This image is my version of Edward Adelson’s checkershadow illusion (with a little inspiration from Magritte). It’s a photograph of a real, physical scene.

Take a look at the central square of the checkerboard, and the square indicated by the arrow. Which is lighter? Quite clearly, it’s the central square, isn’t it?

Remarkably, the central square actually emits less light than the square indicated by the arrow! You could use a light meter to check this claim, but it’s easier to verify it directly by using a piece of card with two holes cut in it to mask off the rest of the image.

Some people will tell you that this image shows you how easy it is to fool your brain. But it does the exact opposite: it shows you what a marvellous piece of equipment your brain is.

Think about the checkerboard itself, and the materials it’s made of. The arrowed square is coated with dark grey paint, and the central square is coated with light grey paint—and that’s exactly what you perceive.

The shadow cast by the pipe means that the light-grey central square is more dimly lit than the dark-grey arrowed square, so much so that it actually reflects less light into your eye than the arrowed square. But your brain cleverly manages to determine the actual lightnesses of the physical surfaces, despite the uneven lighting. Isn’t that a good thing for your brain to do?

If you still don’t believe me, try this thought experiment. Imagine that you live in a forest where there are two kinds of fruit. One is light grey and poisonous, and the other is dark grey and nutritious. Two of these fruits hang next to each other, but in the dappled forest light the (light grey) poisonous fruit is in shadow, and the (dark grey) nutritious fruit is in bright light. Suppose that the depth of the shadow is such that the light-grey poisonous fruit actually reflects slightly less light into your eye than the dark-grey nutritious fruit, just as with the two squares in the picture above. Would you really want your vision to tell you that the poisonous fruit was the dark one and therefore the one to pick? Or would you want it to discount the irrelevant effect of the shadow and tell you which fruit was actually dark and which was actually light (and would kill you)? I know what I’d want.

I think that it is wrong to call this effect an illusion (and so does Adelson). There is nothing illusory about what you see. You perceive the useful truth about the scene in front of you.

Be careful about how much water you put in your kettle and ‘do your bit’ to save energy!

That’s what we’re often told, and boiling only as much water as you need for your cup of tea can only be a good thing. But how much impact does it really have on your overall energy consumption?

Consider this: if you’re driving along in a car at 50 mph, you’re using enough energy to make a cup of tea every few seconds.

1, 2, 3, tea, 1, 2, 3, tea, 1, 2, 3, tea…

As most people don’t think twice about driving their cars for an extra few minutes, never mind an extra few seconds, it’s clear that the energy savings made by being frugal with your kettle are very small compared to your overall energy use.

Does this mean that we shouldn’t be careful with our kettles after all? No. It all matters. But having boiled exactly one cup’s worth of water for your refreshing cuppa, you shouldn’t put your feet up and think that your energy consumption issues are sorted. You’ve much bigger fish to fry.

The calculation

Imagine that you’re travelling in a car that’s doing 50 miles per hour, with a rate of fuel consumption of 50 miles to the gallon (the units are customary in a UK motoring context). This is a realistic situation; if anything it’s optimistic in terms of energy consumption.

In this scenario, your car will burn a gallon of petrol, that’s 4.5 litres approximately, every hour.

With 3600 seconds in an hour, that’s 4.5L/3600s = 0.00125 litres per second. The density of petrol is about 0.8 kg per litre, so the rate of petrol consumption comes to 0.00125 L s^{-1} × 0.8 kg L^{-1} = 0.001 kilograms per second.

The energy density of hydrocarbon fuels is about 46 megajoules per kilogram, so the rate of use of energy by your car is 0.001 kg s^{-1} × 46×10^{6} J kg^{-1} = 46000 joules per second (watts).

Now let’s think about the tea. The energy E required to produce a temperature change of ΔT in a mass m of a substance whose specific heat capacity is C is

E = m C ΔT

A typical mug of tea contains about 250 ml, or 0.25 kg, of water. For water, C = 4200 J kg^{-1} K^{-1}. The temperature change is about 90 K, if the water comes out of the tap at about 10°C and is raised to boiling point. So the amount of energy required to heat the water for a cup of tea is 0.25 kg × 4200 J kg^{-1} K^{-1}× 90 K = 94500 joules.

