Dork Scratchings

The Theoretical Minimum series by Leonard Susskind et al. My favourite books by a long way.  They do exactly what it says on the tin: Provide the minimum level of theory to start doing Physics properly. Prior to reading these books I had read a lot of pop-science and was always left with an uneasy feeling that I'd been duped. Having chosen to keep the audience broad the authors of most pop science struggle to communicate the concepts, and usually fall back on analogy - or wonder! - creating ambiguity and misunderstanding. (There is one notable exception - Feynman's QED: The Strange Theory of Light and Matter .) The Theoretical Minimum series side steps the problems faced by most science communicators by assuming the readers do have some tools under their belt.  Specifically basic calculus.   It's then able to take the quickest possible route to the edge of science .  (Okay, maybe not the edge, but it feels like it.)  Just don't skip any bits because t...

A non-euclidean surface with coordinate system Metrics are important in both the calculus of surfaces and in General Relativity.  These objects are essentially NxN matrices whose values are defined by a) the coordinate system used to parametize the space, and b) the particular point $\boldsymbol{x} = (x^i)$.  In general, theorems can be categorized as either intrinsic or extrinsic , where extrinsic means that information about how the space is embedded in a higher dimensional space is used, and intrinsic means that nothing other than the metric and it's partial derivatives were used.  The word intrinsic comes from the idea that a being which "lives" in the space can calculate the metric. So how does such a being find the metric?   2 & 3 dimensional spaces In three dimensions it's pretty straightforward: Impose a coordinate system $x^i = x^1, x^2, x^3$ on the space Define the covariant basis $\boldsymbol{e_i} = \frac{\partial{\boldsymbol{R}}...

The last time I visited King's College Chapel Cambridge I noticed a small poster which talked about the construction.  It turns out that just before they built it they discovered an innovation which allowed them to use much less masonry.  The poster said that if you build an arch such that it is possible to draw a catenary between the inside and outside wall, then the arch will be stable.  This discovery meant that they could make the walls of King's College Chapel as thin as they wanted, subject to their ability to measure accurately. This got me thinking.  Why should that be?  Then I realised that it all has to do with the name: catenary .  A catenary is a hyperbolic cosine like $cosh(x)$.  (Obviously if you stretch or translate along the x or y axes it is still a catenary.)  The name comes from catena or "chain", because it is the shape a chain makes if held in two places at the same height. Now imagine a chain being held in that manne...

I remember once watching a wildlife documentary which had some amazing footage of a raptor catching a small bird mid air.  I can't remember the name of the program now, and I've completely failed to find a reusable picture online - it must be quite a rare thing to capture on film.  (The picture above shows a bird catching an insect, which is less impressive but illustrates the same thing.) I was impressed at how the bird of prey changed from a diving posture to one in which its wings were spread and its claws were forward, at exactly the right time.  There were no land marks of any kind and it was impossible for the hunter to know how fast it was moving how fast its prey was moving how large its prey was  Without these key bits of information, how is it possible to judge when to put the on (air) brakes and open the claws?  I decided to try and work it out: If the size of the prey is $h$, the distance to the prey is $r(t)$, and the angle subtended by ...

Why opening the box breaks the spell Quantum computers can answer questions like "what are the factors of this number?"   If the numbers are large enough these are questions that classical computers could never have the resources to answer.  For example you could ask it "what numbers divide 621405631250025693248096949484035695232" and it might respond "1932840132984031923 divides 621405631250025693248096949484035695232" as this is one of the divisors. Except not yet.  At the moment quantum computers are pretty puny and only have a few qubits each.  A quantum computer with 4 qubits might be able to tell you what the divisors of 15 are, but it couldn't go any higher without needing more qubits.  And the problem that is making it difficult to build bigger, better , quantum computers is decoherence . Before describing decoherence, let's look at how a quantum computer works.  In the following diagram points on the paper represent orthoganal stat...

When I first learned there are just 5 platonic solids I was surprised and a bit confused.  Why 5?  0 or 1 or $\infty$ all seem like reasonable possibilities, but 5 is just so ... arbitrary.  And how on earth can you ever be sure there aren't more that just haven't been found yet? The first proof I saw went like this:   If 6 equilateral triangles share a vertex the total angle subtended at the vertex is $60^{\circ} \times 6 = 360^{\circ}$ meaning it's flat (they're in a plane).  So platonic solids with equilateral triangles must have 3, 4, or 5 trangles sharing each vertex.  That's 3 platonic solids! If 4 squares share a vertex the total angle subtended at the vertex is $90^{\circ} \times 4 = 360^{\circ}$, and likewise that means it's in a plane.  So there can only one platonic solid made from squares, namely one with 3 squares sharing each vertex. One to go! Since $108^{\circ} \times 3 < 360^{\circ} < 108^{\circ} \times 4$ the onl...

I was looking up David Deutsch when I stumbled across this amazing lecture by the gloriously dorky Sidney Coleman.  It's a grainy recording of a talk he gave to the American Physical Society in 1994.  It's full of witty repartee as he talks about his philosophical struggles with QM prior to embracing Many Worlds.  He wonders, for example, who or what exactly can collapse a wavefunction.  The Copenhagen Interpretation doesn't - cannot - answer this.  Coleman describes going to a colleague and sharing his concern that maybe solipsism is the only consistent interpretation - that maybe he is the only one capable of collapsing wavefunctions.  The colleague puffs on his pipe and says: tell me, before you were born, was your father able to collapse wavefunctions?   Wonderful! Quantum Physics meets Psychoanalysis. After watching the video I made some notes on my favorite app. I'd wanted to understand Bell's Inequality/Bell's Theorem for some time, bu...

