## Pints

Although we live in a 3D world, we aren’t always very good at judging the volumes of things. A few years ago I had the idea of exploring our (mis)judgements of volume by making a collection of differently-shaped objects, all of which had a volume of a pint. I didn’t do anything about it at the time, but when I discovered recently that Pint of Science in Glasgow was holding Creative Reactions, an art exhibition, I decided to take the hint and get to work.

## Cuboids

These cuboids all have a volume of a pint.

## Optimal shapes

These shapes not only have a volume of a pint, but they are all optimal in terms of surface area:

Of all the cuboids with a volume of a pint, a cube has the smallest surface area.
Of all the cylinders with a volume of a pint, a cylinder whose height and diameter are equal has the smallest surface area.
Of all the cones with a volume of a pint, a cone whose height is the square root of 2 times its base diameter has the smallest surface area.

And of all solid shapes with a volume of a pint, a sphere has the smallest surface area.

I made the cylinder, sphere and cone on a lathe, and the sphere on a bandsaw.

## A quiet pint

Here we have the lowest (most distal) pint of my arm and hand about to pour a pint of beer into the front pint of my face. All the beer-glass shapes are casts of the interiors of pint glasses. I was slightly disappointed by the beer-glass casts; I was hoping that they might seem strikingly small compared to the actual filled beer glasses, but they don’t.

Casting was a new venture for me. My thanks to Amy Grogan and Alys Owen of the casting workshop at Glasgow School of Art for their help and advice.

My thanks also to Laura McCaughey, Marta Campillo Poveda, and Danielle Leibnitz, who organised the exhibition.

A pint of my face.

## Red, green, and blue make…black!

In the previous post I looked at how coloured shadows are formed. As I wrote it, I realised how much there is to learn from the coloured shadows demonstration; that’s what this post is about. The image above shows coloured shadows cast by a white paper disc with a grey surround.

In the image at the top, we’re mixing shadows. If we were mixing lights in the normal way, it would look like the picture on the right.

So what do we learn from the coloured shadows image?

#### Red, green and blue add to make white

The white paper disc has all three lights shining on it, and it appears white. Mixing lights shows us this too.

#### The three lights still exist independently when they are mixed

Some descriptions of colour light mixing could leave you with the impression that when we mix red, green, and blue lights together to make white, they combine to make a new kind of light, a bit like the way butter, eggs, flour and sugar combine to create something completely different: a cake.

But if that were so, we’d only get a black shadow. The fact that we get three coloured shadows show that the three coloured lights maintain their independence even though they’re passing through the same region of space. It’s very like ripples on a pond: if you throw two pebbles into a pond, the two sets of ripples spread through the same region of water, each one travelling through the water as if the other pebble’s ripples weren’t there.

#### Coloured shadows obey subtractive colour mixing rules

When we mix coloured lights, additive colour mixing rules apply:

• Red and blue make magenta.
• Red and green make (surprisingly) yellow.
• Blue and green make cyan.
• All three colours add to make white.

In the coloured shadows image, it looks at first glance as if we are adding together coloured lights. But if we were, we’d expect the centre of the pattern, where all the lights overlap, to be white, as it is in the light-mixing image. Instead, the centre of the coloured shadows pattern is black.

The reason is that we aren’t adding coloured lights, we’re adding coloured shadows, and now subtractive colour mixing rules – the rules of mixing paints – apply.

In the cyan shadow, red has been blocked by the disc, leaving green and blue. In the yellow shadow, blue has been blocked by the disc, leaving red and green. Where the cyan and yellow shadows overlap, the only colour that has not been blocked by one disc or the other is green, so that’s the colour we see. We get the same result when we mix blue and yellow paints: the only colour that both paints reflect well is green. (If the blue and yellow paints reflected only blue and only yellow respectively, the mixture would appear black.)

In the black centre of the pattern, all three lights are blocked by the disc. Something similar happens when you mix every colour in your paint box together.

Colour printing is based on these subtractive colour mixing rules.

#### Brightness matters, and blue isn’t very bright

In either of the light- or shadow-mixing images, the boundaries between the regions aren’t all equally distinct. The least distinct ones are:

• magenta and red
• green and cyan
• blue and black
• yellow and white

In each case, the difference in the colours is the presence or absence of blue light.

