Dork Scratchings

A while ago I was pondering the symmetries of Platonic solids and a strange thing occurred to me.  The number of rotational symmetries for the tetrahedron, cube, octahedron, dodecahedron, and icosahedron are 12, 24, 24, 60, and 60 respectively $^\dagger$.  These happen to match the number of hours in an afternoon (I'm told mornings have the same number), the number of hours in a day, the number of minutes in an hour,  and the number of seconds in a minute. This is probably a coincidence due to the fact that the Babylonians (who created our time system) used base 60, and 60 is divisible by a lot of small numbers.  But it's nice to imagine, and just possible, that they were thinking about regular solids when then came up with the system for measuring time.  It's just turned midnight When this occurred to me a picture jumped into my head, which I've tried to recreate above.  It's a clock built from platonic solids.  Every second one or more of them rotates and the numb

Generator symmetries c, b, and a, applied sequentially to the octahedron, resulting in a composite operation of order 2 I've become a bit obsessed with Cayley graphs recently. I figured out a way to construct a graph for the rotational symmetries of the 5 Platonic solids, and the result was quite elegant, I thought.  And it seemed to me that extending this to the complete set of  orthogonal symmetries (which includes reflections) should be quite simple - it is just twice the number of nodes after all.  However,  it took a surprising amount of staring into the middle distance and mumbling to myself to come up with the answer. And I'm going to describe its construction in this post.  As an example I'm using the octahedron here,  but the construction works in exactly the same way whichever solid you choose.  My generators I'm calling $a$, $b$, and $c$, where $a$ is a clockwise rotation of a face,  $b$ is a clockwise rotation around an adjoining vertex,  and $c$ is a reflec

In the previous post I used a short python program to draw a Cayley graph for the rotational symmetries of the cube.  The result was a rhombicuboctahedron, a kind of cross between a cube and an octahedron. Now,  these two platonic solids are "duals" of each other, which is to say that if you start with one and draw a node for each face,  and an edge for each pair of faces that meet, you end up with the other! Is there a pattern here? Can we choose generators for the rotational symmetry groups of the other platonic solids so that the resulting Cayley graphs look like crosses between the original solids and their duals? First let's see if we can find a generic representation for the rotational symmetry group.  Let's assume our solid has $n$ sided faces and the vertices are all of degree $m$. If $a$ is a clockwise rotation about one face and $b$ is a clockwise rotation about an adjoining vertex then $ab$ flips an edge about its midpoint, which implies $(ab)^2=1$. This su

Definitions: A Cayley graph is a directed graph showing how to get from any element of a group to any other using only a finite number of "generator" elements.  For example, in the group $\mathbb{Z}_2\times\mathbb{Z}_2$ we could use the generators $a = (1,0)$ and $b = (0,1)$ and we'd end up with a graph that looks like a square with arrows going round the edges. A Group presentation defines a group by specifying generators and relations between them.  For example the presentation $\langle a,b \vert a^2=1,b^2=1,aba^{-1}b^{-1}=1\rangle$ specifies a group.  In fact the group it specifies is isomorphic to  $\mathbb{Z}_2\times\mathbb{Z}_2$ and we can prove this using the facts that $a$ and $b$ commute and both have order 2.  In this case it's easy to see what group we have but in general it's a bit more difficult.  A useful first step would be to be able to draw the Cayley graph automatically. Question: Is it possible to automatically generate a Cayley graph from a g

  How do we change our minds?  We like to think we're reasonable people - that we listen to the argument and if it is good enough we change our minds!  According to Carl Sagan that barely ever happens outside of science: "In science it often happens that scientists say, 'You know that's a really good argument; my position is mistaken,' and then they actually change their minds and you never hear that old view from them again. They really do it. It doesn't happen as often as it should, because scientists are human and change is sometimes painful. But it happens every day. I cannot recall the last time something like that happened in politics or religion." -- Carl Sagan, 1987 CSICOP keynote address So what about the rest of the world?  People do change their minds, so how do they go about it?  It occurred to me that a good metaphor for the mechanism is a concept from general relativity.  The concept is Parallel Transport and the animation above illustrates

