Dork Scratchings

Theorem of the week: The Hairy Ball Theorem This says you can't comb a hairy ball without introducing discontinuities such as partings or whorls, unless there's a bald spot.  (There's a more mathematical statement below under the heading "theorem".)  The proof is from An Extremely Short Proof of the Hairy Ball Theorem, by P McGrath , but I've put it into my own words, completely removed all maths notation, and added pictures to make it as accessible as possible.  In addition to being extremely short, it's extremely elegant, and somewhat reminiscent of the Ham Sandwich Theorem . Theorem It is not possible to impose a continuous vector field onto a sphere, such that the vectors are all tangential to the surface, unless the field is zero somewhere Proof Let's assume the sphere does have a continuous, tangential, everywhere non-zero vector field, and attempt to derive a contradiction. Draw a small circle around a point p.  Do one lap around

It may not be entirely obvious from the photograph (which I took at night, while hashing ) but this is a wrought iron gate. It opens on to a front garden on Maid's Causeway, Cambridge.  Why?  I don't know.  Has an important physicist lived there?  There's no Blue Plaque , so maybe it's just an enthusiast, like me! What does it mean?  I don't know for certain, but I suspect it is a reference to something similar to the GHSZ variant of the Bell Inequality Test .  The results of this test demonstrate that there are no hidden variables in quantum mechanics.  I say "similar" because, in GHSZ instead of 0 and 1 the spins $\downarrow$ and $\uparrow$ are used, and there's a minus instead of a plus. If you know anything more, please tell me!

Top left box evolves into a superposition of the other two When introduced to quantum physics the first example we encounter is usually that of a single particle.  We are shown that experiment demonstrates a particle left alone evolves into a superposition of states.  These states may be position states or they may be momentum states, or if the particle has spin it may be a superposition of spin states.  It doesn't matter, the point is that fundamental particles can be in a superposition of states.  That's because they are small, we are told, so you wouldn't expect them to behave like big things do, we are told. The next thing we are shown is how the Schroedinger equation governs the evolution of this superposition.  The particle is not usually in every state equally, it is more in one position state (or momentum state, or spin state) than it is in another.  The distribution over these so-called basis states evolves with time and the Schroedinger equa

Is Our Universe Finite? A while ago I drew the pictures above to try to understand current ideas about the size of the universe. The diagrams are based on some pictures I saw in the book " Our Mathematical Universe ". The diagrams show two dimensional slices of four dimensional spacetime. The blue stuff is "inflationary material" which expands at an enormous rate. The current theory of inflation states that universes like ours form as bubbles in the inflationary material as some of the inflationary material changes phase and "evaporates" out as non-inflationary material. An important point is that the sides of this bubble are moving away from each other way too fast for anything - even light - to travel from one side to the other.  The 1st diagram illustrates the point that in this model there is room for more, far more, than one universe. The yellow region in the 2nd and 3rd diagrams is what is known as a light cone. The point in the middle

Credit: Teodomiro, wikipedia.org The ancient Greeks were obsessed with ruler and compass constructions, and they had a lot of successes.  They bisected angles, constructed pentagons, and much more.  One thing that eluded them was finding a general method for trisecting an angle.  Although they could trisect certain angles, e.g. $90^{\circ}$, they tried in vain to come up with a general recipe given just three starting points. It turns out that trisecting an angle is in general impossible, but the proof that this is the case had to wait for some mathematics developed by Galois.  In this post I'll give a proof using polynomials, fields , and vector spaces . Most impossibility proofs work the same way.  First you identify some property which remains invariant with each step, and then you show that the property would need to change to get to the final state.  In this case the property is very abstract.... The Invariant Property Let $\mathbb{F}_0$ be the minimal subfield of

Clockwise from top left: Occam, Deutsch, Everett, and Dirac Occam's razor The anthropic principle Forward reasoning Lack of any consistent alternative 1. Occam's razor Hugh Everett's 1956 thesis The Theory of the Universal Wavefunction opens with a mathematical summary of the then widely accepted Copenhagen Interpretation. "... there are two fundamentally different ways in which the state function can change:    Process 1:  The discontinuous change brought about by the observation of a quantity with eigenstates $\phi_1, \phi_2,...,$ in which the state $\psi$ will be changed to the state $\phi_j$ with probability $\lvert(\psi,\phi_j)\rvert^2$. Process 2: The continuous, deterministic change of state of the (isolated) system with time according to a wave equation $\frac{\partial \psi}{\partial t} = U\psi$, where $U$ is a linear operator." The 1st process is commonly known as the "collapse of the wavefunction" and

