides a very satisfying counterpoint to the abstract work that is the standard fare of a research mathematician.

Being one of this year’s Mead Honored Faculty is an opportunity for me to share with students the experience of creating something functional and beautiful, and at the same time sharing a part of my own idea of what mathematics is.

**ACTIVITIES**

The students and I will have regular meetings over pizza throughout the 2013-14AY. In the initial phase, our meetings will focus on discussing the interaction of art and mathematics, and people’s perception of both. We will decide on the pieces to be constructed and plan their designs and timelines. I would like each installation to reinforce the idea that math happens, naturally, in the simplest constructions, in the absence of technology. We will choose designs based on their ability to illustrate these ideas in a beautiful and/or entertaining way, and I would like us to decide on two or three such pieces.

Once designs are in place, we will enter the building phase, which will involve longer meetings at my home workshop (again accompanied by food) on weekends to implement the designs. Towards the end of construction, the students and I will make arrangements to the display the pieces for a period of time at appropriate locations. Certainly, the lobby of the mathematics department would be a good site, but the students will explore potential locations on grounds and other art and education centers in the city and county. Also, depending on their portability, one or all of the pieces could eventually be incorporated into existing outreach programs I have with local middle schools.

I’d like the participating students to make the choice in the projects we construct, but here I give several concrete options that capture the type of piece I am interested in building.

**– Galton box**

The omnipresence of the normal distribution (that is, the bell curve) is quantified in mathematics via the Central Limit Theorem — the statement that says that the distribution of a large number of independent observations of the same experiment will be approximated by the normal distribution. That is, if I record the height of a large number of people in a large population and graph that data in a histogram, the resulting graph will be approximately bell-shaped. A Galton box gives a physical demonstration of this phenomenon.

A Galton box consists of a triangular arrangement of pins and a mechanism for introducing balls at the top. When a ball is dropped and hits a pin, it can continue either to the left or right with equal probability. It then drops down to the next row of pins where it again continues to the left or right, again with equal probability. Eventually, when the ball has bounced its way down all the rows of pins, it drops into a channel where it is collected. For a large enough number of pins and balls, one is able to actually watch the formation of an approximately bell-shaped curve.

**– Uncoupled pendulums**

A simple pendulum is a freely swinging weight suspended from a pivot. Its motion is very regular: when displaced from equilibrium, the restoring force from gravity combined with the pendulum’s mass causes it to oscillate. Watching a single pendulum swing would be a boring way to spend any amount of time. However, a series of uncoupled simple pendulums of increasing lengths swinging together creates a seemingly random motion, while simultaneously demonstrating traveling waves, standing waves, and beats. The effect is beautiful, and, especially when produced on a large scale, very impressive.

**– Coupled pendulums: Harmonograph**

Again, a pendulum is a simple thing, swinging back and forth in a predictable way. Tracing the trajectory of its endpoint, by say attaching a pen to the tip, would record only a straight line. However, attaching a second pendulum to the pen, swinging at right angles to the first pendulum, produces beautiful and surprisingly complex drawings. Although patterns appear initially to repeat, as the energy stored in the swinging pendulums is expended in the bearings of the machine, the size of the pattern gradually decays, increasing the beauty and complexity of the figures. This essentially is the construction called the Harmonograph, a simple machine of two, three, or four pendulums, that can create an amazing array of drawings.

**SELECTION OF STUDENTS**

Given the hands-on aspects of this project, I think a relatively small group, around five students, would work best. My hope is that such a project, based on more aesthetic principles, may additionally attract students who are not typically drawn to mathematics and thus with whom I would not normally have the opportunity to interact through standard courses.

**BUDGET**

I am requesting $750 for food and snacks for our regular meetings and weekend shop sessions. I am also requesting $2000 for safety gear (like goggles and gloves) for all participants, materials (like furniture-grade hardwoods, wire/cable, glass, metal stock, metal balls), finishing products (like wood stains and varnish, abrasives), as well as any specialized tools which I do not currently have access to.

**SUMMARY**

After drafting the proposal for this dream idea, I noticed that it was similar in some respects to Cass Sackett’s (Physics) dream idea from 2004. Professor Sackett proposed to pair groups of physics students with groups of art students and mentor these groups through the process of creating an art display, which would then be submitted to a contest for display in the Physics building. The present proposal has a quite different emphasis, but one which is complementary to Professor Sackett’s: whereas he proposed displays which would use science as a medium for creating art, I propose to use art as a medium for demonstrating science. Additionally, I will be working hands-on in collaboration with the students to build the exhibits.

There is a unique sense of camaraderie created by physically building something together. The communication and cooperation required for projects of this kind, and even the physical labor to put it all together, builds a particular closeness that cannot be simulated in the instructor-student paradigm. I would look forward to the opportunity to work with students in this unique way, and I would be honored if the Mead Endowment funds my dream idea.