What is grace? I think of it as a sort of beauty associated with movement. A dance can be graceful, but a symphony or a painting cannot. They have a different sort of beauty.

It can also refer to human interactions. When done tactfully and considerately, leaving nobody feeling awkward, or worse than they need to feel, they are graceful. Somebody that deals with others in a way that is unnecessarily rough and hurtful ‘lacks grace’.

I think there may be a connection between these two. I’ll think about that later. But for now let’s think about the grace of movement.

I am a huge fan of graceful movement. It doesn’t just have to be dance, which is often designed to be graceful. It can be found in the most unexpected places.

Since my third year of high school I have enjoyed physical activity and being fit. In my youth that included going in cycling and running races. In later high school I trained hard on my bicycle, and the fitness gained from this equipped me to win our annual school cross country race. That then put me into the team for the inter-school races for our region – the Southern Districts of New South Wales. I was at a Catholic school and I think we competed against the other catholic and non-government schools in that region.

I nearly always came third in these inter-school competitions. There was a boy from another Canberra school that came second. I think his name was David Rowe, but I am not sure. What I am completely sure of is that the winner was always Andrew Reardon, from Saint Patrick’s boarding school in Goulburn.

I didn’t see much of Andrew in those cross country races. Just a pair of heels disappearing into the distance as soon as the starting gun went off. If we were running on trails in the pine forest, as we often were, my only goal was to keep him in sight so I could follow his route and thereby avoid taking a wrong turn.

In summer we would have inter-school athletics. I was chosen to represent my school at the middle distance events, of which I usually chose the 1500m and 3000m races. Again I usually came third, but this time I got to see Andrew Reardon in action from closer quarters, and not just from behind. We often raced on lovely, smooth grass 300m tracks that belonged to the richer private schools. A 3000m race was ten laps, which was enough time for Andrew to get to being on the exact opposite side of the track from me – 150m ahead – so I could see him running from the side. And what a gorgeous sight it was! He seemed to just float over the ground in an effortless manner with a grace that words cannot describe. It felt like watching a gazelle or a cheetah in a David Attenborough film, except that cheetahs are sprinters and would probably keel over if asked to run further than 400m.

Any feelings of envy or competitive resentment just leached out of me, as I just felt so privileged to watch this graceful performance. One would say it was poetry in motion if it hadn’t been said a million times before. But the loss of my competitive urge didn’t make me slow down. Rather I increased my pace so that the distance between us didn’t get to more than half a lap and thereby degrade my view of this majestic performance. I imagine Andrew was just cruising at what was a comfortable pace for him, while I was gasping and spluttering. I expect he could easily have accelerated and lapped me quite soon had he a mind to do so. If so, it was a demonstration of the other sort of grace to not subject me to that humiliation. Noblesse oblige.

What was it about his running style that touched me so? I want to say rhythm and symmetry, but that has a connotation of mechanistic, and it was anything but mechanistic. Relaxation was another key aspect, and machines are not relaxed. Andrew looked like he was playing, or floating. It was like a Brandenburg Concerto in vision. You had to be there.

Thereafter I worked on making my own running style as relaxed, symmetric and rhythmic as I could. This wasn’t just vanity. I also believed that running that way would use less energy and allow me to run faster. Perhaps it worked a bit. I did get much faster over the next few years, and some people were even kind enough to say that I had a ‘nice running style’.

The last I saw of Andrew Reardon was in late 1980, when I saw him on telly, which was showing a NSW schools championship athletic meet at Hensley Field in Sydney, which was then a lovely, smooth grass track (now it’s synthetic). It was a 1500m event, which he won reasonably easily, I think in about 3:52. Watching it on telly, without being distracted by my own attempts to run, I could revel in the joy of this exhibition of perfect movement. It was great.

I sometimes wonder what became of Andrew Reardon. Did he become a farmer, as many of the boys at that rural college might have done, or did he move to the city and become a businessman? Did he grow a middle-age paunch as most men do (Oh no!), or did he keep himself trim? Does he still run?

Shortly thereafter I was struck by the running of another Andrew – this time the Australian representative Andrew Lloyd. I saw him on telly, I think running some national championship meet, at perhaps 5k or 10k. He too had a beautiful, relaxed style, seeming to glide along as if his feet weren’t even touching the ground. I remember he was wearing a cap, which runners would generally avoid as an encumbrance, and making it hard to dissipate heat. But it didn’t seem to trouble him. He looked so cool!

