Dork Scratchings

This post will attempt to explain a strange phenomenon in nature.  If you look into a sunflower, daisy, cactus or fir cone you always see a spiral pattern like the one shown above.  If you look carefully you in fact see two: one spiralling clockwise and the other spiralling anti-clockwise.  And, bizarrely, if you count the number of spiral arms of each you always find they form a consecutive pair from the Fibonacci sequence. To recap, the Fibonacci sequence is the sequence you get if you start off with $F_1 =1$ and $F_2 = 1$ and then iterate using $F_{n+2} = F_{n+1}+F_n$.  So: $$ (F_n) = 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,... $$ This all seems highly contrived, but somehow turns up in seed and leaf patterns.  How? The answer has to do with Phyllotaxis.  This is the manner in which new leaves or seeds are formed.  Imagine a long stem plant growing upwards and generating new leaves as it grows.  The tip of the plant is known as the bud and embryonic leaves known as primordia

This letter from Einstein to Roosevelt is kept at the Hiroshima Peace Memorial Museum.  (See here for transcript.) Dated 2nd August 1939, it outlines the feasibility of both nuclear power and nuclear weaponry, and points to the evidence that the Germans were also investigating these. (It turns out the Japanese were too.) 6 years and 4 days later Little Boy was dropped, killing around 80,000 on the first day. One of the survivors, Sadako Sasaki (佐々木 禎子), just two years old at the time, feel ill with leukaemia years later. A saying went that if you folded 1000 paper cranes you would be granted a wish, so she started making paper cranes furiously and reached 1300 cranes, but died anyway in 1955. Today, schoolchildren from around the world send paper cranes to the peace park in tribute. The picture below is a photograph of her memorial in the Hiroshima peace park The final death toll was at least 140,000.

Little thank-yous Seen Kanazawa, Japan

These are ubiquitous now in Japan. Technology good => more technology better!?

When the 2nd law of thermodynamics was discovered a lot of theorists were understandably suspicious.  It said that entropy always increases with time, or $\frac{dS}{dt} > 0$, but this seems to go against time reversal symmetry, which is a feature of all physical laws. (Okay... in QM you also have to reverse charge and parity in the system too....)  Amongst those who challenged the new theory was James Clerk Maxwell (he who shed light on light) and he did so with a thought experiment.  Imagine you have a box full of air at a given temperature, and it is divided into two parts.  Between the two is a little door which can be opened and shut by a daemon.  This daemon watches the air molecules speeding towards the door and if the molecule is faster than average and on the LHS he opens the door briefly to let it pass through to the right hand side.  Conversely if the molecule is slower than average and on the RHS the daemon will open the door to let is pass to the left hand side. 

A proof that the net force due to pressure on a (fully or partially) submerged object is equal to the weight of the water displaced. The facts/approximations used were pressure is isotropic density is constant gravitational force is constant And the method of proof was to apply the Divergence Theorem. POSTSCRIPT I was just wondering why pressure is isotropic when I found this post and  I realised that the scratchings above actually prove it!  Imagine a blob of water submerged in water: the above proof shows that the net force on it is zero, but only if pressure is isotropic !

I heard somewhere or another the claim that soap film forms a "surface of zero mean curvature".  I wasn't sure exactly what that meant until I read this book which gave me the tools to understand what that means, and prove it.  It turns out to be a simple consequence of Gauss's Divergence theorem.  In tensor notation Gauss's Divergence theorem states $$ \int_{\Omega}\nabla_iT^idV = \int_{\partial{\Omega}}T_iN^idA $$ where $\Omega$ is some volume of space $\nabla_i$ is the covariant derivative along the $i$th coordinate $z^i$ $\delta{\Omega}$ is the surface of the volume $T^i$ is any single index tensor defined over the whole volume $N^i$ is the unit vector normal to the surface in contravariant form  This works in any number of dimensions, so if you take surface embedded in 3 dimensional euclidean space $\vec{z} = \vec{z}(s^1,s^2)$ and cut it, then the theorem tells us $$ \int_{S}\nabla_{\alpha}T^{\alpha}dA = \int_{\partial{S}}T_{\alpha}n^{\alph

