Disappearing lighthouses: atmospheric refraction at Portobello beach

Look out to sea at night from the beach at Portobello in Edinburgh, and you’ll often see lighthouses blinking in the blackness. Here’s a picture of those lights, taken in such a way as to convert time into space so that the different flashing sequences become apparent.

I pointed my camera out to sea, on a tripod, and slowly panned upwards over what was effectively a long exposure of about a minute. The movement of the camera smears out any light source into a near-vertical streak (the tripod wasn’t very level). The two thick streaks are ships at anchor in the Firth of Forth.

A flashing light leaves a broken streak; this reveals the lighthouses. At the left is Elie Ness on the Fife coast, flashing once every 6 seconds. Second from the right is the Isle of May in the Firth of Forth, flashing twice every 15 seconds. And at far right we have Fidra, off North Berwick, flashing 4 times every 30 seconds.

And the green streak? A passing dog with a luminous collar!

Lighthouses are lovely, romantic, things. But what’s extra lovely about the lighthouses in this image is that you shouldn’t be able to see two of them!

Both the Elie Ness and Isle of May lighthouses are, geometrically, beyond the horizon. If light travelled in straight lines, we wouldn’t be able to see either of them. As it is, whether we can see them or not depends upon the weather: if I’d taken the picture in cold weather, only the Fidra light would have been visible.

Isle of May lighthouse. Image: Jeremy Atherton.

The reason that we can see them at all is that light travelling through the atmosphere is refracted by the air: its path is (usually) slightly curved downwards. This means that we can see slightly “over the hill”, a bit further than simple geometry would suggest. Things near the horizon appear higher in the sky than they really are.

It’s not a small effect. For example, when we see the Sun sitting with its lower edge just touching the horizon, the geometric reality is that all of it is actually below the horizon.

The refraction happens because the air gets thinner as you go upwards. Just as light passing through a prism is bent towards the thicker end of the prism, so light passing through the atmosphere is bent towards the thicker (denser) part of the air.

Elie Ness lighthouse. Image: Francis Webb.

The amount of atmospheric refraction depends upon the weather. It depends upon the pressure and temperature, and the temperature gradient (how quickly the temperature drops as you go upwards). When it’s cold, the Elie Ness and Isle of May lights disappear. If it warms up, they pop back into view again. And when it’s really cold and the air is colder than the sea, the refraction can be reversed, and we see mirages along the coast. But that’s for another time.

Mirages on the Fife coast, seen from Portobello (distance about 8 miles/13km). Early April, with air temperature forecast to be about 0° C. The sea temperature was about 7° C.

How horizons work, and how we see things that are beyond the horizon

The horizon exists because the Earth’s surface is curved. In these diagrams the circular arc represents the surface of the Earth. If my eye is at the point O, I can see the Earth’s surface up to the point H (where my line of sight just grazes the surface) but no further. The point H is on my horizon. Its distance depends upon the height of O. It can be surprisingly close: if I stand at the water’s edge, my horizon is only 4.5 km away.


If the thing I’m looking at (a lighthouse L for example) is raised above the surface of the Earth, I’ll be able to see it when it is further away. In the diagram below, I’ll just be able to see the lighthouse. The rays from the lighthouse will just kiss the wave tops in the darkness on their way to my eye. The point is on the lighthouse’s horizon as well as mine. The higher the lighthouse (or me) is, the further away it can be and I’ll still see it.


But if it’s too far away (below), I won’t be able to see it. There’ll be a region of the sea that is not in sight from either my eye or the lighthouse. This “no-man’s land” is the region between my horizon H_O and the lighthouse’s horizon H_L. For example, there’s about 6 km of no-man’s land between me on the promenade at Portobello and the Isle of May lighthouse.


All of these diagrams assume that rays of light are perfectly straight. But rays of light passing through the air aren’t perfectly straight: they have a downward curvature because of atmospheric refraction. This means that rays from an object that is geometrically beyond the horizon might actually reach my eye. That’s why I can sometimes see the Elie Ness and Isle of May lights. 

