Maths

Posts on probability, geometry, combinatorics, and visualization

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With or Without Order

probability
puzzle
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”…
Nov 27, 2025

Self-Fibonacci Numbers

number theory
research
open
fibonacci
exploration
Here’s a seemingly simple question that lacks a clean solution:
Aug 11, 2025

Bayes Theorem

probability
visualization
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…
Jun 2, 2025

Sperner’s Lemma - Geometric Proof

triangles
coloring
visualization
interactive
determinant
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…
May 25, 2025

Sperner’s Lemma - Combinatorial Proof

combinatorics
triangles
coloring
visualization
interactive
This picture is a combinatorial proof of Sperner’s Lemma.
May 16, 2025

Sperner’s Lemma - Statement

combinatorics
triangles
coloring
visualization
interactive
 
May 14, 2025

Random points in a triangle

randomness
triangles
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…
May 12, 2025

Barycentric Coordinates

triangles
interactive
visualization
Barycentric coordinates are a coordinate system for describing points inside a triangle.
May 11, 2025

Pythagoras using Hilbert 90

abstract algebra
Dummit and Foote has a fun way of classifying Pythagorean triples. It is a classic fact that if \((a, b, c)\) is a reduced Pythagorean triple then there exists integers \(m…
Jun 8, 2024

Closed Convex Cones in LP

analysis
linear programming
convex analysis
I was formalizing some linear programming in Lean and came across one of the most surprisingly difficult to prove theorems.
Jan 2, 2024

Combinatorial Nullstellensatz

combinatorics
problems
todo
I came across this fun theorem in a talk about tree colorings:
Nov 26, 2023

Logarithms and primes

todo
infinite series
number theory
riemann zeta
I came across this video by NumberPhile and 3Blue1Brown about the density of primes that are 1 mod 4 vs the density of primes that are 3 mod 4. The main fact is quite cool…
Oct 26, 2023
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