Counting pebbles

Figure 1

Contents

• Introduction

• Bricks

• Fabrics

• Counting pebbles

• Counting the pebbles you can see

• There is no right answer

• A novel counting strategy

• Some experiments

• Consciousness

• Conclusions

• References

Introduction

I have posted on a number of occasions about counting things, most recently at reference 1. These posts were about one-dimensional counting, where the things to be counted were in a stack – like courses of bricks or the storeys of tower blocks – or in a row – like beads on a string. 

Here I think some more about the much more difficult problem of counting two dimensionally – things like the pebbles on the beach – something that some people, possibly challenged in other ways, are said to be able to manage. 

Figure 2

 

I have already set out some preliminaries at reference 6, where I introduced count sets (which pebbles are to be counted), count paths (what route shall we take across the two-dimensional beach for the purpose of one-dimensional counting) and count lists (the choices we make about vocalising the count). People do not behave in the orderly way suggested in the figure above – but it does serve to get us away from the idea of there being a right answer. The count is what you come to on the day; the count is what you choose to count on the day.

We start with a glance at two other two-dimensional counting opportunities, bricks and coarsely woven fabrics. With bricks easy compared with pebbles and fabrics intermediate.

Bricks

 

Figure 3

Two dimensional counting of the bricks in a wall is not too difficult, leaving boundary problems aside for the moment. Provided the bricks have been laid reasonably neatly, it is possible to scan the rows of bricks, known in the trade as courses, counting as one goes. Traversing one row from right to left, then dropping a row and traversing back from left to right. So long as one can hold one’s attention on the current brick and remember in which one direction one is going, the thing can be done. Getting into a rhythm seems to help, perhaps nodding the head slightly in time with the count, and one needs to proceed at a steady, even pace, otherwise one is apt to forget in which direction one is going. One might allow a short pause at the end of each row, to give the eyes and brain something of a rest – but not so much of a pause that one loses track of the next row.

In terms of the count sets and count paths of reference 6, there is little to do: we count the bricks by rows. We might do better with some count list designed for the job, but will we stick with regular count list, the numbers we learned as children and have used ever since. But if we ever get up to speed, maybe the auction people at references 7 and 8 would be worth another look.

 

Figure 4

Counting is still possible when the bricklaying gets a bit ragged, as the rows survive, but damage to the bricks can easily result in mistakes. Count set problems have not vanished.

Figure 5

 

But it gets much harder when the rows get longer, such as that snapped above. Much concentration is needed to keep one’s attention on the current row, concentration which is hard to maintain for any length of time.

Fabrics

 

Figure 6

Counting the squares on this fabric is much the same as counting bricks, although the lack of variation make it harder to keep one’s place – although counting chequerboard fashion or missing out every other row and column seem to help with that.

Figure 7

 

This one is quite a lot harder, with the rows and columns being badly degraded. One might do better counting some subset, perhaps the vertical red stitches, although one then has the boundary problem of red stitches morphing into red blobs, and reds morphing into purples and blues.

Figure 8a

 

I found this one harder still. Rows and columns degraded and not much else to go on.

Figure 8b

And it does not get any better when magnified.

 

Figure 9

Here we are back with something more like the bricks, but with the individual elements too small to keep track of with any comfort.

Note that in the first and last of these five examples, there is, in some sense at least, a right answer. 

There is very little fudging of the count set variety. But this is not the case in the second example. The third and fourth examples – the latter being a magnified version of a portion of the former – are intermediate in this respect.

Clearly, different materials present different problems, possibly requiring different techniques and strategies to count their features. The weave at Figure 7 above, for example, comes close to pebbles on the beach, while that at Figure 6 above comes close to bricks in a wall. But our present interest is pebbles on the beach.

Counting pebbles

I turn back now to those pebbles, the sort of smooth, rounded stones, mostly flints, that one gets on beaches, where there are no rows, ragged or otherwise, although counting is made a lot easier than it might otherwise be by the pebbles sorting themselves out, with sensible sized, washed pebbles sitting on top of the sand below (or whatever else might be below), rather than being mixed up with it. At least most of the time.

We do a sum to give us an order of magnitude. Suppose 100 metres of beach, on average 10 metres deep, promenade to sea, and 1 metre thick. 1,000 cubic metres. Suppose 50^3 pebbles to the cubic metre. Which gives us 125 times 10^6 or 125 million pebbles – which is a large number, but not unmanageably so. Maybe roughly twice the number of people who live in England. And very small indeed when compared with things like the number of molecules or atoms in a cubic metre of air – which Bing suggests might be of the order of 3 times 10^25.

Figure 10

One approach would be to take a digger to the beach. Dig it all up. Put the pebbles through some screens and sieves to get rid of all the unwanted material – small stones, sand, litter and debris of one sort or another. Perhaps wash them. So we now have all the pebbles in the qualifying size band – say around one centimetre to ten centimetres in diameter – and we can put them through a counting machine. Which we suppose can deliver a reliable count; that you get the same answer if you put them through again and two such machines always give us the same answer.

