Dork Scratchings

Why is it tides are semi-diurnal ? That is, why do they occur every 12 hours when they are caused by the gravitational pull of a moon that we turn to face once a day?  The straightforward answer is that water bulges out on both the side nearest and the side furthest from the moon.  (We'll ignore the Sun for simplicity, but adding it in doesn't change anything.) The Earth rotates while the bulges remain in place, causing tides 12 hours apart.  However, this doesn't explain why there should be two bulges.  The reason becomes clear when you change the frame of reference.  Instead of thinking about a frame in which both the moon and the Earth rotate, set the origin to be the centre of mass of the two objects, and choose a rotating frame in which the moon is stationary.  In this frame there is a "fictional" centrifugal force of $\omega^2 r$ which combines with the gravitational force from the moon, $GM_{\text{moon}}/(r-r_{\text{moon}})^2$ The two forces match at the c

In 1964 John Stewart Bell proposed an experiment to determine whether the results of quantum measurements were truly random, or governed by hidden variables, i.e. state that exists prior to the measurement, but which we don’t have access to. The experiment involved creating a large number of EPR pairs, and firing them at two observers, Alice and Bob, who measure their photon’s polarisation, choosing the $\updownarrow$ direction or the $\nearrow\llap\swarrow$ direction at random. Determining the result of the experiment involves doing a complex statistical calculation to see if something called Bell’s inequality is satisfied or violated. The Bell experiment was first performed by in 1982 by Alain Aspect, and the result, as most commonly interpreted, is that hidden variables can only exist if Quantum Mechanics is non-local, i.e. if it supports faster-than-light causality! Some time after Bell proposed his experiment, Greenberger, Horne, and Zeilinger suggested an alternativ

In this post I argue that economic growth is not the normal state of affairs, it is a blip caused by the discovery of a one-off resource of non-renewable energy.   The logistic equation $$ \frac{dy}{dt} = y(1-y) $$ Here's a simple model.  Humans discover an exploitable but limited resource and start consuming it.  The amount consumed, $y(t)$, is a function of time.  The general form of the equation is $\frac{dy}{dt} = \alpha y (\beta - y)$, but if you choose the right units $\alpha$ and $\beta$ both become $1$. Why should this work, in principle? Early on, the factor $1-y$ is approximately $1$ and can be ignored.  So the model states that annual consumption $dy/dt$ starts off proportional to $y$.  In other words, $y$ grows exponentially at first.  This could happen if exploiting the resource enables further exploitation of the resource.  For example, suppose a few humans are shipwrecked on an island with 1000 trees: they take ages to cut down the first tree as they are using their

My first neural net Like everybody else I've been playing a lot with ChatGPT recently and I'm gobsmacked by how good it is!  This has led me to start researching how exactly these things work.  In particular I wanted to understand how neural nets - a core component of the technology - are trained. Running a neural net is simple.  The neural net consists of layers of nodes connected by weights and biases.  The first layer is an input layer and setting the activation levels of each node in that layer causes the next layer to adopt values determined by the weights and biases.  The second layer determines the activation levels in the next layer and so on until the final layer, which is interpreted as the output. 

    I was given a new version of Dobble for Xmas called Dobble Connect.  Dobble consists of stack of cards with pictures on it where each pair of cards has one match, and each pair of symbols appears on one card.  This is a new version which has 91 symbols in total, 10 symbols per card, and 91 cards.  The fingerprint of the original Dobble was (57 symbols, 8 symbols, 57 cards). An additional innovation in Dobble Connect is that the cards are hexagonal and grouped into a colour per player.  Players build a tiling on the tabletop, placing cards next to each other as soon as they spot a matching symbol between their top card and one of the cards already placed.  The first player to get four of their own colour in a row wins. How is it possible for each pair of cards to have one and only one symbol in common, and for each pair of symbols to appear on one and only one card?  Does that have something to do with the strange number 91?  The answer is "yes" and it turns out this is o

Rolling marbles  Here's a puzzle: Suppose you have N identical marbles rolling along a one dimensional table-top.  Each marble is randomly rolling to the left or to the right, all with the same speed.  Collisions are elastic, which means the marbles just change direction.  What is the maximum amount of time before all the marbles have rolled off the table? 8 marbles with speed 1 on a table of length 1 Answering this question is really difficult if you simply pick an individual marble and try to work out how long it might stay on the table as it bounces back and forth.  But there's a simpler way to look at it. Prior to each collision you have one marble rolling to the left and one to the right, and afterwards you still have one rolling to the left and one to the right.  If we swap labels following each collision then the labels never change direction.  Now it's easy to see that the answer is the same whether there are 100 marbles or just one.  And all we did was switch our P

Contents Intro Simple stratified model Alternative convection model Tropospheric temperature gradient Approach used in convection model Calculation of Radiative Forcing from CO₂ increase Calculation of Radiative Forcing from H₂O increase Calculation of zero-feedback ECS Effect of including feedbacks Effect of water vapour Effect of cloud cover change Effect of sea ice loss Calculation of ECS including feedbacks Limitations One more thing: Changes to Troposphere depth Intro The goal of this post is to see if we can come up with an estimate of Equilibrium Climate Sensitivity using a simple model in which the atmosphere is treated as well-mixed.  By "simple" I mean that using the model will not require any advanced mathematics or computation, but will still be realistic enough to come up with an estimate within the likely range of 2.5 - 4.0°C predicted by the IPCC Assessment Report 6 .  I will try to take as little on trust as possible and show how the results are arrived at.

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