As the car is using 46000 joules per second, it follows that the car is using enough energy to heat the water for a cup of tea every two seconds.

Now to be fair to the car, we need to recognise that in real life there is more energy used to heat the water for the tea than actually goes into the water. There are heat losses in energy generation and transmission, and from the kettle itself.

Things get very complicated here. A thermal electricity generating plant (coal, oil, gas, or nuclear) converts energy in the fuel to electrical energy with about 35-45% efficiency. But in the UK, about 20% of our electricity is generated from renewables, where the concept of efficiency is harder to pin down. For example, I could use some oil to heat my water directly instead of doing it indirectly by generating electricity, but I couldn’t do the same with the wind. I’m going to assume that 80% of the electricity that I used was generated in thermal plants at 40% efficiency, and 20% of it was generated at 100% efficiency, giving an overall efficiency of 52%. But be aware that this is a very rough figure, and that there is no right answer.

According to this document, about 93% of the energy in generated electricity makes it unscathed through the transmission and distribution systems to the end user.

I did an experiment and estimated the efficiency of my kettle to be about 88%. I give details later.

This gives an overall efficiency of boiling water as 52% × 93% × 88% = 43%. So instead of using 94500 joules to boil water for a cup of tea, we’re actually using 94500J/0.43 = 220000 joules, which means that the car’s rate of energy use is equivalent to a cup of tea roughly every 5 seconds. Because of all the uncertainties involved (not least the size of a cup of tea), we should treat this figure as very approximate.

Efficiency of my kettle

My kettle took 180 s to raise 1 litre of water from 17°C to boiling.

The kettle is rated at 1850-2200 W for supply voltages in the range 220-240 V. When the kettle was running, I measured the supply voltage to be 242 V, so I assumed that the kettle was operating at the top of its power range, ie 2200 W.

The heat supplied by the kettle element was therefore 2200 W × 180 s = 396000 J.

Using the same equation as earlier, the energy required to raise the temperature of 1 kg of water by 83 K is 1 kg × 4200 J kg^{-1} K^{-1} × 83 K = 348600 J.

The kettle was therefore heating the water with efficiency 348600/396000 = 0.88 or 88%.

I made the machine in the video below for no other reason than that I felt like making a vehicle that was propelled in a non-standard way. The idea arose when I wondered about setting my Product Design Engineering students the challenge of making a vehicle that wasn’t driven through its wheels. In the end we did a different project, but I still couldn’t resist having a go myself.

The vehicle is mainly made of corrugated cardboard with other odd bits and pieces. It’s driven by two servos controlled by an Arduino microcontroller board.

Originally there was going to be a separate weight that was shifted back and forth to make the vehicle tip. I was rather pleased when I realised that I could make the batteries, the Arduino, and one of the motors do double duty as the weight.

I’ve just spent the last day and a half making another irregular polyhedron to use as a lampshade in the room where I work. As I worked, I was reminded how useful it is to have a table with motorised height adjustment. The thing that I was making started off flat, and finished 40 cm in diameter, so a table that was the right height at the beginning would be far too high towards the end of the construction.

But it’s not just about workpieces that grow. Different assembly operations are best done at different heights: cutting and folding need a lower table than glueing, for example, and if I need to take a close look at something, it’s useful to be able to raise it as close as possible to eye level. (I do nearly all making tasks standing up, by the way.)

There are many different sit-stand desks out there. Mine is an Ikea BEKANT electric sit-stand desk. The height adjusts electrically from 65cm to 120 cm, and it takes about 20 seconds to cover the full range. It seems to be reasonably solidly built, though I wouldn’t try to do woodwork on it.

It seemed a bit of an extravagance when I bought it, but I love it. By enabling me to keep a better posture, it’s much more comfortable to work at than a fixed-height desk, and the ability to move the workpiece to the best height for any given operation materially improves the quality of the things I make.