The curvature of the rays of light is hugely exaggerated in this diagram. Otherwise it would be imperceptible. The rays deviate from straightness by only a few metres over a journey of several tens of kilometres.

The curvature of the rays varies according to temperature, pressure etc. It’s a happy accident (for me) that both lighthouses are only marginally out of view over the horizon, so that they can disappear and reappear according to the whims of the weather.

How I took the picture

The picture isn’t actually a single long exposure. I thought that I might end up with a pale sky rather than a black one if I did that. Instead, I took a one-minute video, slowly panning the camera upwards. I then wrote a program in Python to produce a composite image derived from all of the video frames. For each pixel position in the image, the program inspected that pixel position in all of the video frames and chose the brightest value.

Calculating the distance of the horizon

We can use Pythagoras’ theorem to work out how far away the horizon is. In the diagram below, the circle represents the surface of the Earth, with centre at C and radius r. You, the observer, are at O, a height h (greatly exaggerated) above the surface. Your horizon is at H, a distance d away.

The triangle OHC is right-angled at H. Applying Pythagoras’ Theorem, we get
(r+h)^2 = d^2 + r^2
and so
d^2 = r^2 + 2rh + h^2 - r^2

Where h is very small compared to r, as it will be for the heights we’re dealing with, h^2 \ll 2rh, so we can neglect the term h^2 and get, to a very good approximation (within centimetres)
d = \sqrt{2rh}

Calculating the visibility of the lighthouses

There is a Wikipedia article with formulae for the curvature of near-horizontal light rays in the atmosphere.

I’m only going to give a summary of results here, principally because although I’ve got the spreadsheet that does the calculations, I can’t find the notebook where I worked out the geometry. Here are the details of the lighthouses. Heights are the height of the lamp in metres above mean sea level.

LighthouseDistance (km)Height above sea level (m)
Isle of May4473
Elie Ness3115

Consider the next table as an example, based on roughly typical weather conditions for this time of year (March). The figures assume that I’m standing on Portobello promenade. The Ordnance Survey map shows a 3 metre spot height marked on the prom, so that would make my eye about 4.5 metres above mean sea level.

The first three numerical columns of the table shows how high above the horizon each lighthouse would be in the absence of refraction, what the estimated refraction is, and thus how high above the horizon the light should appear. The heights are expressed as angles subtended at your eye. There’s a lot of uncertainty in the estimated refraction (because of uncertainty about the input values such as temperature gradient), which is why the middle two columns are given to only one significant figure.

LighthouseAngle above horizon with no refraction (degrees)Estimated refraction (degrees)Estimated angle above horizon with refraction (degrees)Water depth over direct line of sight (m)
Isle of May-0.0400.0490.015.3
Elie Ness-0.0510.035-0.026.8

Thus we see that the Isle of May and Elie Ness have negative heights above the horizon without refraction, ie they’re geometrically below the horizon. In the conditions given, refraction is enough to raise Isle of May into visibility, but not Elie Ness – the angle with refraction is still negative. This accords with my experience: I’m more likely to be able to see Isle of May than Elie Ness. Fidra is above the horizon, refraction or no refraction.

Note that the angles above and below the horizon are tiny. For comparison, an object 1 mm across held at arm’s length subtends an angle at your eye of about 0.1 degrees. Most of the angles in the table are less than half that.

The rightmost column is there to help understand how the tide can affect things. Saying that the Isle of May and Elie Ness lights are beyond the horizon is saying that there’s water between my eye and their lamps. Imagine that the light from the lamps could travel through the water completely unimpeded and in a perfect straight line to my eye. This column shows how far under the water the light rays would be at their deepest. As you can see, they are single-digit numbers of metres. Now the tidal range in the Firth of Forth is about 4 metres. What this shows us that that the state of the tide could easily make the difference between seeing or not seeing a given lighthouse. It also brings home how slight a curvature of the rays is produced by refraction: in Isle of May’s case, there’s just enough curvature to get the rays over a 5-metre bump in a 44-kilometre journey.


Image credits

Isle of May: Jeremy Atherton; Elie Ness: Francis Webb. Both under CCA license.