There are two things at work here. Firstly, more than we might think, our vision is based on brightness, not on colour. We happily watch black-and-white movies; after a while we hardly notice the absence of colour.

Secondly, our sensation of brightness is largely due to the red-yellow-green end of the spectrum – blue makes a very small contribution, if any. So although the presence or absence of blue light can have a strong effect on colour, it has a weak effect on brightness. So boundaries defined by the presence or absence of blue light tend to be relatively indistinct compared to those defined by the presence or absence of red or green light.

This is a photograph that I took as a response to a challenge that was set by photographer Kim Ayres as part of his weekly podcast Understanding Photography. The challenge was to produce a photo where the main interest was provided by shadows. I lit a rose cutting using red, green, and blue lights that were about 3 metres away and 30 centimetres apart from each other. The result is a gorgeous display of coloured shadows. Coloured shadows are nothing new, but they are always lovely.

Kim suggested that I do a blog post to explain more about how coloured shadows arise. To do this I set up an arrangement for creating simple coloured shadows. One part of the arrangement is three lights: red, green, and blue arranged in a triangle.

The lights shine upon a white screen set up about 3 metres away. In front of the screen, a wire rod supports a small black disc.

First of all, let’s turn on the red light only. The screen appears red, and we can see the shadow of the disc on it. The shadow occupies the parts of the screen that the red light can’t reach because the disc is in the way.

Next, we’ll turn on the green light only. Now the screen appears green, and for the same reasons as before, there’s a shadow on it. The shadow is further to the left than it was with the red light; this is because the green light is to the right of the red light as you face the screen.

Next, we’ll turn on the blue light only, with the expected result. The blue light is lower than the red and green ones, so the shadow appears higher on the screen. (The shadow is less sharp than the previous two. This is because my blue light happens to be larger than the red or green lights).

Now we’re going to turn on both the red and green lights. Perhaps unsurprisingly, we see two shadows. They are in the same places as the shadows we got with the red and green lights on their own. But now they are coloured. The shadow cast by the red light is green. This is because, although the disc blocks red light from this part of the screen, it doesn’t block green light, so the green light fills in the red light’s shadow. Similarly, the shadow cast by the green light is red.

The screen itself appears yellow. This is because, by the rules of mixing coloured lights (which aren’t the same as the rules for mixing coloured paints), red light added to green light gives yellow light.

We can do the same with the other possible pairs of lights: red & blue, and green & blue. (The green shadow looks yellowish here. It does in real life too. I think this is because it’s being seen against the bluish background.)

We’re now going to turn on all three lights. As you might expect, we get three shadows. The colours of the shadows are more complicated now. The shadow cast by the red light is filled in with light from both of the other lights – green and blue – so it has the greeny-blue colour traditionally referred to as cyan. The shadow cast by the green light is filled in with light from the red and blue lights, so it is the colour traditionally called magenta. And the shadow cast by the blue light is filled in with light from the red and green lights, and thus appears yellow.

The rest of the screen, which is illuminated by all three lights, is white, because the laws for mixing coloured lights tell us that red + green + blue = white. The white is uneven because my lights had rather narrow and uneven beams.

Finally, let’s add further richness by using a larger disc, so that the shadows of the three lights overlap. Now we get shadows in seven colours, as follows.

Where the disc blocks one light and allows two lights to illuminate the screen, we see the colours of the three pairwise mixtures of the lights: yellow  (red+green), magenta (red+blue), and cyan (green+blue).

Where the disc blocks two lights and allows only one light to illuminate the screen, we see the colours of the three individual lights: red, green, or blue.

And in the middle, there’s a region where the disc blocks the light from all three lights, so here we get a good old-fashioned black shadow.

If it’s a bit hard to wrap your head around this, let’s trying looking at things from the screen’s point of view. Here I’ve replaced the screen with a thin piece of paper so that the shadows are visible from both sides. I’ve made holes in the screen in the middle of each of the coloured regions, so that we can look back through the screen towards the lights.

Here’s what you see when we look back through the magenta shadow. We can see the red light and the blue light, but not the green one – it’s hidden behind the disc.

This is the view looking back through the green shadow. We can see only the green light. The red and blue lights are hidden behind the disc.

And so on…

## 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…

…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.

## Making the micro macro

On the right is the processor package from my old laptop. The numbers associated with microelectronic devices like this one are beyond comprehension. The actual processor – the grey rectangle in the middle – measures only 11 mm by 13 mm and yet, according to the manufacturer, it contains 291 million transistors. That’s about 2 million transistors per square millimetre.