 

It may be a sign we're close to one or more tipping points The latest report from the Intergovernmental Panel on Climate Change is known as Assessment Report 6, or IPCC AR6 for short.  Rather than contain new research, AR6 summarizes the latest work of climate scientists, and synthesizes thousands of papers.  An important element of this are the climate models.  The climate models examined by AR6 are called CMIP6 models, for Coupled Model Intercomparison Project v6. Climate models can provide answers to "what if?" questions.  Some of these What Ifs are described by the IPCC in their Shared Socioeconomic Pathways, or SSPs for short.   But these What Ifs combine questions about physics (how the Earth will respond) with assumptions about our future behaviour.  What we would like is to be able to factor out the human influence and get a single number that measures just how sensitive the Earth's climate is to CO2?  That's where Equilibrium Climate Sensitivity (ECS) com

I got well and truly nerd sniped the other day when my friend showed my this XKCD cartoon Credit: XKCD.com

  Credit: Colony of Gamers/Flickr   My family settled down to a perfectly civil game of Risk this Christmas.  At one stage my wife, who controlled most of Asia, challenged my total control of Australia by attacking my territory of Indonesia from her territory of South East Asia, with about 100 armies. Attacks in Risk consist of a series of "battles".  In the initial battles two armies are always lost: either two attackers, two defenders, or one of each.  After enough battles either one defending army remains or two attackers.  From this point on only one army is lost per battle.  Eventually either the defender is completely obliterated or the attacker reduced to one army (which is needed to defend the originating territory) and the attack is over. In the initial battles (when the attacker has at least 3 armies remaining - not including one set aside to defend the originating territory - and the defender at least 2) the attacker rolls 3 red dice and the defender 2 blue dice. 

A Pythagorean triple is a set of three non-zero integers $a,b,c$ satisfying Pythagoras' formula $$ c^2 = a^2 + b^2 $$ Pythagoras certainly wasn't the first to know that this formula applied to the sides of right angled triangles, or to compile lists of Pythagorean triples, but he may have been the first to present a proof .  In this post I'm going to show a simple method of finding all Pythagorean triples, using complex numbers. First we need to introduce Gaussian Integers .  These are complex numbers $z=a+ib$ where $a,b$ are integers.  If $z$ is any complex number then, since multiplication is commutative, $$ (zz^*)^2 = z^2(z^2)^* $$ However, in the case where $z$ is a Gaussian Integer the LHS is a square of an ordinary integer, and the RHS is the sum of two squares of integers.  For it to be a sum of two non-zero squares we need $z^2$ to be neither purely real or purely imaginary, or equivalently, that $z$ is neither purely real, purely imaginary, or on a diagonal $a =

Is it possible to bring a temperature below absolute zero? And if so, what does, say, -1K look like? Surprisingly, negative temperatures are in fact possible, but only in some circumstances. To understand how, we need to get a better idea of what temperature actually means from a statistical thermodynamics point of view. Suppose you have a system of $N$ particles$^{\dagger}$, each of which can have an energy level from a discrete list $$ 0 < E_1 < E_2 < E_3 < ... $$ Now suppose that there are $N_1$ particles in state $E_1$, $N_2$ in $E_2$ and so on. Using combinatorics$^{\dagger_2}$ we can see that the number of ways this particular configuration can be achieved is given by $$ \Omega = \frac{N!}{N_1!N_2!N_3!...} $$ The next question is: if we know the total energy of the system $E_{total}$ can we work out the distribution of particles across the energy states? Well, we know the most likely distribution is that which can be achieved in the greatest number of ways. I.e. we

