Welcome to my blog! Posts are in chronological order. You can also browse by category.

# The Schrödinger equation on the cheap

April 17, 2021. A quick post explaining how matter waves lead to natural definitions of momentum and energy operators, and hence the Schrödinger equation. There is nothing new, just (I hope) a clear development of the topic.

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# Indescribably boring numbers

March 23, 2021. I turn the old joke about interesting numbers into a proof that most real numbers are indescribably boring. In turn, this implies that there is no explicit well-ordering of the reals. The axiom of choice, however, implies all are relatively interesting.

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# Taking half a derivative

March 13, 2021. Can you take half a derivative? Or π derivatives? Or even √–1 derivatives? It turns out the answer is yes, and there are two simple but apparently different ways to do it. I show that one implies the other!

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# Why does E = mc²?

February 19, 2021. A self-contained derivation of the most famous equation in physics. I start with a crash course on special relativity, emphasizing the invariance of spacetime lengths, move on to conservation laws, and end by considering the relativistic mechanics of an exploding bowling ball.

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# The statistical basis of Fermi estimates

February 12, 2021. Why are Fermi approximations so effective? One important factor is log normality, which occurs for large random products. Another element is variance-reduction through judicious subestimates. I discuss both and give a simple heuristic for the latter.

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# Reductionism, order and patterns

February 8, 2021. Some philosophical reflections on the nature of scientific explanation, structure, emergence, and the unreasonable effectiveness of mathematics.

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# Binomial party tricks

February 6, 2021. Sketchy hacker notes on the binomial approximation. The flashy payoff: party trick arithmetic for estimating roots in your head.

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# A simplicial generalisation of the Bloch ball

February 5, 2021. I explore unitary orbits of density matrices for finite-dimensional quantum systems. The upshot is a neat scheme for representing orbits using simplices.

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# Turning a thermometer into a sundial

January 28, 2021. I attempt to turn a thermometer (or more specifically, data about the maximum daily temperature) into a sundial. Though it fails on earth, it works on Mercury!

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# A simple proof of the bus paradox

January 26, 2021. The bus paradox states that, if buses arrive randomly but on average every ten minutes, the expected waiting time is ten minutes rather than five. I give a simple proof involving no integrals or formal probability theory.

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# Cashing a blank check

January 26, 2021. Suppose you find a blank check on the ground, and unscrupulously decide to cash it in. If overdrawing gets you nothing, how much should you cash it in for? Assuming wealth follows the 80-20 rule, the answer is: almost nothing!

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# Integrals from pyramids

January 22, 2021. I present an elementary, first-principles trick for integrating polynomials: splitting a hypercube into congruent pyramids.

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# Dicing with chaos

January 20, 2021. Why are dice and coins good sources of randomness? The word “symmetry” is bandied about, but symmetry is not enough to explain why starting with very similar initial conditions and evolving deterministically leads to random outcomes. I explore the relevant factors—chaos and jitter—and use them to build deterministic dice.

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# Eternal recurrence and brainjam

January 15, 2021. In a previous post, I advanced a four-dimensionalist version of eternal recurrence which I call “brainjam”. In this post, I discuss the moral dimensions of eternal recurrence, its relation to fatalism and free will, and end by discussing some subtle but important differences from brainjam.

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# Generalising Spot It!

January 10, 2021. I discuss the mathematics of Spot It! (aka Dobble in the UK) and its various generalisations, including projective planes, combinatorial designs, and an entertaining polytopal turducken.

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# Central limits for unlike variables

January 3, 2021. In the real world, properties like height come from a sum of independent but not identically distributed variables. To explain why height lies on a Bell curve, we need a slightly more sophisticated version of the central limit theorem (CLT). I give a simple heuristic proof using characteristic functions.

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# Why is the sky blue?

December 31, 2020. Why is the sky blue? The conventional answer invokes Rayleigh scattering, but isn’t quite right! Here, we give a fuller answer, which involves a surprising combination of dimensional analysis, thermodynamics and physiology.

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# Pairing random socks

December 27, 2020. If you have a jumbled pile of socks, how many do you need to draw on average before getting a pair? The answer turns out to be surprisingly tricky!

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# The shower curtain effect

December 27, 2020. Another bathroom-themed post, this time on the mysterious billowing of shower curtains. This is related to the fact that planes can fly upside down!

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# From noodles to woodles

December 23, 2020. Buffon asked how likely it is that a needle thrown onto ruled paper will cross one of the lines. The 1860 solution of Barbier is well-known. Equally simple but less well-known are the extensions to noodles and shapes of constant width, which I discuss informally, as well as a fanciful application to polymers.

