Dork Scratchings

See below for code This was the most up to date thing we proved in my maths degree.  It's a lovely little theorem in the field of Fractal Geometry that enables you to create images like the ones above.  But first some primers.... A fixed point of any map $f:X\to X$ is a point $x\in X$ such that $f(x) = x$.  If $X=\mathbb{R}^N$ we can define a contraction as a map $f$ which brings pairs of points closer together, i.e. we can say $f$ is a contraction if there's some $\lambda < 1$ such that for any $x_1,x_2$ we have  $|f(x_1)-f(x_2)| < \lambda |x_1-x_2|$.  Now, it's easy to see that if $f$ is a contraction then it has a unique fixed point.  All you have to do is note that for any $x$ the following sequence converges $$ x, f(x), f^2(x), f^3(x), ... $$ Why's that?  Well if we let $\epsilon = |x-f(x)|$ then $\epsilon\lambda^{n-1}$ is an upper bound for the distance between the $n^{th}$ and $n+1^{th}$ members of the sequence.  Since $\sum \epsi...

Background ECS - Equilibrium Climate Sensitivity - is the temperature change on Earth as a result of a doubling of atmospheric CO2.  That means going from 280 ppm (the level prior to the industrial revolution) past 420 (where we are now) and on to 560 ppm.  Most people probably understand why this would increase temperatures by now, but I'm going to repeat the argument briefly anyway, for completeness

Short answer: If there were a simple formula to follow there would also be one for proving or disproving any mathematical statement. Before setting out a sketch of a proof I need to be more clear about the problem.  I want to rule out the trivial method of untangling where you simply thread one end backwards through the tangle till it meets the other end.  So we're going to imagine it's a loop of string that's knotted, and "untangling" means getting back to a simple loop, without magically passing one bit through another. For most closed loops it's not possible to untangle to a loop.  For example if you look at the Trefoil Knot above (a.k.a. Granny Knot) it's obvious that no amount of fiddling with it is going to get rid of the knotty bit and leave just a loop.  But if your loop $L$ can be deformed into a simple loop then it follows that the complement in 3D space $\mathbb{R}^3 - L$ can be deformed continuously into the complement of the simple loop $\...

  I picked up this fantastic little book in Cambridge market a few days ago.  Published in 1908 by the Society for the Promotion of Christian Knowledge , it summarizes a lecture performed by Professor J Perry in 1890 in Leeds. The theme of the Romance of Science series is that remarkable and unexpected things can be discovered from the study of the apparently mundane.  In this lecture Professor Perry performs a series of experiments on various designs of spinning top and shows i.) What causes the precessional motion of a spinning top ii.) That a subterranean race could determine that we live on a spinning planet without ever seeing the stars iii.) Why the precessional motion of a top is in the same orientation as the spin, but the precessional motion of the Earth is in the opposite orientation iv.) Why some spinning tops stand up when you spin them on their side and even v.) A model of how a magnetic field might cause the observed rotation of plane polarized light The q...

  Sometimes it helps to express things in human scales

  I'm getting arty now. Following my previous post on greyscale, I wondered whether the numbers 0..63 can be made to look pretty too.  So I ran this in a terminal and then rotated and stretched the result in an image manipulation program. for i in range(64): s = f "{i:06b}" print(s.replace( '0' , ' ' ).replace( '1' , '\u2588' )) Mesmerizing!

  Posting a pretty picture of some greyscale in which the fractal nature of it is clearly visible.  Black rectangles represent zeros and white rectangles represent ones. Read it as an array of columns going from left to right.  In each column exactly one bit changes relative to the previous, making it an ideal scheme for a head to determine its position. An $n$-bit greyscale sequence has $2^n$ members, and the last member always differs from the first in just 1 bit position.  This makes greyscale ideal for determining position around a cylinder. Here's the code: #!/usr/bin/env python3 def __greyscale_fwd(nbits, preset_bits): if nbits == 1: yield 0 + preset_bits yield 1 + preset_bits else : for x in __greyscale_fwd(nbits-1, preset_bits): yield x for x in __greyscale_rev(nbits-1, preset_bits | (1 << (nbits-1))): yield x def __greyscale_rev(nbits, preset_bits): if nbits == 1: yield 1 + preset_...