Decisions of cricket umpires

In this post I offer a suggestion for a practically imperceptible change to the laws of cricket that might eliminate controversies to do with adjudications by match officials. The suggestion could apply to any other sport, so even if you aren’t a cricket lover, please read on.

My suggestion doesn’t affect the way the game is played in the slightest. It simply takes a more realistic philosophical angle on umpires’ decisions.

Cricket is a bat-and-ball ‘hitting and running’ game in the same family as baseball and rounders. In these games, each player on the side that is batting can carry on playing (and potentially scoring) until they are “out” as a result of certain events happening. For example, in all of these games, if a batsman* hits the ball and the ball is caught by a member of the other team before it hits the ground, the batsman is out.

In cricket, there are several ways that a batsman can be out. Some of these need no adjudication (eg bowled), but most require the umpire to judge whether the conditions for that mode of dismissal have been met. In the case of a catch, for example, the umpire must decide whether the ball has hit the bat, and whether it was caught before touching the ground. Contact with the bat is most often the source of contention, because catches are often made after the ball has only lightly grazed the edge of the bat.

The umpire’s position is unenviable. They have to make a decision on the basis of a single, real-time, view of the events, and their decisions matter a great deal. The outcome of a whole match (and with it, possibly the course of players’ careers) can hinge on one decision. It’s not surprising that umpire’s decisions are the cause of much controversy.

For most of the history of cricket, the on-field umpire’s judgement has been the sole basis for deciding whether a batsman is out. This is still true today for nearly all games of cricket, but at the highest levels of the game, an off-field umpire operates, using slow motion video, computer ball-tracking, and audio (to hear subtle contact of the ball with the bat). The on-field umpires (of which there are two) can refer a decision to the off-field umpire, and the players have limited opportunities to appeal against the decisions of the on-field umpires. From now on we’ll call the off-field umpire the “3rd umpire”, as is commonly done.

One of the intentions behind all of this was to relieve the pressure on the on-field umpires, but it appears that the opposite has been the case. In a recent Test Match between England and Australia, one of the umpires had 8 of his decisions overturned on appeal to the 3rd umpire. This led to much criticism and must have been excruciating for him.

Here’s a suggestion for a small modification to the laws of cricket that wouldn’t change the course of any match that didn’t have a 3rd umpire, but which would put the on-field umpires back in charge and relieve much of the pressure on them. As a bonus, it would settle another thorny issue in the game – whether batsmen should “walk” or not (see later).

The suggestion

I’ll use the judgement “did the ball touch the bat?” as an example, but the same principle applies to any judgement of events in the game. We’ll assume that the ball was clearly caught by a fielder, so that contact with the bat is the only matter at issue.

There are three elements to an umpire’s decision: the physical events, the umpire’s perception of those events, and the decision based on that perception. We can represent these elements in a diagram:

For our specific example, the diagram looks like this:

Because our perceptual systems are imperfect, the umpire’s perception of events doesn’t necessarily correspond to the actual course of events. They may perceive that the ball has hit the bat when it hasn’t, or vice versa. This source of error is represented by linking the left-hand boxes by a dashed arrow.

On the other hand, the umpire has perfect access to their own perceptions, so the final decision (out/not out) follows inevitably from those perceptions (provided that the umpire is honest). This inevitable relationship is represented by linking the right-hand boxes by a solid line.

Now, at present, the law is specified in terms of the physical events that occurred. This means that, because the umpire’s perception is imperfect, the umpire can make an incorrect decision: one that is not in accord with those physical events.

However, in any match without a 3rd umpire (ie practically all cricket) the umpire is the sole arbiter of whether a batsman is out or not. So regardless of the actual laws, the de facto condition for whether a batsman is out is the umpire’s perceptions, not the physical events, like this:

My suggestion is simply to be honest about this state of affairs and enshrine it in the laws.

