Tinkertoys aren’t just for tots, they’re for mathematicians too. UT-Austin researcher Samuel Payne and alumnus David Jensen developed models that turn geometric shapes into games, providing insight into longstanding mathematical mysteries.
Payne and Jensen are algebraic geometrists, meaning they study the graphs of polynomials, which are mathematical expressions with two or more algebraic terms that drive many processes in the physical world, according to Jensen. More specifically, the two study tropical geometry, a version of algebra that represents complex shapes with simpler, sticklike models. This branch of mathematics has applications in a range of fields, including machine learning.
These concepts are expressed in a game called “Firing Chips.” This turns problems about geometry into a game, making complicated mathematical concepts easier to understand, according to Jensen.
This two player game is played on a graph with poker chips placed at each intersection. Removing a chip creates debt, but adding a chip adds value to adjacent chips. If at the end of the game, no intersections have negative values, you win.
The graphs of each shape create a unique game board, helping the game break down geometric shapes into polynomials. These polynomials form curves, and the more complicated the curve, the more complicated the polynomial equation, Payne said. Polynomials have applications in everyday life, from designing roller coasters to modeling economic growth. The game helps people understand the complexity of curves and how they can be applied to our dimensional space.
According to Payne, the game focuses on a specific algebraic curve called a general curve. The general curve is both a curve and a set of theorems that apply to all curves, meaning that an understanding of the general curve leads to a better understanding of most curves.
“In the past, people have always broken (the general curve) into interesting and complicated objects,” Payne said. “What we’re doing is simpler. We’re breaking this object into something completely trivial. That’s why people call it a tinkertoy model. By analyzing the pattern of how these pieces are linked together … we are able to produce information about the original object.”
By breaking down the general curve into simpler pieces, the game helps researchers gain an understanding of the general curve and its applications.
“What’s fascinating about this game is that it has something to say about the geometry of algebraic curves,” Jensen said. “My most recent work with Sam Payne is on some long outstanding problem in algebraic geometry about how many equations it takes to write down an algebraic curve.”
In the end, algebraic curves are everywhere, Jensen said. This research helps to understand more about how these curves function in space, and provides new information about the general curve.
“Curves defined by polynomial equations are one of the oldest concepts in mathematics,” Payne said. “It’s something a lot of people have been interested in for a long time. In the past 80 years or so, the general curve has become important in mathematical physics. But this is important not only in mathematics, but also other places in science. ”