In my university days I trained sometimes with athletes at the Australian Institute of Sport, since that was in Canberra and so was I. So I got to see Andrew Lloyd up close while training, and to admire his easy style.

He was involved in a horrible road accident in the early 1980s in which his wife was killed and his elbow was smashed. When he recovered, his elbow was, if not fused, swollen and difficult to move, so his style became lopsided and a bit awkward. But he was still very fast. I think he won the City to Surf a few years after that.

Grace is not a pre-requisite for being a fast runner. Contemporary with Lloyd was Laurie Whitty, a runner with a famously ungainly style, but who won national championships and represented Australia. One of the most famous ever distance runners was the Czech Emil Zatopek, who apparently had a very ungainly style. There is not much film of him running because his heyday was in the fifties. Australia’s most prestigious 10k race is named after him because he won in the 1956 Melbourne olympics and apparently really liked Australia.

Have you noticed that the lululemon logo looks very similar to a capital Omega: Ω? It also looks a bit like the emblem on the Torres Strait islander flag.

In the early eighties I was heavily influenced by a book by Percy Cerutty, who coached a number of brilliant Australian distance runners, including Herb Elliott, who held the world 1500m and mile records and won the 1500m at the 1960 olympics in Rome. He advocated a very nature-based training regime, involving only natural foods – mostly raw – and running on sand hills, beaches and in forests rather than on athletic tracks. But the most memorable – to me – aspect of his philosophy was his claim that through too much soft living, adult humans had forgotten how to move naturally. So to learn how to run properly, and fast, we should watch how other animals do it.

Cerutty disdained symmetry. I don’t know what he would have thought of Andrew Reardon. Percy thought human running should have different modes like a horse – trot, canter and gallop, in order of increasing speed. While trotting is symmetric and may be suitable for marathons, cantering and galloping are not, and he thought they should be used for distances of 10k and shorter. I remember running on the beach when on summer holidays trying to imagine myself as a two-legged horse and transition from trot to canter and then to gallop as I sped up. It seemed to work but maybe it was all psychological. If you imagine yourself galloping then you feel fast and, to some extent, that makes you go faster.

I remember seeing some visiting African athletes jogging about in tracksuits on the training track in Canberra, while preparing for a race on the main track that was next door. They just looked so flexible and bouncy, as if every movement was joyful play. That was another manifestation of grace.

Enough about athletics. That is just one example of where grace can crop up unexpectedly. It is there in hurdling and high jump and pole vault as well as in running. Maybe we could even see it in shot put, but we might have to look a little harder.

Grace seems important in Zen, although it doesn’t seem to be identified or named as such. In the Japanese tea ceremony, great importance is placed on the way one moves in preparing the tea, in serving it, and in how one drinks it. I love the way the cup is offered with both hands and a bow, and is received in the same way. This translates to the way that business cards are presented, and even how purchases are handed across in a shop. I try my best to remember to participate in such small but special rituals. When in doubt, use two hands and make a slight bow!

I have never mindfully raked pebbles as Zen monks sometimes do, but I imagine grace plays a role in that as well – watching the intricate patterns made by the pebbles as they are disturbed by the rake tines and then resettle in their wake.

I think if we look hard enough we can find grace in many things that move around us – humans, other animals, trees and bushes in the wind, even inanimate objects. I try to find this when I feel disheartened. It helps a bit.

I think again about the role of grace in human interaction. The grace is in the speech acts, in the words said, the tone in which they are said, and in accompanying gestures and facial expressions. I suppose all of these are movements. On a simple level, they are movements because speech comes from movement of body parts – lips, tongue, larynx, lungs – and of the intervening air that carries the sound waves. On a more abstract level, they are movements because they are expressed over time, and movement is defined in terms of time. They cannot be captured by a still picture – although a skilful snapshot can hint at it. Even more abstractly, they are movements of emotion – a communication of feeling from one being to another.

