Dork Scratchings

I decided to watch Contagion on the telly the other night.  I thought it might be fun to see how Hollywood's idea of a pandemic matched reality....  Anyway, can you spot the glaring mistake in this clip? That's right!  He said math  when he should have said maths .  Math ith a roman catholic thervith! There was another problem too.... In the exponential phase the numbers infected increases by the same factor each day, rather than squaring each day.  Or to put it another way, the sequence should have been $$ x_n = 2^n $$ and not what he said, which was $$ x_n = 2^{2^{n-1}} $$ Dear Hollywood, I am happy to offer my services as on set math(s) consultant.  I am very cheap.

When it comes to getting action on the climate crisis I think no-one has been more successful than Extinction Rebellion .  I remember at a party one time showing the NASA CO2 graph to a lady who didn't believe it could be real.  "If this were true people would be running around in the streets screaming" she told me.  That's why XR's tactics actually work.  You can hear day in day out about the urgency and severity of the crisis, but while people are carrying on as before it's difficult for the information to break out from the intellectual part of your brain, and occupy the bit that can actually make a difference. I don't have an issue with XR's tactics, but I do worry about how we often talk to the public when we do have their ear. The pie chart above is completely made up, but I think it's about right. A tiny proportion of the general public make up their minds by looking up facts and figures; a slightly larger - but still tiny - number ar

[Or was it the editing...?]  At first glance it appears that this recent news article on the BBC is going to be a complete hatchet job on Professor Jem Bendell, author of the Deep Adaptation paper.  The paper is an honest, if uncomfortable appraisal of the likelihood of civilisational collapse caused by the climate crisis, and a blueprint for how we can work together to survive it as best we can. The first indication it's going to be hatchet job is the title: "The 'climate doomers' preparing for society to fall apart" Clearly, calling someone a "doomer" is a way of dismissing their point of view without actually challenging it.  Of course the BBC have taken the approach of distancing themselves from any responsibility by putting 'climate doomers' in quotes.  That's a standard trick for when you want to say what you think without having to justify it. The first paragraph of the article is in bold , and is highly dismissive of Ben

This reminder has been on a wall in my house for ~15 years As a programmer for 22 years I've fixed thousands of bugs, and created many times more.  Very often it appeared that the problem I was trying to fix had multiple independent causes.  However I have found - almost invariably - that if you dig long enough you'll find a single cause for all the problems you are seeing.  In fact the moment you hit on the right theory is often really obvious because it suddenly explains a whole bunch of other things that have been going wrong!  But, I wondered, can this observation be proven mathematically?  It turns out it can! The Debugger's Theorem If a system that usually works currently isn't working, then it is more likely than not that there's just one thing causing all the observed problems. Proof Let $P_0$ be the probability that the system has no problems, $P_1$ be the probability that one independent problem has occurred, $P_{2+}$ the probability that two

Article in the Telegraph... And now, the punchline...

How fast would you have to fire a cannonball for it to never hit the ground? Newton's very first ideas about gravitational orbits are said to have come about from a thought experiment. A cannonball was known to lose 5 metres of altitude a second after being fired horizontally, but the Earth - being round - curves away from the cannonball as it flies forward. So it occurred to Newton to ask: How fast would the cannonball have to be fired for the curvature to completely compensate for the vertical loss? If a cannonball was fired at this speed it would never lose any altitude, and end up orbiting the Earth. The diagram above shows that the answer can be found using simply trigonometry and comes to $$ \begin{align} v &= \sqrt{gr} \\ &= \sqrt{9.81 ms^{-2} \times 6.371\times 10^6 m} \\ &= 7868\space ms^{-1} \\ &= 17603\space mph \end{align} $$ In general, (non-relativistic) orbits are elliptical The next stage was to look at more general orbit

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