Dork Scratchings

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 $\mathbb{F}_0$ b...

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 t...

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 th...

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 o...

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!

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(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 e...

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...

Most Quantum Mechanics courses try to avoid imposing any kind of interpretation. This makes sense since the interpretation of QM is controversial but the mathematics is not. Unfortunately a little bit of interpretation always sneaks in through the back door. Whether you're being taught the Schroedinger Equation, Feynman path integrals, or QFT, the assumption is always that you can divide reality into That Which Is Under Study and the Rest Of The World... and that the nature of reality in the two parts is entirely different. If it's the Schroedinger Equation being taught That Which Is Under Study is represented by a state vector in a Hilbert Space that evolves with time; with Feynman Path integrals That Which Is Under Study is the set of all legal Feynman diagrams which complete the picture by joining neatly with the  diagram for The Rest Of The World; if it's QFT then the nature of reality inside That Which Is Under Study is a single state vector which can be converted in...

Our holiday cottage in Lindos has this wonderful mosaic in the floor. It's made from black and white pebbles each about 2cm in diameter which gently massage your feet as you walk over them. In the doorway the same pebbles write out the date 1908. Looking at a design like this I can't help trying to reverse engineer it.  The first thing you notice is that the edges of the white quadrilaterals share a straight line with the edges of the black ones.  In fact all the shapes other than the concentric circles are formed by intersecting chords of the outer circle. The next observation is that these chords come in parallel pairs which are tangent to the inner circle on opposite sides.  These observations are enough to generate a recipe: Draw the outer circle Mark every 20 degrees to split into 18 equal parts Draw a chord between point n and point n+8 Repeat for n = 0...8 Draw the inner circle I had a go at this using Handwrite Pro on my phone. I ended up using ...

Bio-Energy Carbon Capture & Storage I've been reading a lot about climate change recently.  I'd like to help move this issue further up the political and news agenda, but the problem is that any claim can easily be dismissed as coming from a fringe source and countered with a claim from another source.  So it really helps that the IPCC exists and makes regular reports to the UN.  No one can dismiss the IPCC as "fringe".  Many scientists believe they take a too conservative line for political reasons.  But this too is quite handy as it means that when the IPCC say "we need to do at least this", then everyone(*) agrees we need to do at least that. I've been reading the IPCC's Special Report: Global Warming of 1.5 ºC Summary for Policymakers .  The UN asked the IPCC to report on the differences between a +1.5ºC future and a +2ºC one, and in 2018 they did.  A lot of the report consists of bland qualitative statements along the lines of ...