The Saudi British Joint Business Council and... ...a Perpetual Motion machine!   I spent yesterday wandering around the London Climate Technology Show, and I had many interesting conversations.  There were way too many middlemen selling questionable climate offsets.  These all provided an account and a fancy online dashboard for companies wanting to demonstrate their carbon neutrality.  At least 4 of the companies were using Blockchain as a buzzword.  The idea is that you write a contract with the farmer/landowner/whatever to improve soil carbon/not cut down trees/whatever and then you sign that contract and put it in an online ledger like the Ethereum blockchain.  This prevents the existence of the contract being denied and enables ownership to be transferred, like with digital coins.  Although in principle this can work it requires a single joined up database.  With multiple companies offering the same service, but on incompatible systems, there's nothing to stop an offset bein

Programmers: first they ruined chess, now they ruin ... Wordle (Okay, so ruining chess is a bit harder) Step 4 was a sacrifice move I got stuck yesterday doing Wordle and I couldn't think of a single word that might work.  So... I wrote a short script to prompt me with suggestions.  I've noted that, so far, the solution has never been a plural.  So it's probably a good idea to take the first suggestion that doesn't end in 's'. And here it is:

Here's a neat trick for estimating the distance to objects.  Stick out your thumb and look at it with just one eye open, then with just the other eye open.  The distance to the object is the distance it appears to jump multiplied by 10 .  This works because the distance from your eyes to your thumb on your outstretched arm is about 10 times the distance between your eyes.   I thought I'd have a go at using this to measure the distance to the moon

Here's a nice theorem by Gauss The (2D) shoelace theorem Suppose $\mathbf{x_1},\mathbf{x_2},...,\mathbf{x_n}$ are ordered vertices of a polygon.  Let $x_i, y_i$ be the coordinates of $\mathbf{x_i}$ and define $\mathbf{x_{n+1}}$ to be $\mathbf{x_1}$.   Then the area of the polygon is plus or minus $$ \frac{1}{2}\sum_i{x_iy_{i+1} - x_{i+1}y_i} $$ This formula gives rise to the name of the theorem as shown by the illustration below Proof We're going start off by proving the simple case where the polygon is a triangle.  The first thing to notice is that if you take any two adjacent vertices, .e.g. $\mathbf{x_1}$ and $\mathbf{x_3}$, they form a triangle with the origin.  This has an area equal to half of the parallelpiped formed by the two vertices (treated as vectors from the origin). And the area of the parallelpiped is plus or minus $det(\mathbf{x_1}\vert \mathbf{x_3})$, with the sign being positive if the columns taken in order produce an anti-clockwise motion around the origin.

Rough and ready order of magnitude calculation: Cows burp 100kg methane each year, which lasts 10 years before decaying in the atmosphere, and has about 100x the warming potential of carbon dioxide while it's up there. Cows (and other ruminants such as sheep) continuously burp CH$_4$, a powerful greenhouse gas.  In fact, gram for gram methane warms the planet 87 times more than carbon dioxide .  However, CH$_4$ reacts in the atmosphere so that the carbon atom eventually finds itself in a CO$_2$ molecule.  We can think of it as having an atmospheric half life of about 8.6 years .  For this reason, the Global Warming Potential (GWP) of methane is sometimes averaged over 100 years to give a GWP100 figure of 27-30, i.e. twenty seven to thirty times as powerful (over 100 years) as carbon dioxide. Methane enters the atmosphere from a number of sources.  Two significant ones are biogenic sources such as cows, and fossil sources, such as when unwanted methane is flared by oil rigs, or when

An overhead free aligned memory allocator (Well, almost overhead free)      I designed this memory allocator for a Linux device driver I wrote 20 years ago .  It has the property that if you ask for 64 bytes you get a 64 byte aligned block; if you ask for 128, you get a 128 byte aligned block; and so on.  The other nice thing about it is that it automatically defragments, which is to say that if, for example, a 128 byte aligned 128 byte block is free then it can be allocated as a 128 byte block, even if it one or both halves have previously been allocated as 64 byte blocks and later freed. The key to the design is the data structure.  Each free block has a 64 byte header consisting of 16 pointers (I wrote this for a 32-bit architecture, although it could easily be ported).  For this reason 64 bytes is the smallest block size that can be allocated.  The pointers are all either NULL or point to another free block of the same size, forming a 16-ary tree of equal-sized free blocks.  At th