I would like to cite an example of a well-known graceful interaction, but my memory fails me (I imagine there are lots from Barack Obama. He is a very graceful person). Nevertheless, we all know what they are and have witnessed and valued them. They catch our attention particularly in difficult circumstances – when somebody turns aside aggression or insult, or rejects a crude suggestion, without aggression and without making anybody feel bad. When somebody finds a way to include somebody that is excluded by their difference, without making a big deal of it. When somebody finds a way to show solidarity and support for somebody that is grieving, without patronising them or putting them in a position where they are obliged to respond.

Then there is grace shown by somebody under extreme pressure – be it their own tragedy, anger, fear or anxiety. When they surprise us by expressing and taking care for things beyond themselves and their worries, despite all.

I don’t know whether it’s the same sort of grace. Classifications rarely matter anyway. But it seemed worth mentioning.

I resolve to try to be more graceful in my relations to other living beings, rather than just in how I run.

Andrew Kirk

Bondi Junction, August 2019

PS I just remembered cricket. I couldn’t send this off without mentioning the joy of watching a truly graceful batter. How they can deal with a heavy red projectile fired at them at up to 160 kph by a small, subtle flick of the wrists that sends the ball to the boundary for four runs. Watching really good batting is like watching a brilliant dance. It’s not for nothing that cricket enthusiasts, more than in any other sport I know, keep photos of their heros in action – in the execution or the aftermath of one of the wide variety of elegant shots available to them.

Pointing the Camera at the Monitor – An Answer

Here are my answers to the puzzle about what happens when we point a camera at its monitor. The problem has a flavour of infinite regress about it, and sounds a little like a Buddhist koan – a question designed to make us realise (amongst other things) the limitations of logic. But whereas a ‘correct’ answer to a koan is usually something bizarre like barking like a dog, or hitting the questioner with a stick, the four questions I posed do actually have perfectly logical answers. And I find them quite interesting.

Let’s kick things off by showing a picture of what one might see when one looks into the webcam that is perched on top of the monitor, looking outwards. This is a screenshot of what my monitor showed when I did that (Figure 1).


The image is framed within the window labelled ‘Cheese’ – which is the name of the webcam program I was using. That’s me wearing the red cravat.

Question 1

When we turn the camera around and point it at the monitor, we will see an infinite regress of windows within windows, as the whole picture will be reduced and fitted into the image area where I am above. Then that reduce-and-insert step will be repeated as many times as it takes until the reduced image gets down to a single pixel and can contain no more strip-images. Here’s an image I made of what it should look like (Figure 2):


In every window, the green desktop background, the desktop icons and the file explorer window to the left are reproduced, and the series shrinks off into the distance. I can tell you it was pretty fiddly putting the tiniest innermost parts into that picture. Infinitely small objects are notoriously difficult to manipulate.

The picture looks a little like a classical picture that uses perspective to show a long, straight road disappearing into the distance, with the point of disappearance at which all the lines converge being in the top-right quadrant of the screen.

But there’s an important and fascinating difference: the dimension along which the images disappear here is of time, not distance. That’s because each nested window is an image captured by the camera’s sensor a short interval earlier than the window that contains it. That interval will vary slightly as we move through the nested sequence, based on the relationship between the rate at which the monitor screen is redrawn (the ‘refresh rate’) and the number of snapshot images captured per second by the camera sensor (the ‘frame rate’). But it will always be more than some minimum value that depends on how long the information takes to travel along the wire from the sensor, through any processor chips in the camera, along another wire to the computer, through any processing algorithms in the computer, and then through the video cable to the monitor. Even without knowing anything about the computer, we know that that time – called the ‘lag’ – will be greater than the lengths of the wires involved, divided by the speed of light (because electrical signals cannot travel faster than light).

So each nested window is earlier than the one outside it, and as we look through the sequence of windows towards the point of convergence at infinity, we are looking back through time!

Question 2

Now we maximise the Cheese window. First let’s see what it looks like with the camera the correct way around, pointing at me (Figure 3):


You can tell from my expression that I’m quite enjoying this little exercise, can’t you?

Here there is nothing outside the Cheese frame, but the Cheese frame still has a broad, non-image bar at the bottom, a narrower bar at the top, and black vertical bars at either side, which are needed to preserve the image’s ‘aspect ratio’ – the ratio of its width to its height.