Thus, the relevant part of the law, instead of reading (as it does at present):

…if [the] … ball … touches his bat…

would read

…if the ball appears to the umpire to touch the bat (regardless of whether it did actually touch the bat)…

This may seem like a strange way to word the law, but it’s just codifying what happens anyway in nearly all cricket. The course of all cricket matches that don’t have 3rd umpires, past, present, and future, would be entirely unchanged. We’d be playing exactly the same game. The only difference is that all umpires’ decisions would, by law, be correct, and so the pressure on them would be removed.

The other main advantage of my proposal would that it would render 3rd umpires and all their technology irrelevant, and we could get on with the game instead of waiting through endless appeals and reviews. Cricket would once again accord with the principle that a good game is one that can be played satisfactorily at all levels with the same equipment. And the status of the umpires would be restored to being arbiters of everything, rather than being in danger of being relegated to mere ball-counters and cloakroom attendants.

The opposition

I have to confess that no-one I’ve spoken to thinks that this is a good idea. There seem to be two counterarguments. The first is somewhat vague – that there’s something a bit airy-fairy about casting the law in terms of events in someone’s brain rather than what actually happened to balls and bats. I might agree with this argument if my proposal actually changed the decisions that umpires make, but it doesn’t – the only things that change are the newspaper reports and the mental health of umpires.

The second counterargument is more substantial. Under my proposal, even an umpire with spectacularly deficient vision could never make an incorrect decision. Likewise, a corrupt umpire would have a field day (so to speak). Yet quite clearly, we do only want to employ umpires whose decisions are generally “accurate”, in the sense that they reflect what actually happened. My proposal is quite consistent with maintaining high umpiring standards. At the beginning of any match, we appoint umpires, and by doing so we define their decisions to be correct for that match. That doesn’t stop us later (say, at the end of the season) reviewing their decisions en masse and offering training (or unemployment) if the decisions appear to consistently misrepresent what actually happened. Again, this is roughly what actually happens at the moment. Players (usually) accept the umpire’s decision as it comes, but at the end of the game, the captains report on the standard of umpiring. All I’m doing is changing the way we regard the individual decisions.

To walk or not to walk?

My proposal eliminates another controversy in the game: what does a batsman do if they know that the ball has touched their bat and been caught, but the umpire doesn’t see the contact and gives them “not out”?

Some people say that the batsman should “walk” – that is, give themself “out” and head for the pavilion. Others say that the batsman should take every umpire’s decision as it comes, never “walking”, but also departing without dissent if they have been wrongly given “out”. It is possible to make a consistent and principled argument for either position.

With my version of the laws, all of this argument vanishes. Only one position is now valid: batsmen should never “walk”. A batsman may feel the ball brush the edge of their bat on its way to the wicket-keeper’s gloves, but if the umpire perceives that no contact occurred, it is not a mistake – the batsman is purely and simply not out under the law.


* Batsman or batter?

In recent years the term batter has come into use alongside batsman, in some cases as a conscious effort to use a gender-neutral term. It’s interesting to note that the women’s cricket community doesn’t seem to be particularly enthusiastic about batter (nor indeed batswoman) and there seems to be a long-standing preference for batsman. See, for example, this blog post, which explores the history of the matter a little. Note also that since 2017 the Laws of Cricket have been written in a gender-neutral style using he/her his/her throughout, but nevertheless retain batsman. My understanding is that this has been done in consultation with the women’s cricket community.


“I’m deuterawhat?” – colour vision at Orkney Science Festival

No need to look so sad, Garry. You're special.
No need to look so sad, Garry. You’re special.

You’re deuteranomalous, Garry.

The distressed man on the right is Garry McDougall. Garry’s just found out that his colour vision is not the standard-issue colour vision that most of us have. He made this discovery while watching my talk on the science of colour vision, in Kirkwall as part of the Orkney International Science Festival 2018.

Garry and I were part of a team funded by the Institute of Physics to perform at the festival.  Also on the team were Siân Hickson (IOP Public Engagement Manager for Scotland) and Beth Godfrey.

Garry needn’t look quite so woebegone: he’s not colour blind, and he’s in plentiful company – about 1 in 20 men have colour vision like his.

Normal metameric lights
To Garry, these two lights looked different.

