In 1666, when the Plague was ripping through England, Isaac Newton went into quarantine outside the city and developed calculus. Calculus is an application of Euclid’s geometry to problems involving time. Geometry models space, calculus models space and time.
The reason the ancient Greeks didn’t develop calculus is because of tools. The ancient Greeks had rulers and compasses but not reliable clocks. They used the rulers and compasses to do geometry, and discovered many eternal truths about space and shape.
Ten years ago, a coworker tried to teach me cribbage on my last day, but I was too drunk to follow. I walked several miles home and lost my phone in a movie theater.
The following year, I met my future wife, and she was a cribbage fan. She taught me, and I’ve loved the game ever since.
Six years ago, I wrote this cribbage score calculator in C. It takes a string like JH 2C 3C 3S JS, and returns the full accounting of the score in a game of cribbage.
Why is human society so flexible? In 1000 years, we have had the Great Schism of 1054, the Norman Invasion of 1066, the Magna Carta (1215), the Mongol Empire, the Ming Dynasty in China, the Portuguese and Spanish maritime empires, the Thirty Years War (1618-1648) and the creation of the idea of “Sovereign Nation-States”, the Dutch Empire, the British Empire and the creation of joint-stock companies. The Reformation led to new religions.
John H. Conway has passed away, his contributions to math are too numerous for me to list in this post. What I wanted to explore in this post is one of his “recreational math” contributions: The Game of Life.
If you haven’t heard, It’s this zero player computer “game”: Not this multi-player board game: I scare-quoted game because there are no actual players. It is an example of a cellular automaton, which is a simple program that evolves over time according to a fixed, simple set of rules.
Since a deadly virus appears to be spreading across the globe, I thought it would be useful to explore how this spread is modeled mathematically, and make some predictions about how quickly this can grow.
The simplest model of disease spreading starts by breaking a population up into compartments:
S (Susceptible) I (Infected) R (Removed️) Then, the model describes the flow between these compartments.
NOTE: This version of the model works over short periods and ignores births and natural deaths.
There are some ideas that are obvious to a few mathematicians, scientists and economists, but which are not widely understood or appreciated. One of the big important ones is compound growth.
“Once you start thinking about growth, it’s hard to think about anything else.”
– Robert Lucas, Nobel prize-winning economist
Before launching into an explanation, I want to start with a question: “Would you rather be 1000 times richer today, or become 2% richer each day for a whole year?
Numbers solve problems. Some problems are so precisely specified that they can be written as equations. When you learn to count, you start being able to solve problems like “how long until the weekend?”, which can produce equations like $$ x + 2 = 5 $$
Natural number equations give rise to a solution space that looks like this:
When we learn how to subtract, we learn how to solve equations like \( x + 1 = 0 \), and when we learn to divide, we learn to solve equations like \( x * 3 = 1 \).
Ported from my old blog through Archive.org (donate to Archive.org here!)
I took a kinematics class last term, and I was surprised to learn from the textbook that the law of conservation of energy was an empirical fact and not a necessary result of something more general. This means that no one has ever found a counterexample to it. (All claims of perpetual motion have been debunked)
After the term ended and I entered winter break I did what every reclusive antisocial nerd who doesn’t play video games does, I pursued my curiosity by reading alternate materials on the subject.
I wrote math-canvas.js so I could write my post explaining why Imaginary Numbers are Real