I was never really that interested in economics until after 2008 when suddenly technical terms like structural deficit started appearing in the news.  Oddly, with anything relating to macro economics, journalists have to pretend they were born understanding all the concepts perfectly and they're not going to patronize you by explaining them.  This is in stark contrast to anything related science where they have to pretend to understand even less than they do!  Somehow this attitude has leaked out to the wider world, so that friends and colleagues down the pub - or politicians on Question time - will b******t eternally about the effect of interest rate rises, but happily or even boastfully admit to knowing nothing about how the rest of the universe works.  I, on the other hand, knew I didn't know anything about economics, but thought it was less important than all the other stuff I didn't know.  But when economics stories started to become the main content of the news rathe

In More Mathematical Puzzles and Diversions , Martin Gardner makes a passing reference to the Ham Sandwich Theorem.  It goes like this Any 3 shapes in 3 dimensional space can be simultaneously bisected by a single plane So imagine you have two roughly cut pieces of bread and a slice of ham, then you can always cut the sandwich in half such that each half has exactly half of each piece of bread and half of the ham, no matter how roughly strewn the pieces are. According to Gardner the generalized version has been proved by Tukey and Stone: any n shapes in $\mathbb{R}^n$ can be simultaneously halved by a single $n-1$ dimensional hyperplane.  But I thought I'd have a go at proving it myself in the 3D case, just for kicks. First observe that there are at least enough degrees of freedom to make it not impossible .  A plane (other than one going through the origin) can be described by the equation $k_xx+k_yy+k_zz = 1$ for some $\boldsymbol{k} \neq \boldsymbol{0}$ so there are 3 p

Q: How do you catch a lion in the Sahara? A: Run along a space filling curve carrying a spear.  ($\dagger$) Surprisingly, it is possible to construct a continuous and surjective map $[0,1]\mapsto  [0,1]^2$.  In less techie language: you can plot a curve that fills space.  But how do you do it?  The answer uses the following four maps $[0,1]^2\mapsto [0,1]^2$. $$ \begin{align} & \phi_0(x,y) = \frac{1}{2}(y,x) & \text{flip about diagonal and shrink}\\ & \phi_1(x,y) = \frac{1}{2}(x,y)+(0,\frac{1}{2}) & \text{shrink and translate to top left}\\ & \phi_2(x,y) = \frac{1}{2}(x,y)+(\frac{1}{2},\frac{1}{2}) & \text{shrink and translate to top right}\\ & \phi_3(x,y) = \frac{1}{2}(1-y,1-x)+(\frac{1}{2},0) & \text{flip, shrink & translate to bottom right}\\ \end{align} $$ In words the recipe goes like this:  start off with any continuous map $\gamma:[0,1]\mapsto[0,1]^2$ with $\gamma(0) = (0,0)$ and $\gamma(1)=(1,0)$. apply al

Photo taken by me at 18:00 BST May 20, 2018 I was out and about in Cambridge and I saw a modern sundial on the side of a building in Tennis Court road .  It occurred to me then that I should be able to build a sundial which tells you both the time - in GMT - and the date.  (Although you do need to know whether it is before or after the summer solstice!).  In fact all you need is to know one out of compass bearing, time, date and you should be able to work out the other two! To run it you need to do the following copy the text into sundial.py and chmod +x sundial.py install pre-requisite packages: sudo apt install python-numpy python-matplotlib run it:  ./sundial.py The result is a printout like the one shown.  If you want to adapt the picture for your locale just edit the parameters passed to plot_fixed_lat_long() . When I printed out my first sundial I was surprised to discover that the trajectory of the shadow is straight on the equinoxes.  After a day of pondering

Gödel's incompleteness theorem says the following: Choose your axioms that uniquely define the integers (you'll need a minimum of about 10) Choose your rules of inference that define what type of statements about integers follows from what other types of statements Then, provided you chose a finite number of each, there will be true statements about the integers that you cannot prove using (1) and (2) What follows is a proof of Gödel's Incompleteness Theorem, using Turing's Halting Problem, and bash scripting to illustrate ($\dagger$). For the purpose of this high level (but consise) proof , imagine a computer running Unix which has infinite memory.  Apart from this limitless resource the computer is the same as any other Unix machine, i.e. all files must be of finite size, there are finitely many instructions and system calls, each instruction has a minimum completion time, etc. etc.. We will consider all files to be executable.  For the vast majority