How could it be improved? Foot switches (or even better, speech control!) would be handy for those stressful times when you need to change the height of the desk quickly, but have both hands occupied holding something together. And being able to tilt the tabletop would be wonderful – a project for the future, maybe…

The lilac chaser is a remarkable visual phenomenon that is normally seen as a computer animation. Dr Rob Jenkins of York University wanted to show people that the effect works with real, honest-to-goodness, physical lights, so he asked me to make him the equipment to do this. The video below shows you how the apparatus works. Note that the limitations of my camera mean that the effect is not as strong in the video as it is in real life.

One useful technique that I developed here was a way of producing an an even spot of light from an LED. Diffused LEDs give an even spread of light but send light in all directions, which is wasteful if you want only a small bright spot. Clear LEDs are available which direct the light in quite a narrow beam, but the distribution of light is very uneven. I found that shining the light from a clear LED down a short white tube, about 10 mm internal diameter and 60 mm long, did a very good job of producing a sharp-edged even spot on a piece of tracing paper placed at the end of the tube. I assume that the many reflections inside the tube thoroughly mix up the light. I found the tubing in the plumbing section of a hardware shop, and lightly roughened the inside of it using fine sandpaper.

To get a spot with a blurred edge, I placed a second tracing-paper screen a short distance away from the end of the tube. By varying the distance of this screen I was able to vary how blurred the patch of light on it was.

Last week I was on holiday in Wales. It wasn’t the driest of weeks, and while I was inside not climbing mountains, I finally got round to doing some mountain-related geometry that I’ve been putting off for the last 30 years or so. It’s about knowing how high you are.

If you’re climbing or descending a mountain, you sometimes want to know roughly how far (vertically) there is still to go. One way to get an idea of your altitude is to use nearby peaks or other points of known altitude as reference points. But how do you judge whether you are above or below another point? It’s not always obvious, and without some sort of rule, the worry is that you’ll make optimistic judgements, leading to disappointment in the long run.

My friend Malcolm once told me that, as a rule of thumb, you should look at the reference point relative to the distant skyline. If the point appears to be above the skyline, you are lower than it, and if it appears to be below the skyline, you are higher than it. I’ve used this skyline-rule ever since, but I’ve never checked how accurate it is.

The fact that there’s any doubt about the rule is because the Earth is not flat. If it was flat, then your line of sight to the (infinitely distant) sea-level horizon would be exactly horizontal, and the rule would work perfectly. If the skyline was made up of mountains, the rule would work perfectly as long as they were as high as your reference point.

But the Earth isn’t flat: it’s a big ball. How does this affect the accuracy of the rule? I used a wet Welsh Wednesday afternoon to find out.

It turns out that the rule is good enough for general hillwalking purposes as long as the reference point is no more than two or three kilometres away (as it usually will be). The errors are smaller if the skyline is distant mountains rather than the horizon at sea level. The rule consistently underestimates your altitude, which, in ascent at least, is probably better than the alternative. Continue reading How high am I ?

How fast does a plucked guitar string move? It’s a complete blur, so surely it’s travelling at a terrific speed. 50 miles per hour? 100 miles per hour? What do you think?

Walking pace. A typical speed for the middle of a guitar string given a good twang is walking pace. And that’s the middle of the string. Near its ends, it’ll be moving much more slowly.

How can that be so? Well, although the string is going back and forth hundreds of times a second, it’s only travelling a few millimetres on each trip. So the distance that it travels in each second isn’t as much as you might expect. It certainly isn’t as far as I expected.

If you think that the string moves slowly, what about the body of the guitar? The string itself radiates very little sound into the air; its job is to set the body of the guitar vibrating. The body of the guitar, with its much larger area, is much more effective than the string at setting air into motion. Yet we can’t even see the body vibrating. At what snail’s pace must it be moving?

Remember also that the air molecules on which the guitar body acts are already travelling at something like 500 metres per second. Isn’t it astonishing that the sub-pedestrian movements of the guitar affect the movement of the air molecules enough to produce a sound that we can easily hear?

The calculation

Suppose that we look near the centre of the string, where its movement is the greatest. Shortly after being plucked, the width of the blur that we see is going to be something like 5 mm. So for every complete oscillation, the string does a round trip of about 10 mm.