How did Garry’s unusual colour vision come to light? In one of the demos in my talk, I compare two coloured lights. One (at the bottom in the picture on the right) is made only of light from the yellow part of the spectrum. The other (at the top) is made of a mixture of light from the red and green parts of the spectrum. If I adjust the proportions of red and green correctly, the red/green mixture at the top appears identical to the “pure” yellow light at the bottom.

Except that to Garry it didn’t. The mixture (the top light) looked far too red. By turning the red light down, I could get a mixture that matched the “pure” yellow light as far as Garry was concerned. But it no longer matched for the rest of us!  To us, the mixture looked much greener than the “pure” yellow

Garry metameric lights
To Garry, these two lights looked the same.

light; the lower picture on the right shows roughly how big the difference was. This gives us an insight into how different the original pair of lights (that we saw as identical) may have appeared to Garry. It’s not a subtle difference.

We can learn a lot from this experiment.

Firstly, we’re all colour blind. The red/green mixture and the “pure” yellow light are physically very different, but we can’t tell them apart. “Colour normal” people are just one step less colour blind than the people we call colour blind.

Secondly, it shows that there’s no objective reality to colour. People can disagree about how to adjust two lights to look the same colour, and there’s no reason to say who’s right.

Thirdly, it shows that Garry has unusual colour vision. Our colour vision is based on three kinds of light-sensitive cell in our eyes. They’re called cones. The three kinds of cone are sensitive to light from three (overlapping) bands of the spectrum. Comparison of the strengths of the signals from the three cone types is the basis of our ability to tell colours apart. Garry is unusual in that the sensitivity band of one of his three cones is slightly shifted along the spectrum compared to the “normal” version of the cone. This makes him less sensitive to green than the rest of us, which is why the red/green mixture that matches the “pure” yellow to Garry looks distinctly green to nearly everyone else.

Garry isn’t colour blind. He’s colour anomalous. A truly red-green colour blind person has only two types of cone in their eyes. Garry’s kind of colour anomaly is quite common, affecting about 6% of men and 0.4% of women. It’s called deuteranomaly, the deuter- indicating that it’s the second of the three cone types that’s affected, ie the middle one if you think of their sensitivity bands arranged along the spectrum.

My thanks to Siân Hickson for the photographs.

Ben on rocks
Exploring the coast at Rerwick Point.
Showery weather meant that we were treated to many magnificent rainbows, like this one seen at Tankerness.

A note to deuteranomalous readersNormal metameric lights

Please don’t expect the illustrations of the colour matches/mismatches above to work for you as they would have done if you’d seen them live. A computer monitor provides only one way to produce any particular colour, so the lights that appear identical to colour “normal” people (image duplicated on the right) will also appear identical to you, because, in this illustration, they are physically identical.

A machine full of noises

Sarah Kenchington and I made this machine for the Full of Noises festival in Barrow-in-Furness in August 2018.

Sarah designed and made the bicycly bits that raise the table-tennis balls from the pit into the hoppers at the top, and I made the two devices that the balls descend through on their way to the cow bells and glockenspiel.

The complete machine also included other noise-making devices and an exercise-bike powered drive system, both made by Sam Underwood. It was housed in a greenhouse. Here’s a video of the whole thing in action at Full of Noises.

We shot the video in this post in a hurry on a dark damp Tuesday morning before packing the machine up to take it to Barrow, so it comes with apologies for the poor lighting in places.

The peg board (Galton board) that appears from 1:13 to 1:31 is an established classic (see below if you want to make one). The swinging-ramp ball-feeding device (2:09 to 2:18) is a revival of something I designed for the Chain Reactor.

What’s new from me is the arrangement for feeding the balls from the wire chute into the swinging-ramp assembly (1:56 to 2:18). Its operation should be clear from the video, except perhaps for one detail. Because this device may jam if it tries to collect a ball that has not quite arrived at the bottom of the wire chute, and because the timing of the arrival of the balls is erratic, it’s necessary to maintain a queue of balls in the chute to guarantee that there’s always a ball in place at the bottom to be collected. To achieve this, we arranged that the average rate of ball delivery into the chute (determined by the number of spoons on the bicycle chain) was greater than the rate of collection of balls out of the chute, and had an overflow route for the excess balls. Once three balls have accumulated in the chute, any further balls are diverted back into the ball pit (2:30-2:40).

