25 Mathematical Explainers to Explore: Summer of Math Exposition 3

The third Summer of Math Exposition invited people to publish mathematical explanations in any medium. Its goal is more good explanations online, not merely a list of winners. In this video I highlight 25 entries and explain what makes each useful to its intended audience.

Judging is difficult because an excellent explanation is often excellent for a particular audience. The internet can serve specialized audiences especially well, even when an entry assumes knowledge that a general viewer might not have.

Pixel Art Anti-Aliasing explores displaying low-resolution pixel art in a high-resolution 3D environment. Its mathematical motivation can interest a general viewer, but graphics and shader programmers are an especially good audience.

The Math of Saving the Enola Gay examines a plane’s route after dropping a nuclear weapon. The author acknowledges the moral issues; I highlight the detailed physics worked example, which could be useful in a classroom.

Making a Pitch Shifter works through techniques for shifting pitch or changing playback speed without changing pitch, including code and limitations of initial approaches.

Cayley Graphs and Pretty Things is a non-video resource especially useful to group theory students. Its visual explanation of the semidirect product gives a mental picture for a concept that can be difficult in a textbook.

The Longest Increasing Subsequence asks what length to expect for the longest increasing subsequence of a random permutation. Its clear, distinctive presentation motivates surprising techniques for a mathematically curious audience.

Matrix Arcade covers linear-algebra ideas in an interactive format. Even where its topics overlap familiar lessons, the illustrations, text and playground add a useful resource for teachers and learners.

Watching Neural Networks Learn uses visualizations of learning and considers how the choice of features affects a network learning a two-dimensional image. I especially liked the discussion of why Fourier features can improve the result in that setting.

Functions Are Vectors is a thorough article for making the conceptual transition from viewing functions as formulas to viewing them as vectors. The Art of Linear Programming begins with a simple nontrivial example and builds toward more general cases.

The Mosaic Problem uses a personal “backburner problem” to show how curiosity can give context to new mathematics. A problem need not be important to everyone: returning to it while learning can make ideas such as generating functions and inclusion–exclusion meaningful.

Affording a Planet with Geometry uses clay animation to explore what shape a planet might take to provide a desired gravitational experience without using too much mass. It turns an unusual premise into an optimization problem.

When Can’t Math Be Generalized? explains limits of analytic continuation: sometimes a function defined in one domain cannot be extended in the relevant sense. I recommend it especially to students encountering complex analysis.

Rotation + Translation = Rotation illustrates why a nonzero rotation combined with a translation can be expressed as a rotation about some point. A pure translation with zero rotation is the important exception; the proof uses a memorable game-like framing.

Rethinking the Real Line presents connected ways to visualize rational numbers and rational approximations of irrational numbers. Its natural delivery and dense set of ideas make it particularly rewarding for readers already curious about those topics.

An explainer on Egyptian volumes asks how the volume of a truncated pyramid might be found without modern algebraic notation and manipulation. The historical framing makes familiar formulas feel less automatic.

A circular-motion video begins with a ball released from a rope and asks which path it follows. Its hook challenges an intuitive answer and then uses explanation and experiment to investigate.

Minimal Surfaces and the Calculus of Variations studies the shape of a soap film between two circular boundaries. It introduces a general method for optimizing functions and gives viewers room to pause over the details.

A non-video explanation of computing logarithms discusses Taylor series and why a naive approach may be inadequate. Another entry follows the mediant, formed by adding numerators and denominators, into rational approximations and related mathematics; it is not ordinary fraction addition.

The Shadow Game asks whether a three-dimensional shape can be reconstructed from shadows in several directions; square shadows need not imply a cube. Chasing Fixed Points presents fixed-point theorems with a distinctive style that incorporates outdoor settings.

How Computers Use Numbers explains floating-point representation through a sequence of attempted designs. Each imperfect idea motivates the next, making apparently arbitrary choices easier to understand.

Mathematical Magic Mirror Ball shows how an image of a spherical mirror can help reconstruct a 360-degree view, including the uses and tradeoffs of the approach.

The Mathematics of String Art follows the construction of a machine that winds thread around pegs to reproduce images through varying line density. Its experiments show that finding the right formulation of a problem can matter more than applying an elegant technique to the wrong formulation.

A final illustrated entry explores different meanings of infinity and the relationships among them. I appreciate both its visual style and its compact treatment of large ideas.

The selected winners are The Mathematics of String Art, Minimal Surfaces and the Calculus of Variations, Rethinking the Real Line, Pixel Art Anti-Aliasing, and the non-video How Computers Use Numbers. The video description links to the full video playlist and the non-video collection.

I thank organizer James Schloss, web developer Frédéric Crozatier, sponsor Jane Street, the guest judges, peer reviewers and all the creators. Many worthwhile submissions are not covered in this video, so the collections are worth exploring.