With that setting, we turn the camera on the monitor, and this is what we would see (Figure 4):


The lower, upper, left and right bars are reproduced as a series of receding frames, and there is nothing in view other than the receding frames. There is no room left for an actual image of anything other than frames.

Figure 4 gives an even better sense of the ‘time tunnel’ that we mentioned in the previous section. Those white borders really do regress away in a spooky way. It looks like something out of Doctor Who.

The ratio of the height of one window to the height of the window immediately inside it is 1/(1-p) where p is the sum of the heights of the upper and lower margins of the outermost window divided by the total height of the outermost window. The ratios of the widths is the same. In this case it looks like p is around 1/5, so each window will be about 4/5 of the height of the window that contains it.

I used my webcam and shaky hands to try an empirical verification of this. I maximised the Cheese window and, pointed the hand-held webcam at the monitor, centering it as closely as I could. Then I asked my partner to press the Screenshot button on the keyboard to record what the monitor was showing. Below is what we got (Figure 5).


It’s a bit rough, but you can see it does the same sort of thing as Figure 4.

When one is holding the camera like this, the little involuntary movements one makes cause the trail of receding frames to wobble left and right in waves, that remind me of the effects used in 1970s television to produce a psychedelic impression – particularly prevalent in rock music film clips.

Here’s a link to a video I captured of this effect.

Question 3

Question 3 asks what we will see on the monitor after we click the full-screen icon.

When we click the full-screen icon, we have no borders, so there can be no infinite, reducing regress of nested borders. How can we work out what is shown, assuming that we started in non-full-screen mode with a maximised window, so that the monitor was showing Figure 4?

The answer turns out to be remarkably simple. We just note that, when the full-screen icon is clicked, the computer will do some computing and then redraw the screen using the whole monitor area for the image from the camera. When it has finished the computing it will draw the screen using the image received most recently from the sensor and, since that image was captured before the screen redraw, it will be the same as whatever the screen was showing previously. This assumes that the previous image is left in place on the monitor until the computer is ready to draw the new one. We consider later on what happens if that is not the case.

That re-drawn screen will then be captured again by the camera sensor, sent to the computer and then drawn again on the monitor, and so on. So the image will remain exactly as it was before the full-screen icon was clicked! In this case, since the Cheese window was previously maximised, it will continue to show something like Figure 4.

The image remains static until either the camera is pointed away or the computer is switched out of full-screen mode, using the keyboard or mouse. Barring earthquakes, electricity blackouts and such-like, we would expect the monitor to still be displaying Figure 4 if we locked the room it was in, went away and returned to inspect it ten years later.

We can understand this a different way by considering the time tunnel we talked about in the responses to questions 2 and 3. In those cases, as we travel inwards through the tunnel to successively smaller windows, each window’s image was captured a short while earlier than the image of the window around it. The interval between the capture time of those windows will be much less than a second, typically 1/24 seconds. In Figure 4 each window’s height is about 4/5 of the height of the window that contains it. So to see what the monitor was showing t seconds ago we have to go to the nth window in the sequence, counting from the outside, where n=24t. The height of that window, assuming the height of the monitor’s display area is 300mm, is 300 x 0.8-24t mm. The window size reduces rapidly as we go back through time. A little calculation shows that the 26th window in the sequence is the first to have height less than 1mm, and that window shows what the monitor was showing just over one second earlier.

Without constraint, that time tunnel would continue to go back, getting smaller at an increasing rate, window by nested window (like Russian dolls, or the cats in the Cat in the Hat’s hat), until we got to the time before the camera program window was opened on the computer, and that image would show whatever was on the monitor before the window was opened. But it would be indescribably tiny. If we opened the window – in maximised form – five minutes ago, the height of the window that now showed that image from back then would be 300mm x 0.8–7200, which is approximately 10-688 mm. This is indescribably smaller than the smallest atom (Hydrogen, with diameter 10-7 mm) or even just a proton (diameter 10-11 mm).

I expect our eyes probably could discern no more than the first twenty windows in the sequence. Further, since my screen has about three pixels per mm, the windows would reach the size of a single pixel by the 31st window in the sequence, and the regress would stop there. Hence the sequence would look back in time no more than 1.3 seconds.