The frequencies of the strings on a standard 6-string acoustic guitar are (to the nearest whole number) 82, 110, 147, 196, 247 and 330 hertz (one hertz is one oscillation per second). If we multiply these frequencies by the 10 mm round trip, it tells us how far the centre of each string travels in one second, that is, its average speed. I’ve converted these speeds into metres per second. For comparison, a brisk walk at 4 mph is about 1.8 ms^{-1}.

Note name

Mean speed of middle of string in metres per second (to 2 sig. fig.)

E

0.82

A

1.1

D

1.5

G

2.0

B

2.5

E

3.3

Our brisk walk is right in the middle of this range. And remember, we’ve done the calculation for the part of each string that’s moving the most. Near its ends, each string will be moving much more slowly than this.

Complication 1 – how long is the round trip really?

So far, we’ve assumed that each part of the string does a simple back-and-forth movement along a straight line, but if you carefully watch a vibrating guitar string you’ll see that the string often moves in an irregular but roughly elliptical orbit. The wire-wound lower strings show this most clearly; you can see a hint of it in the image at the top. This makes the round-trip distance a bit longer than the 10 mm that we used in the calculation earlier. Does this affect the string’s average speed much?

We’ll take the extreme case where each part of the string moves at constant speed in a circle of diameter 5 mm rather than along a straight line 5 mm long. The circumference of this circle will be mm, or about 15 mm. So the speeds of the strings (in this rather unlikely extreme case) will be about 50% greater than the ones listed above. They are still hardly impressive.

Complication 2 – the peak speed

So far, we’ve calculated the mean (average) speed of the string over its round trip. However, unless it’s moving in a perfect circle, its speed changes constantly, and its peak speed will be higher than its mean speed. How much higher?

Imagine that part of the string is vibrating back and forth along a straight line in the simplest possible way. At one end of the movement the string is stationary as it changes direction. It then speeds up, reaching its peak speed at the centre of its range of movement. Then it slows down until it reaches a halt again at the other end of the movement and changes direction again. How do we calculate the peak speed if we know the time taken for the round trip?

You’ll need to know a bit of maths for the next bit. The simplest vibration of the string is where each part undergoes simple harmonic motion, that is, where its position varies sinusoidally with time. This means in turn that the velocity of the string also varies sinusoidally in time. So we need to ask: how does the mean value of a sinusoid compare to its peak value?

Consider the function , for half a cycle, that is, for theta from to radians. We construct a rectangle between these limits, such the rectangle’s area equals the area under the sine curve. The mean value of the sine function is the height of the rectangle, which its area divided by its width (which is ). So we need to work out the area under the sine curve between and , which we can do by integration:

We constructed the rectangle to have area . Its width is , so its height, and therefore the mean value of the sine function, is . The height to the peak of the sine curve is 1, so the peak value of the sine function is times its mean value. This means that the peak speed of our guitar string is times, or 50% more than, its average speed. Again, nothing to write home about.

(Rather pleasingly, this ratio of is the same ratio that we got earlier in Complication 1. It means that the peak speed of a particle doing simple harmonic motion with a given amplitude and period is exactly the same as the (constant) speed of a particle moving in a circular orbit of radius with period . This isn’t a coincidence. It arises because the circular motion can be considered as two linear simple harmonic motions at right angles to each other.)

It’s sometimes said that if you sit an immortal monkey in front of an equally durable typewriter and leave it to tap randomly away at the keys, then eventually it will produce the entire text of Richard III (or any other Shakespeare play of your choice), completely by chance. All you have to do is wait long enough.

I was thinking about this one day, and also thinking about air molecules. In the room I’m sitting in at the moment, there are at least 1,000,000,000,000,000,000,000,000,000 air molecules, all frantically dashing around bumping into each other. How often, I wondered, do little clusters of these molecules fleetingly arrange themselves, by chance, in arrangements that we would regard as being somehow regular or remarkable?

It’s not possible for me to directly observe air molecules, so instead I used my computer to make a 2-dimensional simulation of some atoms of gas doing what atoms of gas do. I set it going, and waited…and waited…and waited…

I promise you that my simulation didn’t involve any secret forces drawing atoms towards certain positions. The movements and collisions of the atoms all occurred in accordance with the laws of mechanics. But I did cheat a little. Can you figure out how?

