Geometry · Calculus
Manifolds, differential forms, and multivariable calculus
How do you do calculus on the surface of a sphere, and what does \(dx\) really mean?
Hidden Phenomena
One of the most wonderful parts of mathematics is observing the interesting phenomena of the mathematical world. And, historically, it seems humans have found the most interesting phenomena when in pursuit of solutions to concrete mathematical problems. The goal of this blog is to use problems as a way to show off interesting phenomena which, without the problem guiding our attention, would be hidden from view.
Latest articles
Newest firstAlgebra
Syzygies and higher groupoids
In mathematics, sometimes things can be equal for multiple reasons.
Physics
Mechanics II: symmetry is conservation
Why symmetries of a physical system produce conservation laws.
Physics
Mechanics I: nature is lazy
Derives the principle of least action, Lagrange's lovely alternative approach to physics.
Topology · Calculus
Möbius strips and differential equations
How do strange shapes like Möbius strips show up in more concrete mathematical situations?
Number theory
Fractal patterns hidden inside Pascal's triangle
If you color the entries in Pascal's triangle according to whether they are even or odd, a remarkable pattern appears!
Algebra
How to solve cubic and quartic equations II: cubics
Using the mathematics of symmetry, we derive the cubic formula.
Algebra
How to solve cubic and quartic equations I: quartics
Using the mathematics of symmetry, we derive the quartic formula.
Calculus
What is the factorial of 3.5?
Remarkably, \(\pi\) appears!
Number theory · Calculus
Infinitely many primes end in a 7
An explanation of Dirichlet's theorem on primes in arithmetic progressions.
Number theory · Analysis
The Riemann hypothesis (or, how to earn $1 million)
How is the Riemann zeta function actually related to the distribution of primes?
Number theory
Why study Diophantine equations?
An introduction to our mathematical philosophy.
Number theory
What is \(\log(3)\) modulo \(7\)?
Remarkably, logarithms still make sense in modular arithmetic, and are very useful!
Number theory
What is \(\sqrt{-1}\) modulo \(5\)?
When does a number have a square root modulo a prime?
Number theory · Calculus
Using calculus to do number theory
Using Newton's method from calculus, we can solve a tricky Diophantine equation!
No articles in this area yet.
Suggested starts
What is \(\sqrt{-1}\) modulo \(5\)?
What does it mean to solve the equation \(x^2=-1\) modulo \(5\)?
What is the factorial of 3.5?
Factorials are typically defined only for whole numbers... but is there some way to make the definition make sense for numbers like 3.5?
Möbius strips and differential equations
Have you been going around in circles? Well, you might have travelled further than you think!