To try to bring these numbers within my comprehension, I asked the following question:

If I were to magnify the processor – the grey rectangle – so that I could just make out the features on its active surface with my unaided eye, how big would it be?

The answer is that the processor would be something like 15 metres across.

Consider that for a moment: an area slightly larger than a singles tennis court, packed with detail so fine that you can only just make it out.

The package that the processor is part of would be over 50 metres across, and the pins on the back of the package (right) would be 3 metres tall, half a metre thick, and about 2 metres apart.

#### Caveat

The result above is rather approximate, as you’ll see if you read the details of the calculation below. However, if it inadvertently overstates the case for my processor, which is 10 years old, the error is made irrelevant by progress in microprocessor fabrication. Processors are available today that are similar in physical size but on which the features are nearly 5 times smaller. If my processor had that density of features, the magnified version would be around 70 metres across, on a package 225 metres across. And those pins would be 13 metres tall and 2.25 metres thick.

#### The calculation

The processor is an Intel T7500. According to the manufacturer, the chip is made by the 65-nanometre process. Exactly what this means in terms of the size of the features on the chip is quite hard to pin down. Printed line widths can be as low as 25 nm, but the pitch of successive lines may be greater than the 130 nm that you might expect. I’ve assumed that the lines on the chip and the gaps between them are all 65 nm across.

“The finest detail that we can make out” isn’t well defined either. It depends, among other things, on the contrast.  But roughly, the unaided human visual system can resolve details as small as 1 minute of arc subtended at the eye in conditions of high contrast. This is about 3 × 10-4 radians. At a comfortable viewing distance of 30 cm, this corresponds to 0.09 mm.

So to make the features on the processor just visible (taking high contrast for granted) we need to magnify them from 65 nm to 0.09 mm, which is a magnification factor of 1385.

Applying this magnification factor to the whole processor, its dimensions of 11 by 13 millimetres become 15 by 18 metres. The pins are 2 mm high, so they become 2.8 metres high and about half a metre thick.

Some processors are now made using 14 nm technology. This increases the required magnification factor by a factor of 65/14, to 6430, yielding the results given in Caveat above.

## Anticrepuscular rays

I took this photograph at dusk recently from the beach at Portobello, where Edinburgh meets the sea. As sunset pictures go, it’s not much to look at. But what caught my attention was the faint radiating pattern of light and dark in the sky.  The light areas are where the sun’s rays are illuminating suspended particles in the air. The dark areas are where the air is unlit, because a cloud is casting a shadow.  You may have seen similar crepuscular rays when the sun has disappeared behind the skyline and the landscape features on the skyline cast shadows in the air.

The rays in my picture appear to radiate from a point below the horizon, because that’s where the sun is…isn’t it?

No! Portobello beach faces north-east, not west. The sun is actually just about to set behind me! So why do the rays appear to come from a point in front of me? Shouldn’t they appear to diverge from the unseen sun behind me?

To understand why, we need to realise that the rays aren’t really diverging at all. The Sun is a very long way away (about 150 million kilometres), so its rays are to all intents and purposes parallel. But just as a pair of parallel railway tracks appear to diverge from a point in the distance, so the parallel rays of light appear to diverge from a point near the horizon.

The point from which the rays seem to diverge is the antisolar point, the point in the sky exactly opposite the sun, from my point of view. It’s where the shadow of my head would be. When I took the photograph, the sun was just above the horizon in the sky behind me, so the antisolar point, and hence the point of apparent divergence, is just below the horizon in the sky ahead of me.

For normal crepuscular rays, the (obscured) sun is ahead, and the light is travelling generally* towards the observer. The rays in the picture are anticrepuscular rays, because the light is generally travelling away from me. This was the first time that I had knowingly seen anticrepuscular rays.

*I say “generally” because the almost all of the rays aren’t travelling directly towards the observer. An analogy would be standing on a railway station platform as a train approaches: you’d say that it was travelling generally towards you even though it isn’t actually going to hit you.

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

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.

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

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.

#### A note to deuteranomalous readers

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.

## Faces 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.

The 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.

## This is not an illusion

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.

## Lilac chaser

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.

I used an Arduino microcontroller board to control the LEDs.

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.