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# Hairy shadows

December 13, 2029. A short, equation-free post on the fun physics of hairy shadows.

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# The continually interesting exponential

November 28, 2020. I discuss some of the key properties of the exponential function without (explicitly) invoking calculus. Starting with its relation to compound interest, we learn about its series expansion, Stirling’s approximation, Euler’s formula, the Basel problem, and the sum of all positive numbers, among other fun facts.

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# Asymmetric dice are unfair

November 3, 2020. Symmetric dice are fair since you cannot tell faces apart. This obviates the need to consider the mechanics of rolling. Here, I analyse the mechanics of asymmetric, two-dimensional dice and show they can always be “dynamically loaded”, i.e. loaded by rolling at different speeds. So fairness implies symmetry!

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# A HyperLaunch oompoc

October 25, 2020. In which I pretend to be Elon Musk, and launch rockets into space using empty cylinders. We’ll learn a little about atmospheric physics, rockets, and building giant cylinders, in order to sketch an order of magnitude proof-of-concept (oompoc).

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# Uncertainty and virtual particles

October 23, 2020. According to physics folklore, virtual particles are objects which can flit in and out of existence, governed by Heisenberg’s uncertainty principle. But what is uncertainty in time? And what are virtual particles? I offer a semi-rigorous connection using the Mandelstam-Tamm and Robertson-Schrödinger uncertainty relations.

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# The battle of the bulge

October 21, 2020. The earth spins and the equator bulges, turning from a sphere into a flattened ellipsoid. To estimate just how flattened, we assume that the potential energy is the same at the equator and the poles, and find an equatorial bulge only a few hundred metres off the correct result!

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# Hacking Kepler's problem

October 16, 2020. Night thoughts on Kepler’s first law. Main takeaway: an elliptical orbit is just simple harmonic motion with respect to inverse radius! I review Kepler’s laws and derive the second and third for good measure as well.

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# Parallel axes and wise crowds

October 7, 2020. A quick post showing that the parallel axis theorem from first-year mechanics is mathematically equivalent to the wisdom of the crowd, aka diversity of prediction.

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# Why is quantum gravity hard?

October 2, 2020. People often say that quantum gravity is hard because quantum mechanics and gravity are incompatible. I’ll give a brief, non-technical explanation of the real problem: powerful microscopes make black holes. In an appendix, we’ll also see why there is no problem combining gravity and quantum mechanics as long as we stick to sufficiently weak microscopes.

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# The Bloch sphere and Hopf fibrations

August 9, 2020. The Bloch sphere encodes the geometry of single qubit states. Remarkably, it is equivalent to the Hopf fibration of the 3-sphere! I prove this result and briefly touch on the generalisation to two and three qubits.

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# An equinoctial experiment

August 6, 2020. Eratosthenes measured the size of the earth using three data points: two shadows and the distance between them. During the vernal equinox in March, I performed a less elegant version of the same experiment and found the latitude of Vancouver. Lacking one data point, I invoke an imaginary friend in Whistler to help me.

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# Solipsism and emergent time

July 8, 2020. Why can we remember the past but not the future? And why do objects persist? I discuss how folk metaphysics draws attention to genuine puzzles about the nature of time, and outline some desiderata for a physical explanation.

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# From straight lines to random fractals

July 6, 2020. A derivative is a local linear approximation. Linear approximations are natural candidates for the function at “infinite zoom” since they are self-similar, i.e. fixed points of scaling. Here, I make the natural generalization to local approximation by an arbitrary self-similar curve. I also introduce random fractal approximations, and use these to motivate Brownian motion and more general Itô processes from an elementary perspective.

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# A hacker's guide to the Chandrasekhar limit

April 6, 2020. If you throw too many electrons into a box, it collapses under its own weight to form a black hole. But how many is too many? We will hack our way towards an order-of-magnitude estimate, and comment on the implications for astrophysics.

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# Hacking physics from the back of a napkin

February 24, 2020. The computational power of a humble napkin is awesome. I discuss three napkin algorithms — dimensional analysis, Fermi estimates, and random walks — and use them to figure out why rain falls, the length of the E. coli genome, and the mass of a proton, among other things. These examples suggest a napkin-based approach to teaching physics.

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# The scrubland manifesto

September 04, 2019. In the 21st century, mathematical skills are more important than ever before, but high school maths classes are dull, alienating and disempower students. How do we improve them, and plug some of the holes in the pipeline? I propose we make maths interesting! I illustrate the approach for teaching derivatives.