When we move to full screen mode, we still have a time tunnel of nested windows, but each one is exactly the same size as the one before it – the height ratio is 1, rather than 0.8. That means that how far we look back in time is no longer limited by shrinking to the size of a pixel, and the sequence will go all the way back to the last image the monitor showed before drawing its first screen in full-screen mode – which will be Figure 4.

In practice, as my friend Moonbi points out, the slight distortions in the image arising from imperfections in the camera lenses, although they may be imperceptible at first, will compound on each other with each layer of nesting so that what is actually shown will be a distorted mess. Like a secret whispered from one person to another around a large circle, or a notice copied from copies of itself dozens of times recursively, the distortions – however tiny they may be at first – will grow exponentially to eventually dominate and destroy the image. One minute after full-screen mode has been commenced, the monitor will be showing a 600 times recopied image of Figure 4, which will be more than enough to obliterate the image.

But this is a thought experiment, so we allow ourselves the luxury of assuming that are lenses are somehow perfect, that there is zero distortion, and each copy is indistinguishable from the original.

What if the screen blacks out?

Above we assumed that the display does not change until the computer is ready to redraw the full-screen image. If you go to YouTube, start playing a video and then click the full-screen icon (at bottom right of the image area) you will see that is not what it does. It actually makes the whole screen go black for a considerable portion of a second, and only then redraws the screen. If the camera program we are using does that then the screen will go black and remain black indefinitely.

If the black-out is shorter than YouTube’s, different behaviour may arise. It depends on four things:

  • the time from image capture to display, which we call the lag and denote by L,
  • the interval between image captures, which we call the frame period and denote by T, and
  • the time the blackout period commences and the time it ends, both measured in milliseconds from the last image capture before the blackout. We’ll denote these by t1 and t2.

If no image capture occurs during the blackout, which will happen if t2<T, the blackout will have no effect on the final image and we can ignore it. The eventual image will still be Figure 4.

If images are captured during the blackout, and the first image shown on the monitor after the blackout was captured during the blackout, the screen will thereafter remain black indefinitely. This will be the case if both L and T are less than t2.

The other possibility is that T<t2 and L>t2. In this case the first image shown after the blackout will be a picture of the pre-blackout monitor, ie Figure 4, but it will be followed sooner or later by one or more black images captured during the blackout.  What will follow then will be an alternation between images showing Figure 4 and black images. It will look like a stroboscopic Figure 4. The strobe cycle will have period approximately equal to L, and the dark period will have approximately the same length as the blackout, ie  t2t1. In essence, the monitor will indefinitely replay what it showed in the period of length L ending at the end of the blackout.

This black-screen issue will also arise if the monitor is an old-style, boxy, CRT (Cathode Ray Tube) rather than a LCD, Plasma or LED device. CRT screens typically draw around 75 images per second, made up of bright dots on a photo-sensitive screen, by shooting electrons at it. In between those drawings, the screen is black. That’s why those screens sometimes appear to flicker, especially viewed through a video camera.

For a CRT screen, the image captured immediately before the redraw may be Figure 4 or black, or something in-between – a partial Figure 4, with a complex dependency based on four parameters: the length of exposure used by the sensor (shutter speed), the length of time taken for a single redraw of the CRT screen and the refresh rate and frame rate. Unless there were a particularly unusual and fortuitous relationship between those four numbers, the image on the monitor would not be Figure 4. I think instead it would either be just black or an unpredictable mess. But one would need more knowledge of CRT technology than I have, to predict that.

Anyway, we don’t want to get bogged down in practical technology. This is principally a thought experiment. And for that ideal situation, we assume an LCD monitor with a computer that, upon receiving a full-screen-mode command, leaves the prior image in place until it is ready to redraw the image on the full screen. And the answer in that situation is Figure 4.

Doing a careful experimental verification of this is beyond me because, amongst other things, I don’t have a camera program that has a full-screen mode. But just for fun, I made a video like the one above under Question 4, where I focused the camera on the area of the monitor that displayed the image, trying to exclude the borders. It wobbles about, partly because of my shaky hands. It is mostly black, but there’s a blue smudge that appears in the lower half and wobbles around. I think that is a degraded version of the regressing images of the lower border. But under such uncontrolled conditions, who knows?

Here’s the video.