Source: https://www.pexels.com/photo/black-and-yellow-snake-65296/ Why does black and yellow signal danger?  Is it just an arbitrary convention or is there some explanation for this particular choice of colours? The answer may lie in Game Theory.  We can think of evolution as an iterative game played by the following 3 players: The genome of poisonous prey-like animals The genome of non-poisonous prey The genome of predators Most non-poisonous prey camouflage themselves to some degree or another, for example by adopting the colour of chlorophyll or the colour of mud.  Poisonous animals often do the exact opposite and choose colours that stand out, like yellow and black. Why shouldn't poisonous animals use camouflage colours too?  It is often claimed that this is done to help the predators for spot them, but that doesn't make much sense.  If the snake shown in the picture were green and brown it would still be possible for predators to learn not to eat it because

The End Is Always Nigh I suspect I am not the only person to feel this way, but for me the present moment seems to have a strong five minutes to midnight character to it.  Another way of putting it is the end is nigh , although I prefer five minutes to midnight because it carries fewer religious connotations.  (And also because it gives a nod to the famous Doomsday Clock started by the Bulletin of Atomic Scientists following the bombings of Hiroshima and Nagasaki.) From xkcd.com First and foremost in my mind is the climate crisis.  We're told by the IPCC the tipping point could be as low as 1.5 °C, which is only 0.4 °C away from where we are now.  And that (making optimistic assumptions) we have a CO 2 budget equivalent to 10 years of emissions at today's rates.  Beyond the tipping point the future looks so bleak that even if humans do survive it's hard to see there being very many of them, or their lives being very nice. But it isn't just the climat

How it should look How it does look

P(l)ayoff table What's the best strategy for taking or defending penalty kicks?  Essentially there are three options for each player: go left, right or centre.  But what proportion of the time should each player take each of these options? We can try to answer this with a toy model and see what happens.  In this model the goal is always saved if the choices match and is always conceded if they don't.  What makes this model interesting is the "utility" of each outcome to each player. We could choose 1:0 to the striker for a goal and 1:0 to the goalie for a save, but we can make things more realistic by applying a bit of psychology.  Let's face it: if the goalie doesn't move and the ball goes left or right s/he will look pretty stupid.  For that reason I have given the goalie a minus one in this case.  Likewise if the striker shoots towards the middle and the goal is saved then the striker will look silly, so in that case he or she gets a minus one inste

Can you cover the 34 squares with 17 2x1 dominoes? Here's a puzzle I was shown by my lecturer Professor Schofield back in Bristol, last century: Make a 6x6 grid and remove two diagonally opposite squares so that you're left with 34 squares.  Given 17 two by one dominoes, can you cover the remaining area?  The illustration above shows one failed attempt. S C R O L L D O W N F O R T H E S O L U T I O N Sorry peeps.  It's an impossipuzzle.  To see why apply a checker pattern to the squares: 18 black squares but only 16 white squares The two removed squares were both white.  So there are fewer white squares than black ones.  But each domino placed covers exactly one white and one black square.  If you could cover it perfectly with 17 dominoes there would be the same number of black and white squares.  There isn't, so you can't!

Yay! Clickbait for nerds! What's this?  Read on to find out!

(Probably mad, but worth saving for posterity) Today's post follows on from this previous post  in which I argued that the entire universe is in a superposition of states. I showed how this can lead to the impression that only the system under study is in a superposition, whilst the rest of the world is in a single - classical - state.  To summarize that argument: Reality is a superposition of states for the entire universe, each of which can be thought of as a complete classical block universe. We don't directly experience the whole universe, so let's arbitrarily place a closed surface around us to demarcate what we "directly" experience.  Remember that this is a surface in 4 dimensional spacetime and so is itself 3 dimensional.  Also remember that it is completely arbitrary: if you like you can let it contain your entire body for your entire life; or it could just be your brain for some duration.  You could even let the surface enclose everythi

It's not much.... Last year the IPCC stated that all anthropogenic emissions to date add up to 2200 GTon, that we have 600 GTon remaining before hitting a 50% chance of exceeding 1.5 degrees of warming, and that were adding 50 GTon each year. The flask in the picture shows what's going on graphically. It's already nearly full with  historical emissions and each year it gets more full. Around 2030 it will overflow. Why 1.5 degrees?  They agree that at around 2 degrees the various feedback loops will mean warming is no longer under human control. We're at around 1 degree already. It would seem that 1.5 was deemed the "safe" limit because it was halfway between the two.  Unfortunately there's at least a couple of reasons to think that the picture painted is overly rosy: Firstly, the "budget" they specify comes with a number of caveats: there are a number of uncertainties in the calculation and they quantify them. These can b