Second Drafts

Doing Hard Things

Casey Neistat has a couple of videos that I just love. In part, because I love running myself and I find it to be a very spiritual experience, but also because I love the…

Why Programming?

I don’t code much these days - AI does all the programming for me. I almost exclusively write programs in Python, and I do not like Python.

With or Without Order

Here’s a question about basic probability theory that a student once asked me in class: When computing probabilities, does “choosing with order” vs “choosing without order”…

Self-Fibonacci Numbers

Here’s a seemingly simple question that lacks a clean solution:

Architects and Gardeners

“I think there are two types of writers, the architects and the gardeners. The architects plan everything ahead of time, like an architect building a house. They know how…

Learning is Remembering

In Greek philosophy there is a theory of “anamnesis,” mostly due to Plato and Socrates, which roughly means “learning is remembering,” that wisdom is already inside you and…

Bayes Theorem

Here’s a classic probability problem that illustrates this challenge. A standard mammogram test has, say, 80% sensitivity and 90% specificity. This means if you have breast…

Sperner’s Lemma - Geometric Proof

This is my new favorite proof of Sperner’s lemma. In the following app the vertices are slowly morphed to the “cardinal” R, G, B vertices with time while preserving the…

Sperner’s Lemma - Combinatorial Proof

This picture is a combinatorial proof of Sperner’s Lemma.

Sperner’s Lemma - Statement

Random points in a triangle

barycentric coordinates

Randomness is a surprisingly hard thing to simulate. Any program, by definition, has a predictable output. The best you can hope for is to predictably generate points that lo…

Barycentric Coordinates

affine geometry

Barycentric coordinates are a coordinate system for describing points inside a triangle.

Begin Again

I opened this page with a clear idea of what I wanted to write. I’ve now spent the last half hour updating quarto, setting up a new git repository, fixing a git merge issue…

Metropolis Random Walk

Explore random walks using the Metropolis algorithm and observe how particles move through space.

Metropolis-Hastings Algorithm

Visualize the Metropolis-Hastings algorithm, a Markov Chain Monte Carlo method for sampling from probability distributions.

Estimating π

Use Monte Carlo methods to estimate the value of π by randomly sampling points in a square.

Buffon’s Needle

Estimate the value of π using Buffon’s needle problem - a classic probability experiment that connects geometry with randomness.