Sarah and I are very grateful to Edinburgh Tool Library for the use of their Portobello workshop, and to Bike for Good and Magic Cycles for donating bicycle parts.

Making the Galton board

Chris Wallace and I discovered while making the Chain Reactor that the horizontal spacing of the pegs on a Galton Board is important. If the spacing is too great, a ball that sets off rightwards will tend to keep going rightwards, and vice versa. To get good randomisation, the ball should rattle between each pair of pegs, and to get this to happen, the gap between the pegs should be only slightly greater than the diameter of the balls. This in turn means that the pegs need to be precisely placed to avoid there being pairs of pegs that don’t let the balls through at all.

In that project we achieved the necessary precision by making the position of each peg (a bolt) adjustable, but with something like 100 bolts, this difficult job was very tedious and sorely tried Chris’s patience.

This time round, I developed a system that let me get every hole in the right place first time. Firstly, I cut the board into four strips so that all parts of it were accessible to a pillar drill. drilling jigThis guaranteed that every hole was accurately perpendicular. Secondly, I made a drilling jig (top right) to get the hole spacing correct. After drilling each hole, I put the peg (the bolt on the right-hand part of the jig) into the just-drilled hole, and the drill for the next hole into the drill hole on the left-hand part of the jig. The spacing between the peg and drill hole is adjustable using the long bolt. ThirdlyPillar drill table, I made a large custom table for the pillar drill (bottom right), with a fence arrangement so that each row of holes was straight.

When I was doing the drilling, the only measurements I had to make were to get the first hole in each row in the right place with respect to the previous row. It took me a few hours to perfect the drilling arrangements, but then only an hour or so to drill 90 holes, all exactly where I wanted them.

peg board

Puff pastry

I was rather taken by the video above, which I first saw on Core77. I started wondering how many times you have to put the roll of silicone material through the machine to get satisfactory mixing of the two colours of material. The people in the video consider the job done after four passes. What does that mean in terms of the thickness of the red and white layers within the material?

The roll is a rather complicated object, so I worked with an idealised version of the real process, where the sheet emerging from the rollers isn’t rolled up, but cut into several pieces which are stacked up before being passed through the rollers again. I came up with the following:

After only 2 passes, the layers in the slab are too thin to see with the naked eye. And by some margin, too: there are over 600 of them and they’re only a fortieth of a millimetre thick. If you made a perpendicular cut through the slab, it wouldn’t appear to have red and white layers in it.

After only 4 passes, a standard compound microscope operating in visible light wouldn’t be able to resolve the layers in the slab.

After only 6 passes, the layers would be thinner than the width of the molecules of the silicone material. At this stage the concept of red and white layers no longer makes sense.

These results will only apply to material near the centre of the roll. It’s easy to see from the video that material near the edges is not mixed so well.

The calculation

From the video, it looks like there are about 9 turns in the roll. Each time the roll is flattened by the rollers, those 9 turns are converted into 18 layers. The resulting sheet is rolled up and passed through the rollers again, multiplying the number of layers by 18, and so on.

This doesn’t work at the sides of the roll. We’ll ignore that complication, and work with a flat analogue of the actual situation. We’ll assume that we start with two long rectangular flat sheets of material, a white one and a red one, laid on top of each other. We’ll cut this assembly into 18 identical pieces, and make a stack of them; this stack will have 36 layers. We now flatten this stack in the rollers, cut it into 18 pieces, stack them up (giving us 648 layers), and repeat.

On emerging from the roller, the sheet appears, by eye, about 1.5 cm thick. We’ll assume that we start with two layers of half this thickness. The table below shows the number of layers and the thickness of each layer after 0, 1, 2, 3… passes through the rollers.

