Dork Scratchings

(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

In this post I will attempt to show that sitting down and having a nice cup of tea is the best approach to solving any new mathematical problem.  So, here's a problem: A plane has 100 seats and 100 passengers each with an allocated seat number.  The first passenger to get on the plane is blind and chooses a seat at random.  Each subsequent passenger to board chooses their own seat if still available, or a seat at random if not.  What is the probability that the 100th passenger gets their allocated seat? This is more subtle than it seems at first.  The last passenger could get their own seat because the blind passenger chooses the correct seat, or because the 2nd passenger chooses the blind person's seat, or because the first 87 passengers occupy the first 87 seats.  In fact there's a huge number of ways in which it could happen. Knuckleheaded Compsci solution Suppose we've forgotten to have a cup of tea.  Then we might just dive in and start modelling

Queqiao and Chang'e-4 Exciting news from the BBC Website: China Moon mission lands Chang'e-4 spacecraft on far side .  The page includes a video with sinister background music as if to suggest they're the baddies (a la Drax in Moonraker).  However, the part that really intrigued me was the mention of the L2 Lagrange point - the first I've ever seen in a news story! As I described in my post Lagrange Points   there are 5 locations in the Earth-moon-Sun plane in which - in the rotating frame of reference and taking centrifugal forces into account - there is no overall force and an object can be parked indefinitely.  One of them is just beyond the moon and is called L2 . Now the problem with landing a probe on the far side of the moon is that you can't talk directly to it: there's a big rock in the way!  So, according to the BBC article the Chinese Space Agency has parked a satellite Queqiao at L2 to relay messages.  This left me a bit confused, as

If the sky were covered in moons it'd be almost a bright as day! Walking to the pub through the Yorkshire Dales on a particularly brilliant moonlit evening, when everything was clearly visible, I wondered just how much less light there was than during daytime?  It turns out to be quite easy to estimate an upper bound - all you need to do is measure the angle subtended by the moon! Let $A_e$ and $A_m$ be the cross sectional areas of the Earth and moon, $d$ be the distance to the moon, and $r$ be the distance to the Sun.  Now, suppose the Sun releases some energy $E_s$ then the amounts $E_{se}$ and $E_{sm}$ which land on the Earth and the moon are given by: $$ \begin{align} E_{se} &= \frac{A_e E_s}{4\pi r^2}\\ E_{sm} &\approx \frac{A_m E_s}{4\pi r^2} \end{align} $$ On a full moon, let's assume for the sake of calculating an upper bound that the moon reflects all of $E_{sm}$ equally in all hemispheric directions.  Then the energy reflected to the Earth is giv