Molecules and atoms

In case you’re fidgeting on your chair wondering why I started talking about air molecules but finished talking about gas atoms, let me explain.

Nearly all of the air is nitrogen and oxygen. Nitrogen and oxygen atoms are essentially spherical. But in the air, nitrogen atoms are bonded together in pairs to form nitrogen molecules that are, roughly speaking, a stubby rod shape. The same goes for oxygen. Now a collision between two moving rods is much more complicated than a collision between two spheres, because the rods can spin end-over-end in a way that spheres can’t. In fact, I’m not sure that I know how to do the calculations. As the point of the video could be made just as well using atoms rather than molecules, I did the simulation using atoms. If you like, you can think of them as atoms of helium or argon, which do go around on their own.

In a previous post I showed some examples of irregular polyhedra that I’ve been making out of paper. These polyhedra were all based on points distributed quasi-randomly over a sphere. At each point, my program placed a plane tangent to the sphere. The result of the intersection of all of those planes was an irregular polyhedron.

The 47-hedron in the picture above was made in a rather different way. It’s certainly irregular, but the process that created it was not at all random. As in the previous polyhedra, the starting point was a sphere with a number of points distributed over it. But this time, the points were placed according to a simple and perfectly regular rule.

Imagine you you draw a line from one pole of a sphere to the opposite pole, winding sort-of-helically around the sphere. Then place a number of points at exactly equal spacings along this line. In the examples below, there are 43 points.

If your quasi-helix had 2.5 turns, the result would look like the diagram on the right. The numbers are where the points are placed on the imaginary sphere; the resulting polyhedron is also drawn. The helical structure is very clear, but notice the irregularity of the faces: for example, the lower edges of faces 26, 27, and 28 are all different. This is because the radius of the quasi-helix is not constant, so turns of the helix near the poles will have fewer faces on them than turns near the equator, which means that the way the numbers on one turn line up with the numbers on the previous and subsequent turns keeps changing.

If we increase the number of turns in the quasi-helix to 5, the helical structure is still clear:With 7.5 turns in the quasi-helix (below), adjacent-numbered faces are now so far apart that the faces on the turns above and below them are starting to intrude into the spaces between them. See, for example, how faces 27 and 28 are almost pushed apart by faces 19 and 35.By the time we have 10 turns (below), consecutively-numbered faces are so far apart that faces from the turns above and below often meet between them, making the helical structure hard to discern (see, for example, how faces 34 and 22 squeeze in between faces 28 and 29). The polyhedron is beginning to look quite irregular, but note that consecutively-numbered faces are still more similar to each other than they are to the other faces.

The 47-hedron on the right (same as at the top of the post) was based on a quasi-helix with about 12.5 turns. At first sight, it looks quite random, and it’s certainly irregular, but there are still visible similarities between faces if you look hard enough. Compare, for example, the mid-blue and light-blue faces at roughly 10 o’clock and 2 o’clock respectively. And what about the dark-blue and very dark-blue faces just below the centre? One looks suspiciously like a rotated version of the other – and it is! In fact the whole polyhedron has an axis of 180° rotational symmetry. This suprised me when I spotted it, but it shouldn’t have. A helix has 180° rotational symmetry (turn a corkscrew upside down and it looks the same) so it’s inevitable that shapes based on a helix will have that symmetry also.

This post is all about answering the question: “How deep is the atmosphere?”. The question doesn’t actually have a simple answer, because there is no altitude where the atmosphere suddenly stops and space starts. Instead, the air gets progressively thinner and thinner as you get higher, gradually giving way to the vacuum of space. What can be startling is how quickly the air gets thinner as you travel upwards. On this page I’ll be finding some ways of getting to grips with the scale of the atmosphere.

At the end, I hope that you might agree with me that the atmosphere is really awfully shallow, and that we definitely ought to be looking after it much more carefully than we do at the moment.

What do we mean by “the air gets thinner”?
A cubic metre of air at sea level contains about 1.2 kilograms of air. Higher in the atmosphere, a cubic metre would contain less air. For example, at 9,000 metres (just above the summit of Mount Everest) a cubic metre of air would contain only about 0.47 kilograms of air, less than half as much as at sea level. It is this decrease of mass per unit volume (density) that I mean when I write of the air getting “thinner”.