Question 4

We are now in a position to work out the answer to question 4, which is ‘what will we see after we point the camera to the right of the monitor and then pan left until it exactly points at the monitor, and stop there?

To start with, we know that, once the camera has panned to the final position of exactly capturing the image of the entire monitor, it will hold that image indefinitely, and that image will be whatever the monitor was showing immediately before the camera finished panning.

We’ll be a little more precise. A video camera captures a number f of images (‘frames’) per second, typically 24. The final image shown by the camera will be whatever the camera captured in the last frame it shot before completing the pan. The nature of that image depends on relationships between the pan speed, the frame rate and the width of the monitor screen, which we will explore shortly. But, to avoid suspense, let’s assume a frame rate of 24, that 32 frames are shot while performing the pan, and that the view to the right of the monitor display area, including the black right-hand frame of the monitor itself, being this (Figure 6):


Then, on completion of the pan, the camera will show, and continue to show indefinitely thereafter, the following image (Figure 7 – note that the image was made by editing, not shot through a camera. My equipment is nowhere near precise enough to do this accurately):


C’est bizarre, non?

Those of you that enjoy exploring intricate patterns may wish to read on, to see the explanation of this phenomenon. I will not be offended if most don’t.

The answer depends on the ratio of the speed at which the camera pans left, to the width of the screen. If we want to be precise, these speeds and widths must be measured in  degrees (angles) rather than millimetres. But millimetres are easier to understand so we’ll use them and ignore the slight inaccuracy it introduces (otherwise I’d need to start using words like ‘subtend’, and we wouldn’t want that would we?).

Say the camera rotates at a speed of s mm per second, and that it shoots f frames per second. Hence it shifts view leftwards by s/f mm per frame. So, if the width of the monitor’s display area is w mm, it shoots N=wf/s frames between when the camera first captures part of the monitor’s display area (on the monitor’s right side) and when the camera is in the final position where it exactly captures the view of the whole monitor display area, not counting the last frame. We label the positions of the camera at each of those frames as 1 to N, going from earliest to latest. We omit the last frame because we know that, once the camera is pointing exactly at the monitor, the image will remain fixed on whatever the monitor is showing at that time, which will be what the camera saw in position N.

By the way, we assume that N is an integer even though in practice it won’t be, because it will have a fractional part. It doesn’t make the calculation significantly more difficult if it’s a non-integer, but it is messier, longer, and the differences are not terribly interesting, so we’ll assume it’s an integer.

Divide the view to the right of the monitor’s display area, as shown in Figure 6 above, into N vertical strips of equal width. Number those strips 1 to N from left to right. We will call these ‘strip-images‘, as each is a tall, thin picture. Next, number the positions the camera has when each frame is shot as follows:

  • Frame 0 is when the left-hand edge of the image captured by the camera coincides with the right-hand edge of the monitor display area, so that the camera captures exactly the image of Figure 6, ie strip-images 1 to N. At that time the monitor will be showing what the camera captured one frame earlier, which will be an image made up of strip-images 2 to N+1 (strip-image N+1 is what we can see in an area the same size as the opther strip-images, immediately to the right of Figure 6)
  • The frames shot after that are labelled 1, 2, etc.

Label the times when the camera is in position 0, 1, 2 etc as ‘time 0’, ‘time 1’, ‘time 2’ etc.

With this scheme, Frame N will be the one that is shot when the camera is in the final position, when it exactly captures what is on the monitor, so that that image remains on the monitor indefinitely thereafter.

The following table depicts what is shown by the monitor and what is captured by the camera at each position, in the situation we used to produce Figure 7, which has N=32.

The rows are labelled by the camera positions/times. The first 32 columns (the ‘left panel’) correspond to the 32 vertical, rectangular strips of the monitor display area, numbered from left to right. The next 32 columns (the ‘right panel’) correspond to the strips of what can be seen to the right of the monitor. The number in each cell shows which strip-image can be seen by looking in that direction. The yellow shading in each row shows what images are captured by the camera at that time, to be shown on the monitor in the next row. (Figure 8):

figure-08-pan-table-fullA few key points of interest are:

  • The numbers in the right panel do not change from one row to the next, because rotating the camera does not change what can be seen to the right of the monitor.
  • The numbers in the left panel change with each row, to reflect that what was captured by the camera at the previous camera position was different from what was captured at the one before that, because of the camera’s movement.
  • The yellow area denoting what the camera captures moves to the left as we go down the table, reflecting the camera’s panning to the left.