Number of passesNumber of layersLayer thickness (m)
027.50 × 10-3
1364.17 × 10-4
26482.31 × 10-5
311 6641.29 × 10-6
4209 9527.14 × 10-8
53 779 1363.97 × 10-9
668 024 4482.21 × 10-10

We can identify various milestones, as follows:

Limit of visual acuity. A person with clinically normal vision can resolve detail that subtends roughly 1 minute of arc at the eye. At a viewing distance of 30 cm, this corresponds to about 0.1 mm (10-4 m). The layers of material are much thinner than this after only 2 passes. If you made a perpendicular cut through the slab of material, after two passes you wouldn’t be able to see the layered structure. (This might not be true if the cut was oblique.)

Limit of standard light microscopy. A compound microscope working in visible light can resolve detail down to about 200 nm (2 × 10-7 m). The layers become thinner than this after only 4 passes.

Single-molecule layers. The question here is the number of passes needed before the layers are less than a molecule thick (at which point the idea of layers fails). The difficulty is that molecules of silicones are long chains, and these chains are almost certainly bent, so their size is ill-defined. This part of the calculation will be hugely approximate. We’ll be as pessimistic as possible, assuming that the molecules are roughly straight and that they lie parallel to the layers in the slab of material.


A common silicone material is polydimethylsiloxane or PDMS. This consists of a silicon-oxygen backbone with methyl groups attached. The lengths of carbon-silicon and carbon-hydrogen bonds are 1.86 × 10-10 m and 1.09 × 10-10 m respectively. So the width of the molecule is going to be, very, very approximately, of the order of 4 × 10-10 m. The layers are thinner than this after only 6 passes.





Faces in the snow

A hollow impression of my face in the snow.

The hollow face illusion is a wonderful visual effect in which a hollow mask of a face appears to be convex, like the face itself. Making a hollow mould of your face (for example using plaster) is difficult and potentially dangerous. However, last weekend my attention was drawn to an easier and safer way.

I was walking down from Coire an Lochain in the Scottish Highlands with a group from the Red Rope club, when I saw my friend Maia standing on the path ahead, chuckling. She’d been making face imprints in a steep snowdrift, and they showed the hollow face illusion beautifully.

Ben face plantThe procedure needs no explanation (see right). The snow needs to be fresh and soft; you’d be surprised how hard it is to push your face into what feels to your hand like very soft snow. The tip of my nose is noticeably flattened in the picture above.

Ben and Matthew on skyline
Near Coire an Lochain on the day in question. (Readers familiar with Highland place names will realise that I’m not giving much away here.)



The kinetic energy of a drifting tectonic plate…

…is broadly similar to the kinetic energy of me and my bike as I pedal along.

Map of tectonic plates (United States Geological Survey) http://pubs.usgs.gov/publications/text/slabs.html

According the the theory of plate tectonics, the outer layer of the Earth is divided into a number of separate plates, which very slowly drift around, opening and closing oceans, causing earthquakes, and thrusting up mountain ranges.

A moving body has energy by virtue of its motion: kinetic energy. Kinetic energy is proportional to a body’s mass and to the square of its speed.

Now tectonic plates move extremely slowly: the usual comparison is with a growing fingernail. But they are also extremely heavy: tens of millions of square kilometres in area, over 100 km thick, and made of rock. I wondered how the minute speed and colossal mass play out against each other: what’s the kinetic energy of a drifting tectonic plate?

There are so many variables, that vary such a lot, that this calculation is going to be extremely approximate. But the answer is delightfully small: the kinetic energy of the tectonic plate on which I live, as observed from one of the plates next door, is about the same as the kinetic energy of me and my bike when I’m going at a reasonable pace: about 1500 joules.

Me struggling up one of the many steep roads in NW Scotland. Here, the kinetic energy of me and my bike is much less than the kinetic energy of a drifting tectonic plate. In fact the speed of me and my bike is probably less than that of a drifting tectonic plate.
Me struggling up one of the many steep roads in north-west Scotland. Here, the kinetic energy of me and my bike is much less than the kinetic energy of a drifting tectonic plate. In fact the speed of me and my bike is probably much less than that of a drifting tectonic plate ;-).