Quantum physics slash Silicon Valley drama Devs features an interesting architectural idea, namely a floating laboratory.  In order make a laboratory float you need very lightweight construction materials and it seems that they've solved this problem by using Menger Sponge bricks, which are very lightweight indeed. A Menger Sponge is a 3D version of this To make a Menger Sponge you need to perform an iteration.  Start with any shape, then make 8 copies each scaled down by a factor of 3, and use them to construct the perimeter of a square.  Then take that shape and do the same operation, and keep repeating.  The picture above shows that you end up with the same thing whether you start with a solid square or a hollow square.  But actually it doesn't matter what shape you start off with (as long as it's bounded). So, what's the area $A$ of the 2D version Menger Sponge shown above?  We can see from the first row in the picture that it's made up of 8 copies of...

Fossil Fuels are dragging your pension down!   At the end of last year I finally managed to switch to a fossil fuel free pension.  The journey wasn't completely straightforward, but it was worth it and hopefully this will become a lot easier in the near future. Why do this? There's a bunch of things you can do to reduce your carbon footprint including giving up flying and eating less meat, and I recommend all of them.  But right at the top has to be to stop investing in fossil fuel companies.  A typical pension pot in the UK is around £88,000 of which - typically - 4% is invested in oil, gas, and coal.  That's three and half grand of your money going directly to the likes of ExxonMobil.  If you simply spent this much on tax free petrol it would buy you about 8.8 tons of fuel which would create about 29 tons of CO2, or about 5x the annual emissions of an average Brit.  But investing probably results in an order of magnitude more emissions $^\dagger$ ...

According to this story on today's BBC news website the UK government is set to mandate the proportion of ethanol in standard petrol is increased from 5% to 10%.  The 5% grade is known as E5 and the new grade will be known as E10. So, how much of the UK's arable land would be needed to make this, if it were made here? The article says " Ethanol is a kind of alcohol manufactured from plants, including sugar beet and wheat ".  Since sugar beet can be grown successfully in the UK, and it has a higher ethanol yield than wheat per square metre, I'm going to assume that we make the ethanol entirely from sugar beet.  So let's see if we can work out what proportion of the UK's 60,000 sq km arable land would be needed to power 10% of it's vehicle miles: In [ 1 ]: beet_ton_per_sq_km_per_year = 1320 In [ 2 ]: litre_biofuel_per_ton_beet = 108 In [ 3 ]: litre_biofuel_per_sq_km_per_year = beet_ton_per_sq_km_per_year*litre_biofuel_per_ton_beet In [ 4 ]: gallon_b...

What's Strange about this clip?   (fair use)   This programme about the future appeared on my TV last night, and it touched on an issue close to my heart: the future viability of life on Earth.     The first 3 minutes 45 seconds delivers some of the stark evidence for what we are doing and projections for where we are heading.  There's no emotion on display, but that's fine - we can't be emotional all the time and there's room for dispassionate analysis as well as appeals to the heart.  But then something strange happens at 3:48 - the presenter Hannah Fry introduces the next subject with an incredibly chirpy "Flying Cars! There - I've said it". Watching this one is left thinking:   Either I'm mad or everyone else is.  We've just been told that the average population decline across all species between 1970 and 2020 will be 67%.   And after dedicating three and bit minutes to that we're now talking about flying cars!  I'm worri...

  Hydrogen is an incredibly inefficient way to heat your home compared with heat pumps.  Heat pumps move about 4 times as much energy in the form of heat as the electrical energy they consume.  With hydrogen boilers the heat is created from chemical energy rather than moved and so there's no such upscaling.  In fact it's worse than that since the hydrogen has to be created in the first place and if you're creating it from renewable electricity about a third is wasted. Despite the horrendous waste, hydrogen boilers may have a role.  Heat pumps require an insulated hot water tank and many houses, especially in the UK, will not have space for an airing cupboard.  So perhaps the solution will be to install heat pumps where-ever possible and then use the much-less efficient hydrogen boilers to mop up the rest.  The risk, however, is that the government's enthusiasm for Hydrogen will slow down the decarbonisation of our heating sector. We are in a deadly ra...