How high is the top of the atmosphere?
To get a feel for the thickness of the atmosphere, we will look at a number of different definitions of the top of the atmosphere. As well as these, we’ll also look at some altitudes with life-and-death significance. I’ve put any calculations at the bottom of the post.

100 km – the Kármán line
The Kármán line is a common definition of the boundary of space. This beautiful and clever idea rests on two points.

The first point: to be in orbit around the Earth (assuming there’s no atmosphere) there’s a certain speed at which you need to travel. If you travel more slowly than this, you’ll fall out of orbit. At the altitudes we’re talking about, the critical speed is nearly 8 kilometres per second.

The second point: an aeroplane stays up in the air because its wings generate lift as they pass through the air. The thinner the air, the faster the aeroplane must fly in order to generate enough lift to stay airborne. So the higher an aeroplane goes, the faster it must fly to stay up there.

Here’s the clever bit: there comes an altitude where the air is so thin that the aeroplane must travel at about 8 kilometres per second for its wings to generate enough lift to stay up. But at this speed, it’s going fast enough to stay in orbit anyway, even if there were no air and it had no wings.

This altitude, which is about 100 km, is the Kármán line. You could say that it’s where an aeronaut becomes an astronaut.

The picture on the right is a scale diagram showing the Earth (grey) and the atmosphere (blue, thickness defined by the Kármán line). 99.99997% of the atmosphere lies in the blue region.

31 km – the 99% line 99% of the atmosphere is below 31 km above the surface of the Earth. The picture on the right shows roughly what part of central London would look like seen from 31 km. It doesn’t look too far away, but from this altitude, you are looking through nearly the entire atmosphere. (Please note that this picture represents roughly how big things would look from 31 km. It doesn’t reflect the optical degradation that viewing through 31 km of air would produce. I scaled the picture from one taken at an unknown altitude.)

19 km – the Armstrong line At this altitude the air pressure is lower than the vapour pressure of the water in your body. Uncontained body fluids (such as saliva) would start boiling at this altitude.

8 km – top of the constant-density atmosphere
Another way to think about the depth of the atmosphere is to ask “How much air are we looking through when we look upwards through the atmosphere?”. To answer this question, imagine that the air in the Earth’s atmosphere is all at sea-level density, instead of getting thinner and thinner with altitude. With the same total amount of air as in the traditional atmosphere, this imaginary atmosphere will come to an abrupt end at a certain altitude. How deep would this constant-density atmosphere be?

The answer is: a little more than 8 kilometres. Mount Everest would just poke out of the top of it. There are details of the calculation later on.

Looked at in this way, the atmosphere is startlingly shallow. You can commonly look horizontally from one place to another place 9 or more kilometres away. When you are doing this, there is more air between you and the not-very-distant object than there is between you and a star overhead.

5.5 km – the habitation line
It appears that no amount of acclimatisation will enable you to survive indefinitely above an altitude of around 6 km. Lambert (1971) reports that in 1961 a team that spent six months at 5,800 m was less fit at the end of this time than newly-arrived people. He also cites an Andean mine at 5,800 m, where the miners chose to walk up daily from 5,300m rather than live at the higher altitude. The current highest permanent human settlement is La Rinconada, at 5,100 m in the Peruvian Andes.

The photograph on the right shows roughly what Buckingham Palace would look like from 6,000 m altitude. It doesn’t look very far away, but at this altitude you wouldn’t last very long. About 50% of the atmosphere is below the habitation limit.

The habitation line on the map The photograph on the right shows an ordinary 1:50,000 Ordnance Survey map, familiar to UK hillwalkers. The grid squares visible in the sea are 1 km across. It shows the town of Aberdeen on the Scottish coast. On this scale, the tip of my thumb is at about 6,500 m, comfortably in the region where the air is too thin to support human life for very long. If you were on the north side of Aberdeen, you’d be closer to the uninhabitable zone than you would be to the south side of Aberdeen. Seen this way, the atmosphere seems very shallow. 55% of the atmosphere is below the level of the tip of my thumb.