Here's a puzzle: Multiply $142857$ by $2$ and you get $285714$ which has the same digits Multiply $142857$ by $3$ and you get $428571$ which has the same digits Multiply $142857$ by $4$ and you get $571428$ which has the same digits  In fact you keep getting permutations of $142857$  till you get to a multiple of $7$ at which point you get $999999$.   Why? I can't remember now how I figured it out, but I do remember not being able to sleep, then having some sort of eureka moment and running downstairs in the middle of the night to scribble down the answer. The solution involves some group theory, which I studied last century when I did my Maths degree at Bristol.  Periodically I have a little panic that I've forgotten everything I learned there, and I refresh my memory.  I think it must have been after one of these episodes that my subconscious linked it to this strange number that I probably first came across at school.  Anyway, the s...

A few years ago I got involved in a conversation with my brother in law who has, shall we say, conspiracy theorist leanings.  He was terribly upset whenever he looked up and saw what he described as "chemtrails" in the sky.  Apparently, the illuminati - or whoever - are controlling our thoughts by spraying mind altering chemicals from jumbo jets. The main argument he made was that they didn't look like what you would expect to come out of an engine. A few days later I was mulling over this while cycling to work.  What, I thought, should it look like?  I realized that it should be really easy to work out and by the time I arrived at work, I'd figured it all out, bar some missing constants that I then looked up.  I quickly knocked up an email which I then sent ... to myself.  Because responding to conspiracy theorists with arithmetic is like poking bears with sticks. I took special care when researching online not to use any search terms that refer ...

If you know me, you'll know I am positively obsessed with quantum physics and the whole Many Worlds Interpretation business.  No day goes by without me thinking about it, and my overriding obsession is to try to find a way to explain it to the less obsessed.  And, that's not easy - as my many mad ramblings at friends in the pub have proved to me. You think you have it simmered down to a perfect elevator pitch, and then, an opportunity! a potential convert! and the words get all jumbled.  Because it is only then that the mathematical knowledge assumed becomes apparent. About a year ago I read The Fabric Of Reality , and I realized that, thankfully, there exists a far, far smarter person with the same obsession as me.  Chapter 2 - Shadows - consists of the most perfect explanation - derivation even - of the reality of parallel worlds.  I wish I'd thought of it. The author David Deutsch starts off with the standard double slit experiment.  Then, li...

I found this during a terribly terribly boring day at work (before I discovered the light and vowed never to work in an open plan office again). https://plus.maths.org/content/magic-19 A lovely little problem.   Can you assign the numbers 1 to 19 to the nodes in the diagram such that the three numbers in every line segment always add up to 22?  I figured it out after about a day (mostly in the office) using logic alone.  But what I couldn't understand is how the problem setter knew a solution would exist?  I posed the question on the site but never got an answer.

I said I was going to use this blog to save my HandWrite Pro scratchings.  Well here is the first.  A statement of Gauss' Divergence Theorem in both traditional euclidean coordinates, and in general coordinates. ​

I ended up buying the book for the Tensor Calculus course.  Almost all the material is available for free in the youtube series, but I wanted to own a trophy.  I was also partly motivated by a sense of guilt - having gatecrashed a complete university course for free I felt I should at least leave the host a bottle of wine.... Having said that, I have delved into it a couple of times since.  The first time was because I was left with a slight sense of incompleteness by the course after it stated but did not prove Gauss' Divergence Theorem in general coordinates.  (The proof is in one of the final chapters.)  The second time was much later. I was preparing to learn GR and I needed a refresher in TC to boost my confidence.

After a few abortive attempts to teach myself General Relativity I eventually realized that the best approach would be to master (as best as possible) Tensor Calculus on its own, and then tackle GR.  That sounds easy doesn't it!  Well, no, as it turns out.  There's certainly lots of material online, and there are books on the subject at the Cambridge University Press Bookshop, but in general they are all very bad at explaining the motivation behind the maths, or they get hung up on relating back to the building blocks of pure maths, thereby obscuring the vision. Eventually I found Pavel Grinfeld's excellent online course.  48 lectures (yes!) that take you by the hand and really do explain what Tensor Calculus is about, and how to do it.  I watched these during my lunch hours over the course of a couple of months.  I'm sure all the other cafe goers thought the man sitting in the corner with his headphones in was a complete dork, but I didn't care - I...

I discovered this app about 6 months ago and I've been in love with it ever since.  I used to be constantly searching around for scraps of paper - and a pen that works - to scribble down some confused thought.  Usually this was the back of an envelope, or the reverse side of a failed attempt to print something (it normally takes 3 attempts to print anything correctly). So the house ends up littered with detritus and Anna would constantly be asking me: "is this important?"  (I.e. can I throw it away?)  And the answer is always, No, it's not (or yes you can.)  I think she thinks that I'm smarter than I am and perhaps some of these squiggles are solutions to outstanding problems of the day. Now that I've discovered HandWrite Pro all that's changed.  Whenever I need to work something out I just pull out my phone and start squiggling.  I never run out of space, and better still I can correct and finesse till I have something I don't actually want to...