  The news here in the UK over the last couple of months has focussed on the unfolding catastrophe of Covid19 in India.  It has been implicit in pretty much all of the coverage that the situation there is much much worse than we have suffered in the UK, and this has been backed up by numbers showing just how many more people have died in India than here. True to form, our media have almost universally failed to take into account the fact that the population of India is nearly 25 times bigger.  I think dividing one number by another is not part of the training.... Here's a chart that puts things in context.  I've used 66.65 million for the UK population and 1.366 billion for India: INDIA OFFICIAL:   These are the Indian government official statistics republished by JHU CSSE COVID-19 Data showing 311,000 deaths INDIA ESTIMATE: These are from David Spiegalhalter and are based on excess mortality.  The best estimate is 1.6 million but could be as low as 600,0...

Simple example from The Theoretical Minimum I recently re-read The Theoretical Minimum - Classical mechanics , an excellent book. The authors give an example of the use of the Hamiltonian formulation of mechanics to illustrate how much easier it makes things (compared to the Newtonian formulation) and that inspired me to have a go myself. The example they give in the book is of a negligible mass charged sphere spinning in a magnetic field pointing in the z direction. In this case the Hamiltonian $H = T + V$ is proportional to $L_z$ where $\mathbf{L}$ is the angular momentum $L_x \hat{i} + L_y \hat{j} + L_z \hat{k}$. Let's say $H=\omega^2L_z$. The book explains that for any function $G(q,p)$ of the generalized coordinates $q_i$ and the conjugate momenta $p_i$ that $$ \dot{G} = \{G,H\} $$ i.e the time derivative of $G$ is the poisson bracket of $G$ and the Hamiltonian. And indeed this does make the solution simple because the components of angular momentum satisfy $$ \begin{align} ...

 I've just found this design I must have drawn when I was 16 I had a girlfriend at the time called Catherine with very long hair.  In fact it was so long she could sit on it.  Anyway, I was amazed at how much time she spent combing - about half an hour every day.  She'd start at the bottom and work her way up, combing all the way to the ends with each stroke. I thought that there had to be a better way, so I came up with this machine.  The two belts rotate in opposite directions and the angle between them gradually decreases so that it starts off just untangling the ends and ends up combing the whole length of the hair. Fortunately it never got built.

... One day. But I don't want to lose this important historical information - my offspring's height vs time! So I've converted it into a nice graph Using some code #!/usr/bin/env python3 from datetime import datetime from matplotlib import pyplot as plt table = [ (132.4, '10/7/11' ), (134.8, '14/9/11' ), (136.9, '2/3/12' ), (138.4, '7/9/12' ), (141.5, '25/5/13' ), (142.6, '23/6/13' ), (145.8, '15/3/14' ), (148.7, '15/11/14' ), (150.0, '20/1/15' ), (153.1, '8/6/15' ), (157.7, '12/11/15' ), (159.0, '21/12/15' ), (160.1, '7/2/16' ), (164.1, '12/6/16' ), (167.6, '27/12/16' ), (169.9, '4/3/17' ), (171.7, '14/11/17' ), (172.1, '6/6/18' ), (172.9, '24/8/19' ), (173.6, '2/4/20' ), ] def labelToX(label: str): d,m,y = label.sp...

Are we approaching the cladding crisis rationally? Cladding is in the news at the moment.  In response to the tragedy at Grenfell the UK government has introduced regulations requiring buildings with "unsafe" cladding to be patrolled.  This is obviously expensive and in most circumstances the cost is borne by the leaseholder.  The average paid per leaseholder is £137 per month (with a median of double that).  If we assume an average of two occupants per flat that comes to £822 per occupant per year. Obviously these actions may save lives.  But there are many other risks that we face day in and day out and it is worth considering whether we price these risks the same.  After all, that money has consequences - it has bankrupted people, particularly those who face the tail end of the cost distribution.  So let's try to work out how much car tax would be if we treated the risk of dying in a traffic accident in the same way, and loaded the cost per life ont...