There are lots of lovely number patterns in the left panel, which I will leave the reader to explore.

Here is a zoomed-in image of just the left panel for those whose eyes, like mine, have trouble making out small numbers (Figure 9):


Referring back to Figure 7 we see that, as we move from the right to the left side, it has a series of eight vertical images of increasing width. The first one is just the monitor’s right-hand frame – a black plastic strip. The next is twice as wide and has the monitor frame plus the strip-image to its right. The one after than is three times the width and so on. This corresponds to the last row of Figure 9:

5 6 7 8, 1 2 3 4 5 6 7, 1 2 3 4 5 6, 1 2 3 4 5, 1 2 3 4, 1 2 3, 1 2, 1

I have put commas between each contiguous set of strip-images. I call each such contiguous set a ‘sub-image‘. The first sub-image is incomplete – being 5 6 7 8 instead of 1 2 3 4 5 6 7 8 – because N is not a triangular number.

The time tunnel applies here, with a slightly different flavour. The newest sub-image is the one on the far right, composed solely of strip-image 1. This was captured from the world outside the monitor one frame period ago. The next, the ‘1 2’, was shot from the monitor last time, and came from the real world outsider the monitor two frame periods ago. The oldest sub-image is the one on the left, which has been through the camera-monitor loop seven times, having first been captured from the real world eight frame periods ago.

It’s quite fun to trace the path of these images as they repeatedly traverse the camera-monitor loop, by the numbers in the above table. Here’s the lower part of the table showing how the first, third and sixth sub-images from the left (using blue, grey and pink shading respectively), make their way from the real world (to the right of the vertical dividing line), into the camera-monitor loop (to the left of that line), around that loop as many times as needed, and finally to the ultimate static image (Figure 10):


My example with N=32 involves a high panning speed. Shooting 24 frames per second, the pan would need to to completed in 32/24 = 1.33 seconds. One would need extremely good equipment to accomplish that without getting a bounce or wobble when the camera stops at the end of the pan – and avoiding wobble is critical to getting the indefinite static picture we have discussed.

It may be that in order to avoid camera bounce one would need a slower pan, giving us a (perhaps much) higher N. What would be the outcome of that? Well, the right-most sub-image contains only one strip-image and, as we move left, each image contains one more strip-image than the one to its right. So, if the number of sub-images is r, then N will be greater than the sum of the numbers from 1 to r-1 (the (r-1)th triangular number) and not exceeding the sum from 1 to r (the rth triangular number). A little maths tells us that this is from (r-1)r/2 to r(r+1)/2. For large N this means that there will be approximately √(2N) sub-images and the largest will be comprised of about √(2N) strip-images. Since for a display width of w the width of a strip-image is w/N, that means the widest sub-image will have width about w√(2N)/N=w√(2/N), which will get smaller as N increases. If N=2048, corresponding to a very slow pan time of about 85 seconds, the widest sub-image would be narrower than the frame of my monitor, so all we would see in the final static image would be black plastic monitor frame, something like this (Figure 11):


The bars of light are from the screen reflecting on the shiny monitor frame. Because I couldn’t hold the camera straight, they are bigger at the bottom than at the top.

I will leave you with my synthesis of the sequence of 32 images that would be seen, given near-perfect equipment, in the 32-step pan that ends with Figure 7 above. They are simply a realisation in pictures of the patterns shown in figure 9, when applied to the image Figure 6. The sequence is in a pdf at this address. If you go to page 1 and then repeatedly hit Page Down rapidly, you will see a slow-motion video representation of what it would look like as the camera panned. You may need to download it first in order to be able to view it in single-page view, which is necessary in order to achieve a video-like effect. Alternatively, it is represented below, albeit somewhat more crudely, as a pretend film strip.

Andrew Kirk

Bondi Junction, February 2017


Mindfulness at the dentist’s

I went to the dentist today. It was just a routine check-up. I’m a recalcitrant patient though, stubbornly resisting the six-monthly reminders they send me. This time I was especially tardy, leaving it for fifteen months, maybe longer.

