Notes

It’s a curious fact that 1 = 0.999… where the 9s repeat forever to the right. An even curiouser fact is that −1 = …999 where the 9s repeat forever to the left. Now that may seem crazy, but in this talk I will explain the twisted logic behind it which leads us down a rabbit hole to the strange world of p-adic numbers.

Walk around inside 3-manifolds built as branched covers. Simple rooms are stitched together along invisible membranes and explored seamlessly in first person. In the knot hall a trefoil, figure-eight knot, or Hopf link floats overhead, and looking through the knot shows you the other sheets of the cover, like the colored rooms of the S₃ cover of the trefoil pictured here.

A tool for visualizing the history of mathematics. Choose mathematicians and events from a list to add them to the timeline, then click any of them for a short description and an image. I made it to give historical context in my classes, so the current selection reflects the occasions where I have used it.

Practice row reduction one elementary row operation at a time, with exact rational arithmetic. Start from a matrix or a linear system, watch each operation animate, and see when you reach row echelon and reduced row echelon form. Work can be shared by link or exported to LaTeX. Inspired by the row reduction demo in Margalit and Rabinoff’s Interactive Linear Algebra.

See what a matrix does to space. Enter a 2 × 2, 2 × 3, 3 × 2, or 3 × 3 matrix and watch it carry a figure, the unit circle or sphere, and a lattice from the domain to the codomain, drawn side by side in two or three dimensions.

Explore Frobenius’s group determinant. Choose a finite group to see its presentation, its multiplication table, the matrix (x_gh), and the factorization of det(x_gh) into irreducible polynomials: one factor for each irreducible representation, raised to the power of its degree.

Draw affine lines over the field with three elements and watch what an element of the affine group Aff₂(𝔽₃) does to them. Points can be shown on a 3 × 3 grid or on the integer lattice, where each line of 𝔽₃² appears together with all of its translates.

A solitaire version of the board game Ricochet Robots. Robots slide until they hit a wall or another robot; find a short route to the target, then compare your solution with an exhaustive solver that always finds an optimal one.

A simple generator for static QR codes. Paste a link or some text, pick foreground and background colors (the presets come from Vassar’s official palette) and a border width, then copy or download the code as a PNG. The codes are static, so they never expire. Scan the one here to open the generator.