Dork Scratchings

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

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 a slig

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 &quo

I got into a debate with a flat-earther the other day.  After a while I grew to respect the position.  After all, not taking anything on trust is a cornerstone of science.  I pointed out that I've verified the Round Earth theory personally, by creating a sundial that tells both the time and the date .  My sundial was created using a mathematical model that assumes Earth is spherical, spinning about it's axis, and orbiting the sun.  And I've tested it works.  (I also had a job writing code for transceivers that talk to satellites, so you could say I'm a fully signed up round-earther!) Unfortunately my argument did not convince because it required either That the flat-earther trusted me, or That he was willing and able to follow the maths behind the model This led to to wonder what's the simplest way to demonstrate the Earth is round?  It has to involve no maths, and be easy to reproduce by skeptics anywhere.  The video is my answer to that question. PS

UBI stands for: Universal Basic Income The idea is pretty simple - it's a benefit$^*$ that is given to everyone (hence Universal) and provides enough to live on (hence Basic Income).  It's very much en vogue with economists at the moment but it tends to be received badly by voters .  "Why should millionaires get benefits?" they ask.  In this post I'll explain why I think they should! Imagine a country in which benefits are means tested, so that a citizen with no income receives £10K, but loses 50p of it for each extra £1 of income up to £20K.  After that they receive no benefits and start to pay tax.  The tax rate is a smooth curve that starts at zero, grows to 25% of a £40K income, 30% of a 50K income, and 40% of a 100K income.  That sounds fair. In the country next door there's a flat 50% tax rate on all income, and a £10K universal basic income.  Taxing people with barely any income at the same rate as higher earners and giving millionaires bene

Any orbit can be represented given enough epicycles. The Geocentric Model The geocentric model was wonderful for human self aggrandizement.  We belonged at the centre of the universe and everything revolved around us.  That early humans believed this isn't that surprising.  After all the Sun, moon, and stars do seem to move in circles around us (albeit different ones).  What's more interesting is how we attempted to cling on to this theory in the light of a) conflicting evidence, and b) a far better explanation. David Deutsch in The Fabric of Reality puts forward the view that the point of a scientific theory is to explain .  The more a theory explains - whilst still remaining consistent with observable facts - the better it is.  This is a philosophical justification for Occam's Razor.  Why is an explanation with fewer postulates better than one with more postulates? because it leaves less unexplained! The earliest theory for why the planets did not move in sim

I discovered the card game Dobble a few days ago whilst visiting relatives in Germany.  There are 57 cards with 8 symbols on each.  To play the most basic version of the game you deal the cards face down between the players such that one card remains (if there are an odd number of players you may need to hold back more than 1 card).  Then you turn face up the remaining card (or one of the remaining cards) and each person turns their top card face up. The first person to spot a symbol on their own card that matches one on the left over card shouts out the name of that symbol (e.g. "car!") and gets to move their card to the top of the shared stack; the others move their top card to the bottom of their own stack.  The winner is the person who gets rid of their cards first. At first sight this doesn't seem very mathematical.  However, after a while I noticed something odd about the cards: every card shares exactly one symbol with each other card in the pack.  I thou

I've been wondering: is it possible to visualize the rules of calculus? 1. Integration by parts To simplify things, in the above visualization $v(x_1) = 0$, so that $\int^{x'}_{x_1}\frac{dv}{dx}dx = v(x')$.  This means that $v(x_2)$ is simply the area of the cross-section facing us on the LHS, and the volume  on the LHS is $u(x_2)v(x_2)$. The RHS shows the same volume split into two parts.  The rectangle embedded in the first is $u\frac{dv}{dx}$, and so its volume is $\int^{x_2}_{x_1}u\frac{dv}{dx}dx$.  The cross section of the 2nd part is $v(x)$, so its volume is $\int^{u_2}_{u_1}vdu$, which becomes $\int^{x_2}_{x_1}v\frac{du}{dx}dx$ when rewritten as an integral over $x$. So the picture is visual proof that the equation in pink holds, provided that $v(x_1) = 0$.  To prove it in general we just need to check that when we replace $v$ with $v+v_1$ in the equation, both sides change by the same amount. 2. The chain rule This picture demonstrates the cha

Connecting Geometry and Algebra Matrix determinants have the following strange definition that seems to have been pulled out of thin air: $$ det(A) = \sum_{\sigma}{sign(\sigma)a_{1\sigma(1)}...a_{n\sigma(n)}} $$ where $A = (a_{ij})$ is a real $n\times n$ matrix $\sigma$ ranges over all permutations of $\{1,...,n\}$ $sign(\sigma)$ is $+1$ if $\sigma$ is a product of an even number of transpositions and $-1$ otherwise$^\dagger$ However, in the geometric world the definition is far more intuitive: The determinant of A is the volume of the parallelepiped formed by its columns, multiplied by minus one if these have the opposite handedness to the unit vectors. Why are these two definitions the same?  To begin to answer this we need to first define elementary matrices and then show that every square matrix can be written as a product of these. Definition The elementary matrices are $E_{i, j}$ for $1 \le i,j \le n$ $E_{i,\lambda}$ for every real $\lambda$ and $1 \l