Dork Scratchings

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

The number $\phi = \frac{1+\sqrt{5}}{2}$, known as the Golden Ratio , turns up in a lot of places.  This one that I read about in National Geographic surprised me.  The following coordinates describe an icosahedron: $$ (\phi,1,0), (-\phi,1,0), (-\phi,-1,0), (\phi,-1,0) \\ (0,\phi,1), (0,-\phi,1), (0,-\phi,-1), (0,\phi,-1) \\ (1,0,\phi), (1,0,-\phi), (-1,0,-\phi), (-1,0,\phi) \\ $$I thought I'd have a go at figuring out why, and the above picture is the result.  All you have to do is observe that the vertices of an icosahedron are the same as those of a set of three identical intersecting rectangles.  The proof then reduces to showing that the ratio of the long side to the short side is $\phi$ - i.e. that these are golden rectangles.  The proof above takes the short side to be of length 1 and the long side to be of length $x$. So, why does $\phi$ turn up so often in maths and geometry?  I think the answer is the same as the answer to the question "why does $\frac{1}{2}$

The idea of wave particle duality has been around for a long time, and is often used to explain phenomenon like the double slit experiment .  The idea goes like this: the particle (electron, photon, whatever) behaves like a wave some of the time - like when it is passing through the slits - and like a particle at other times - such as when it hits the screen and produces a flash in a single location.  This explains how it can behave as if it went through both slits at the same time despite being in just one location whenever we check - e.g. by making it collide with a surface.  The change from wave like behaviour to particle like behaviour is called the collapse of the wavefunction . This idea raises many questions, chiefly: what does and what does not collapse the wavefunction?  For example, if you replace the screen with a mirror then a photon continues to exhibit wave behaviour after bouncing off of it.  The proponents of the wave particle duality theory never answered this q

Proof that $a^2+b^2=c^2$, baked by me at a pottery in France while on holiday

In quantum mechanics a lot of emphasis is placed on the concept of  vector spaces.  One of the key tools is the ability to construct new vector spaces out of existing ones.  However, very often authors construct new vector spaces without explicitly saying what they have done, and the result can be confusing.  In this post I am going to attempt to summarize all the methods I have seen for constructing new vector spaces out of old, and point out where they are used in quantum mechanics. The building blocks What: Hilbert Spaces Why: To represent superpositions of classical states   The building blocks are always Hilbert spaces.  These are vector spaces over the complex numbers $\mathbb{C}$, with inner products and limits.  The pair $(V, \langle\cdot\lvert\cdot\rangle)$ is a Hilbert space if $V$ is a vector space over $\mathbb{C}$ $v\mapsto\langle u\lvert v\rangle$ is linear map $V\mapsto\mathbb{C}$ for any $u$ in $V$ $\langle u\lvert v\rangle = \overline{\langle v\lvert u\rang