Math, physics, and puzzles

Organized by Matthew Hase-Liu

To physicists, math is typically thought of as an indispensable tool used to answer questions. Occasionally, this relationship can be reversed and physical intuition can give insight into math. For instance, one can derive the Pythagorean theorem by spinning a fish tank filled with water, prove the AM-GM inequality by setting up a certain electrical circuit and shorting it, or visualize Euler's answer to the Basel problem via fluid flow. We'll explore these kinds of ideas in this seminar, mostly following "The Mathematical Mechanic" by Mark Levi.

As an undergraduate, I took a similar class taught by Cumrun Vafa called "Physics, Math and Puzzles" that used mathematical puzzles to introduce the fundamental laws of physics; our seminar, however, is about using physical puzzles to introduce mathematical ideas.

Familiarity with Calculus I is required and some knowledge of physics will be very helpful.

References

We will use the following texts in this seminar:

Grading and expectations

This class will meet once a week on Thursdays. You are expected to attend and participate each time, and unexcused absences may result in a deduction from your grade. Aside from participation. Your grade will be based on your presentations.

    • Giving one presentation will put you in the "B" region of grades, i.e. a B-, B, or B+, assuming you have no unexcused absences. Similarly, giving two presentations will put you in the "A" region of grades. If there aren't enough slots to accomodate two presentations per student, you may also write an expository paper (single-spaced, LaTeXed, and five pages excluding references) to replace your second presentation.
    • The topic and date of your presentation should be selected from the schedule below. It should last between 45 minutes and an hour to ensure there is enough time for two presenters each week. You are also encouraged to coordinate and prepare with the person you are presenting with. Moreover, you must meet with me sometime before your talk to discuss logistics.

Schedule

We meet on Thursdays from 9:30 to 11:30 AM in Room 622.

Date Speaker Abstract References
01/30 Matthew Overview: I will give a brief overview of the planned topics of the seminar. [§1, Levi]
02/06 Emory Complex analysis and fluid flow: Complex analysis is the study of single-variable, complex-valued functions which are complex differentiable. The key insight which allows us to apply physical intuition and heuristic arguments to the complex world is the observation that there is a one-to-one correspondence between complex differentiable functions and vector fields describing the flow of an incompressible and irrotational (ideal) fluid. We will see how this insight allows us to interpret complex line integrals as the measure of both the rotation and flux of a fluid along a curve, which we will use to prove Cauchy's theorems. Then, we will examine the connection between harmonic functions (real parts of complex differentiable functions) and heat distribution, culminating in a heuristic argument for the Riemann mapping theorem. [§11, Levi]
02/13 Noa Optimization questions: Optimization questions are usually solved by taking derivatives of quantities to be minimized or maximized and setting them to zero. However, this can lead to tedious, complicated and unenlightening calculations, particularly when constraints and/or multivariate calculus are involved. Instead, we will reinterpret optimization questions across statistics, geometry and optics in terms of physical systems. By representing quantities in terms of potential energy and analyzing such systems at equilibrium, we can often arrive at simple, intuitive solutions to otherwise-laborious problems. We will apply this approach to linear regression, isoperimetric problems, and the behavior of light, including Snell’s law and the optical property of ellipses. [§3, Levi]
02/20 Ben Electrical circuits and inequalities: We will be utilizing basic circuitry to yield a simple proof of the AM-GM-HM inequality (which relates the arithmetic, geometric, and harmonic means of any n numbers), an inequality which mathematically is actually not so easy to establish. [§4, Levi]
02/27 Edgar, Nitya The Pythagorean theorem and some variants, symmetry and conservation: The Pythagorean theorem, a cornerstone of mathematics, reveals itself in surprising ways beyond the familiar.We begin with a "fish tank" filled with still water, demonstrating how zero torque uncovers the theorem's truth. This idea is then refined with a rigorous geometric argument using areas swept by rotation. We also connect this concept to the fundamental theorem of calculus and extend it into three dimensions, where a tetrahedron's orthogonal faces yield a 3D version of the theorem through gas pressure balance.If time permits, I will introduce an additional proof related to equilibrium in mechanical systems. Specifically, we will explore how springs balance a sliding ring and a pivoting rod, each setup elegantly proving the classic relationship. Finally, we will examine how kinetic energy provides further insight—taking us from skating on ice to cutting strings for a final flourish. [§2, Levi]
03/06 Alicia Optimization questions: Using fundamental physics principles, I will explore optimization problems that offer a more tangible understanding of what may already be familiar geometric concepts, such as the centroids of triangles and inscribed angles. I will also demonstrate how certain math problems can be simplified by translating complex algebraic or calculus-based approaches into mechanical systems, making them more accessible through physical intuition. [§3, Levi]
03/13 Mason, Emory Paradoxes of Special Relativity, Hamiltonian mechanics and symmetries: Starting from base principles, we will first explore the basics of special relativity, including time dilation, simultaneity, length contraction, and the Lorentz transformation. Applying the basic principles we derived, we will explore some of the paradoxes arising from special relativity, including the pole-vaulter paradox, the twin paradox, and the addition of velocity. If time permits, we will end with a discussion of the limits of special relativity. [§???, Levi]
03/27 Alicia, Joshua Geometry and kinematics, center of mass equations: Today, we will explore various problems and theorems related to the center of mass of different objects. A key assumption throughout our exercises is the presence of a uniform gravitational field. While such fields do not technically exist in nature, we typically assume that variations in gravitational forces near the Earth’s surface are negligible—hence the existence of a gravitational constant. We will begin by determining the center of mass of a semicircle, then apply our results to find the center of mass of a semi-disk. From there, we will then move on to proving Ceva’s Theorem and examining its applications. [§???, Levi]
04/03 Ben, Kailey Computing weird integrals, more on optics and an uncertainty principle: We will examine the way in which physical systems can be represented by area-preserving mappings, which are functions that transform a shape without changing its area. Assessing the mapping defined by a simple mechanical system of two rings that can slide freely along separate lines, we’ll find that when the system is manipulated by moving each ring in a cyclical motion, the work done upon each ring has an equivalent geometric interpretation which maintains a preserved area. Connections will be drawn between mechanics and analysis, and we will generalize the area-preservation property to mappings in higher dimensions, as well as contextualize the property as a classical mechanical analog of the uncertainty principle. The aforementioned conclusions will allow us to generate insight into fundamental characteristics of optical devices such as telescopes or binoculars. [§???, Levi]
04/17 Edgar Euler-Lagrange via springs: This presentation introduces the Euler-Lagrange equation, covering its standard derivation and focusing on a mechanical interpretation. We explore how this viewpoint treats the Lagrangian functional as potential energy, yielding the Euler-Lagrange equation from force balance and providing a physical derivation for the associated energy conservation law. The aim is to offer intuitive, mechanical insight into this fundamental equation. [§???, Levi]
04/24 Kailey, Noa Undecided (something related to quantum computing), probabilistic heuristics for number theory: ??? [§???, Levi]