This is a fun calculation to do, but we shouldn’t get carried away thinking about the kinetic energy of tectonic plates. Plates are driven by huge forces, and their motion is resisted by equally large forces. The mechanical work done by and against these forces will dominate a plate’s energy budget in comparison to its kinetic energy.

But the calculation does provoke an interesting thought about forces and motion. I can get my bike up to full speed in, say, 10 seconds. If the Eurasian plate were as free to move as my bike, and I were to put my shoulder against it and shove as hard as I could, it would take me about 500 years to get it up to its (very tiny) full speed.

In both cases, I’m giving the moving object roughly 1500 joules of kinetic energy. How come I can give that energy to my bike in a few seconds, but to give it to the plate would take me centuries?

I’ll return to that thought in a later post.

The calculation

Depending on how you count them, there are 6-7 major tectonic plates, 10 minor plates, and many more microplates. The plates vary hugely in size, from the giant Pacific Plate with an area of 100 million km2, to the dinky New Hebridean plate, which is a hundred times smaller. The microplates are smaller still. Plates also vary a lot in speed: 10-40 mm is typical.

I’m going to be parochial, and choose the Eurasian plate for this calculation.

Let’s call the area of the plate a and its mean thickness t. Its volume is then given by at, and if its mean density is ρ, then its mass m is ρat.

A body of mass m moving at a speed v has kinetic energy ½mv2. So our plate will have kinetic energy ½ρatv2.

The area of the Eurasian plate is 67,800,000 km2 or 6.78 × 1013 m2, and its speed relative to the African plate is (the only speed I have) is given as 7-14 mm per year. We’ll use 10 mm per year, which is 3.2 × 10-10 ms-1. The thickness of tectonic plates in general varies roughly in the range 100-200 km depending upon whether we are talking about oceanic or continental lithosphere; let’s call it 150 km or 1.5× 105 m. The density of lithospheric material varies in the range 2700-2900 kg m-3; we’ll use 2800 kg m-3.

Putting all of these numbers into our formula for kinetic energy, we get a value of 1500 joules (to 2 significant figures, which the precision of the input data certainly doesn’t warrant).

Now for me and my bike. I weigh about 57 kg, my bike is probably about 10 kg. Suppose I’m riding at 15 mph, which is 6.7 ms-1. My kinetic energy is almost exactly…

…1500 joules!

The closeness of these two values is unmitigated luck*, and we shouldn’t be  seduced by the coincidence. Just varying the speed of the plate in the range 7-14 mm would cause a 4-fold change in kinetic energy, and there’s the variability in plate thickness and rock density to take into account as well. The choice of bike speed was arbitrary, I guessed the mass of the bike, and I’ve since realised that I didn’t account for the fact that the wheels of my bike rotate as well as translate.

However, what we can say is that the kinetic energy of a drifting continent is definitely on a human scale, which leads to a new question:

Suppose the Eurasian plate were as free to move as my bicycle, and that I put my shoulder against it and shoved, how long would it take me to get it up to speed?

From the figures above, the mass of the plate is 2.85 × 1022 kg. If I can push with a force equal to my own weight (about 560 newtons) then by Newton’s 2nd Law I can give it an acceleration of about 1.96 × 10-20 ms-2. Rearranging the equation of motion v = at, where v is the final speed, a is the acceleration, and t is the time, then t = v/a. Inserting the values for v and a, we get t = 1.6 × 1010 seconds, or about 500 years.


* I didn’t tweak my assumptions: what you see above really is the very first version of the calculation!

Edinburgh Mini Maker Faire 2017


On 16th April Sarah McLeary and I had a busy day at the 2017 Mini Maker Faire in Edinburgh (part of the science festival). We showed the paraboloidal castings that we’ve been working on together, and I showed developments in my irregular polyhedra since I showed them at the 2015 Mini Maker Faire.

As well as plaster casts, we’ve tried slipcasting porcelain using our paraboloidal moulds, squirting the slip on as the mould spins to try and get a lacy structure. You can see one of these near the front left of the table. I’ve also been having a go at making irregular puckered plane-based tilings (standing up at the back of the table). I will write more about both of these projects before too long.