My dentist is an excellent fellow, but he managed to punish me for my negligence by finding that a filling had gone missing, and proposing to replace it today, rather than making an appointment to replace it later.

How could I refuse? When any backward step would be interpreted, quite correctly, as cowardice.

But that’s not all. My worthy dentist informed me that I could choose not to have the local anaesthetic injection if I wanted, so that I didn’t have a numb jaw for the next few hours. He said I might ‘experience some sensitivity’ during the drilling, but that it would probably be OK.

I suppose if I really was the Renaissance Man I pretend to be, I would have said ‘No, give me the anaesthetic’, because I’m so in touch with my feelings and I am not ashamed to admit that I don’t enjoy having my teeth drilled.

But I was brought up in the seventies, when boys were expected to display the characteristics of men as quickly as possible, and men were supposed to grit their teeth stoically in the face of pain (although that wouldn’t work very well at the dentist’s would it?). So of course I said, doing my best Clint Eastwood impersonation, ‘Forget the injection. Let’s just do it!’ (actually the exact words I used, which I can’t remember, were probably much less impressive and manly than that, but the information content was the same).

It wasn’t just about pretending to be macho though. I have been dipping my toe in the water with Buddhist Mindfulness techniques recently, trying to learn some basic meditation skills. I’m too busy at present to take classes and learn it all properly, so I’ve just read something about it and listened to a podcast. It seems to be mostly about focusing attention on physical sensations, and gently but firmly sending away the other thoughts that try to crowd into one’s mind. Breathing is the classic sensation to focus on, but apparently it doesn’t have to be that. I sometimes practice focusing on my breathing when in bed at night, and I think I’d be much better at it by now if it weren’t that I always fall asleep within a minute or two of starting.

Before the drilling, the dentist did some cleaning and scraping, which was really most uncomfortable and provided an excellent opportunity to practice before the Big Event. I tried to focus on the feel of the electrical scraper against and in between my teeth, a feeling of pressure and vibration, and every now and then a minor surge of pain when a tender spot was probed too inquisitively. I found that, for as long as I could focus my attention entirely on those sensations, batting away thoughts such as ‘I wonder what the drilling will feel like’, the experience was not altogether unpleasant. The sensations were objects of interest or wonder. It’s a little like when you realise what a peculiar word ‘kettle’ is and you say it slowly over and over to yourself to contemplate its sudden unfamiliarity, its strangeness.

Sensations are strange things. They are impossible to describe because the only thing they can be like is themselves, or other sensations that are so similar that expressing the likeness gives no further information. It’s like replying to the question ‘Where does Betty live’ with ‘Oh, she lives next door to her neighbour Bob’. My natural tendency is to try to describe sensations visually, such as saying that cool breeze feels ‘white’. For me this is not synaesthesia but rather just an indication of the impossibility of expressing a sensual experience in words. I think the reason I reach for a visual simile is that for most sighted people, vision is the dominant sense, and we tend to primarily think of things in visual terms. I imagine it is very different for someone born blind.

This peculiarity and strangeness of sensation is useful at the dentist because it makes the object on which we are trying to focus attention peculiar enough to be worthy of that attention. It is easier for me to focus attention on a physical sensation than it is to focus on a mental image of sheep vaulting a hurdle in the old, but mostly useless, ‘counting sheep’ technique for getting to sleep. Sorry sheep, but I just don’t find you interesting enough!

Maybe I was a wimp. I don’t know. I can’t objectively evaluate my internal bravery level because I’ve never been anyone else. I did my best to show no outward signs of wimpiness, and the shambolic, amateurish attempt at mindfulness helped with that.

But I really don’t like having my teeth scraped, descaled and whatever other things he was doing to them.

The funny thing was, when he finally got around to drilling for my filling, it didn’t hurt a bit! It would be nice to think that’s because my attempt at mindfulness was working. But realistically, it’s much more likely to be that he was simply telling the truth when he said it would probably be OK.

I think I have a lot of work ahead of me if I ever want to be able to tolerate unanaesthetised wedge resection, passing kidney stones, or maybe even chest-waxing, with equanimity.

Andrew Kirk

Bondi Junction, March 2013