About 4 km – the oxygen-mask line
Above this altitude, in an unpressurised cabin, an aeroplane pilot is required to use an oxygen supply. Having said that, thousands of people (myself included) climb 4,000-metre peaks in the Alps and nobody uses oxygen.

12 metres – top of the liquid atmosphere
Finally, suppose we condensed the entire atmosphere to its liquid form. How deep would the resulting “ocean” be? The answer: just under 12 metres.

So what?
The alarming thing about the altitudes I’ve listed above is how small they are. Compared to the distances that we regularly travel horizontally across the Earth, these distances are tiny. If you could walk vertically upwards, you’d need an oxygen supply after only an hour. Three hours walking and you’d need a pressure suit. Space itself is only a good day’s bike ride away. If you’re in Birmingham, you’re comfortably closer to space than you are to London.

The tall picture
The tall thin picture on the left at the top of the post is a representation of the way the atmosphere gets thinner with altitude. The density of blue dots is proportional to air density at each altitude, starting with solid blue at sea level. The tick marks on the right-hand side are at 10 km intervals. Note that there is air above 60 km, even though there are no dots. It’s just that the air there is extremely thin, and the dots are so widely spaced that you’d need a much wider picture to have a chance of spotting one.

The picture lets you see at a glance roughly how much of the atmosphere is below any given altitude. The aeroplane silhouette is at a typical cruising altitude for airliners – see how much of the atmosphere is below you when you fly.

You could walk across the bottom of the picture in less than two hours.

The wide picture
The picture below is drawn on the same principle, but to a different scale. It shows Edinburgh (E), London (L), and the atmosphere. The broken line is the boundary of space as defined by the Kármán line.

Afterthought
You are in the vastness of space. In all directions in front of you, almost empty star-studded space stretches out for unimaginable distances, giving an near- overwhelming feeling of exposure. Behind you is an apparently limitless hard surface. A strange force presses your back firmly against this surface, almost as if it were magnetic. A transparent layer of air, only a few miles thick, between you and the void gives you something to breathe and protects you from the cold of space.

But as on most clear nights, it’s a bit chilly for lying on your back on the ground, so after a while you stand up and walk home to warm your toes in front of the fire.

The 99% line
The pressure at any level in the atmosphere must be exactly that required to support the overlying layers of the atmosphere. The pressure at sea level is enough to support the entire weight of the atmosphere. If the pressure at a given altitude is, say, half sea-level pressure, then we know that half of the mass of the atmosphere must be above this level. Therefore with a table of atmospheric pressures we can quickly work out what fraction of the atmosphere is above any given altitude.

The Armstrong line…
is not named after Neil.

The constant-density atmosphere
Imagine a column of air, of cross-sectional area 1 square metre, that extends through the full height of the atmosphere. The air pressure at the bottom of this column, at sea level, is very close to 10^{5} newtons per metre squared. This means that all of the air in the column of the atmosphere has a weight W close to 10^{5} newtons. The acceleration due to gravity is, for our purposes, constant throughout the height of the column – let’s use g = 9.8 ms^{-2}. The mass of air in the column is given by W divided by g, which comes to 10200 kg. The density of air at sea level is about 1.22 kg m^{-3}, and therefore the volume of air in the column is 10200/1.22 = 8360 cubic metres. As the column has a cross-sectional area of 1 square metre, this means its height is 8360 metres.

The liquid atmosphere
When working out the depth of the constant-density atmosphere, we established that the mass of a column of air of cross-sectional area 1 m^{2}, extending the entire height of the atmosphere, is about 10200 kg. The density of liquid air is about 870 kg m^{-3}, and so if we liquefied this amount of air it would have a volume of 11.7 m^{3}. Hence if we liquefied the atmosphere, the resulting ocean would be 11.7 m deep.

Acknowledgement
This is a minor reworking of a page from my old website. I’m republishing it after receiving an email from Sarah Bush from the Division of Biological Sciences at the University of Missouri, who used the old page as teaching material.

References
Lambert, D. (1971) Medical appendix in Bonington, C. (1971) Annapurna South Face. Cassell.

West, J.B. (2002) Highest Permanent Human Habitation. High Altitude Medicine & Biology,3, 401-407.