You can’t get there from here. ~Maine saying My father once quoted a saying from Maine, where he spent some of his youth: “You can’t get there from here.” It refers to Maine’s winding road system, which often prevents a traveller from taking a direct route between two places. In physics and math terms, we might say that Maine’s road system is of fractional dimension: Less than two-dimensional, but more than one-dimensional. Integer Dimensionality Traditionally, we define the dimensionality of a space as the number of directions one can move in. For instance, a ski lift lives in a
Mathematics
explanatory articles on math
Mathematics / Physics / Quantum Mechanics / etc.
Resolution, Fourier Analysis, and The Heisenberg Uncertainty Principle
All the effects of nature are only mathematical results of a small number of immutable laws. ~Pierre-Simon Laplace In my discussion last time (corrections here), I discussed how there is a physical limit to how good a recording can sound, whether vinyl or digital. There is a more fundamental limit, however, that I glossed over—a limit that depends not on atoms or compression techniques, but on pure mathematics. This limit was partially discovered by Jean Baptiste Joseph Fourier, and the method we will discuss bears his name. The Superposition Principle Before we discuss Fourier’s discovery, let’s take a brief
Mathematics / Science And Math
Panning For Primes With the Sieve of Eratosthenes
They consider me to have sharp and penetrating vision because I see them through the mesh of a sieve. ~Kahlil Gibran I know that last time I promised to talk more about quantum mechanics. Unfortunately, finals hit me with all the force of a great typhoon; I don’t feel like I have the time to write an article on quantum I’d be happy with. So here’s some filler on prime numbers. Our regular programming returns next week. A prime number is a counting number that is greater than one and only divisible by one and itself. For example, 2
Geometry / Mathematics / Physics / etc.
FTL Part 3: General Relativity Lets us Take Shortcuts
People assume that time is a strict progression of cause to effect, but actually, from a non-linear non-subjective viewpoint,it’s more like a big ball of wibbly-wobbly, timey-wimey… stuff. ~The Tenth Doctor (David Tennant) This is part three of a multipart series on faster-than-light travel. In the first part of the series, I explained why the speed of light is constant, no matter the observer. In part two, I explained why this invariance prevents us from going faster than light. This time, I’ll explain how we might use general relativity to get around this restriction. Fair warning: although general relativity
Analysis / Mathematics / Science And Math
Taming Infinity: infinite sums, infinite primes and the sizes of infinity
The most merciful thing in the world, I think, is the inability of the human mind to correlate all its contents. We live on a placid island of ignorance in the midst of black seas of infinity, and it was not meant that we should voyage far. ~Howard Phillips Lovecraft To infinity, and beyond! ~Buzz Lightyear Infinity: The Early Years When I was about ten, I had the following conversation with my friend: Me: I want Pokemon Yellow the most of anyone! My Friend: No, I do! I want it twice as much as you! Me: I want it