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# Zeta regularisation voodoo

August 13, 2019. A technical post on path integrals, zeta function regularisation of determinants, and an application to hot oscillators.

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# The endless present

July 29, 2019. Modern physics suggests a view of time called four-dimensionalism: just as all places exist simultaneously, all times should exist simultaneously. I examine some consequences for the psychology of time, and contrast with the (more intuitive but ultimately incoherent) philosophy of time called presentism.

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# Imaginary time and black holes

July 28, 2019. Imaginary time is a trick for turning heat into geometry. More precisely, hot systems repeat themselves in imaginary time. This perspective leads to a quick proof of the Unruh effect (empty space looks hot when you accelerate) and with a little more work, to Hawking radiation (black holes are hot since observers near the horizon accelerate).

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# Partition identities in Haskell

September 30, 2018. With a little experimental mathematics, you too can arrive at the insights of Ramanujan! A simple program in Haskell for discovering partition identities.

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# The geometry of coin flips

September 8, 2018. Getting a handle on higher-dimensional objects is hard. In very high dimensions, however, we can view simple shapes as random processes, and convert statistical properties into geometric ones. I’ll explore this approach for hypercubes, loosely speaking the geometry of coin flips.

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# Inverted pendulums

August 17, 2018. I discuss the physics of an inverted pendulum, and prove (using a judicious combination of hand-waving and the Hill determinant) that if you wobble the pivot fast enough, the pendulum will settle into equilbrium upside-down.

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# Incompleteness and provability logic

December 14, 2017. Studying Gödel’s theorems in their original arithmetic context involves a lot of detail and hard work if all you are interested in is the logical content (e.g. Gödel’s Incompleteness Theorems). I talk about an alternative called provability logic, which cleanly extracts all the interesting logical behaviour. In this context, Gödel’s results reduce to a single logical axiom called Löb’s Theorem and the existence of certain propositional fixed points.

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# Replicators and Fisher's Fundamental Theorem

November 30, 2017. One of the basic tenets of evolution is that different organisms (or genes) have different rates of reproduction. Simple models treating these organisms as replicators, competing for population dominance, are surprisingly rich. I give a simple proof in this context of Fisher’s “fundamental theorem” that average fitness increase equals genetic variance.

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# Cigarettes, hard labour and a box full of money

August 11, 2017. Three paradoxes of rational action — the Prisoner’s Dilemma (rational behaviour for you may not be collectively rational), Newcombe’s paradox (correlation is not causation), and the Smoker’s Dilemma (common cause is correlation) — coincide in certain contexts. I explain in more detail what these paradoxes are and how they can be viewed as equivalent.

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# Partitions, pentagons and products

September 30, 2015. Partitions are the additive equivalent of primes, telling us how many ways we can add (rather than multiply) smaller numbers to get a given natural number. Euler proved a lovely result about the number of partitions which, surprisingly, involves pentagons. I’ll show one way to obtain this result from Jacobi’s triple product formula.

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# Black hole thermodynamics

August 31, 2015. An introduction to black hole thermodynamics, based on a talk given to undergraduates at the University of Melbourne.

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# Sums of squares and normed algebras

August 23, 2015. There are nice connections between representations of numbers as sums of squares, and the multiplicative structure of weird number systems. I explain some of these connections, and show why 3 × 21 = 63 tells us that a certain normed division algebra cannot exist!

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# Cumulants from Möbius inversion

May 31, 2015. Cumulants are like the moments of a probability distribution, but better. I discuss a nice technique for calculating them in terms of Möbius functions on partition lattices, and use it to prove a famous result about Gaussian moments attributed to Wick/Isserlis. These are more or less just my notes on a paper by Terry Speed.

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# The functional equation for Riemann's zeta

May 17, 2015. A self-contained proof of the functional equation for Riemann’s zeta function, mainly written for my own edification.

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# Pedagogy, polylogarithms and particles

April 30, 2015. The polylogarithm series often appears as the butt of convergence tests; here, we go further, evaluating it and discussing the application to quantum statistics.

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# Summing the natural numbers

April 1, 2015. According to a now infamous Numberphile video, the sum of all natural numbers is -1/12. I explain two methods for getting this result, one rigorous and one non-rigorous, both due to Ramanujan.

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# Shifted Gaussians and Hermite functions

April 15, 2014. I give a simple method for calculating moments of a shifted Gaussian using the generating function for Hermite polynomials.

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