So we can count what we dig up, but what exactly is it that we are digging up? How do we know when to stop the diggers? And the answer is, on a real beach, that we don’t know. There are always going to be grey areas at the margins, perhaps at the edge of the sea, perhaps where the beach washes over onto the promenade, or over onto the shelves of flat bedded rock which bound part of the beach. Possibly grey areas which only disturb the count in proportion to the area of the beach, rather than disturb the count in proportion to its volume. But disturbance nonetheless. 

And then are we really counting what we dig up? How replicable is the business of moving the pebbles to the screening machinery? In hundreds of digger loads or lorry loads, how many odds and ends are there going to be? I associate to the nice, simple fact that there is supposed to be exactly one national insurance number for everybody, for the twenty million or so people of working age – and to all the odds and ends that exist at the margins of that system. And then there is the screening process itself. What about, for example, all the pebbles near the permitted margins of one centimetre and ten centimetre? Given the numbers involved it seems unlikely that this process is replicable, that one is going to get exactly the same count every time.

Furthermore, our count is destructive. Although we could put the pebbles back on the beach, the boundaries will have been disturbed, the grey areas will have been disturbed, some pebbles will have been lost or damaged – and any recount is likely to come up with a different answer. The count is not replicable.

In any event, in what follows we concentrate on what can be done in the way of non-invasive counting. The sort of counting you can do by looking without touching, or even standing. We suppose our beach to be static in the sense that the pebbles are not moving about and that two such non-invasive counts should, in principle, give the same answer. A count which is replicable – at least, that is what we hope.

All a bit unrealistic but it is a place to start our count.

Counting the pebbles you can see

The first question is which pebbles are to be counted, in the jargon of reference 6, what or where is the count set, with the first answer being that one counts the pebbles that one can see. And the second answer being that there must be a boundary. Perhaps the concrete promenade at the top, the sand at the bottom and two wooden groins at the sides. Bearing in mind that groins may not run all the way from sea to dry land and they are often partially covered, as can be seen by asking Bing or Google for ‘beach groins’.

Figure 11

Another answer might be that one counts a roughly circular, growing patch of pebbles, centred roughly at one’s feet. There may be no end point, but there is a start and there is growth. So in the middle panel of the figure above the count has proceeded, is proceeding, in an anti-clockwise spiral. The count path is spiralling outwards. And even if one does not know when to stop, one does know how many pebbles one has counted so far. And one has done away with the problem of short, covered or missing groins. As they used to tell us at school, it is playing the game that is important, not who wins, who comes top. Who counts all the pebbles. What counts is the process, not the result.

However, just for the moment, we suppose that the task is to count something like all the pebbles in the left hand panel of the above. Perhaps the area has been staked out with pegs and string, as suggested in the right hand panel. String which brings boundary problems of their own, but which are hopefully small relative to the number of pebbles so enclosed. 

Note that by counting we mean counting, not estimating. Not, for example, taking the product of an estimate of the number per cubic metre and multiplying it by an estimate of the volume, which is what we did above, albeit rather crudely. Or an estimate of the number per square meter and multiplying it by an estimate of the area. The latter being more or less equivalent to counting what can be seen, which is what we are after.

Counting with a computer is, at one level at least, straightforward. With the big advantage of the computer being that it can mark up its image of the beach as it goes, something the average brain certainly can’t do in a conscious way. It might do the job in two passes of the image. In the first pass it overlays the image with something like the count set of reference 6, perhaps expressed as a large array of pixels, with each pixel taking one of two values: 1=region and 0=null, with all the regions being separated, one from another, by null pixels. 

Figure 12

Regions do not touch and they certainly do not overlap. The count proceeds by finding a region pixel, incrementing the count and then setting all the pixels for this region to null, easy enough to code as the regions will have the property that one can get from one pixel to any other without leaving the region. The counts stops when all the pixels are null. A process which will work well enough when the pixels are small relative to the pebbles, to the regions. The figure above is a sketch of an array of pixels, towards the end of this destructive counting process. Note that it is only the representation of the count set which is being destroyed, not the underlying image, which can be reused.

Note also that the red blobs do not need to be very accurate. It is probably good enough if they are maximal within the region they represent. With good enough meaning that there is enough there to relate regions back to the raw image of the pebbles, to be sure that one has got a reasonable count set.

Turning back from the computer to people, I recall reading about a chap who, if on the beach and bored, would count the pebbles there to give himself something to do. Sadly, I cannot now find him: there are plenty of curious ‘savant’ skills out there, but I have not yet found this particular one, although I have found a chap who, with what was not much more than a glance, could count the matches which had been emptied out onto the floor. See for example, references 2 and 5.

But the average human brain cannot manage these tricks. Perhaps there is a problem with the amount of working memory needed – sometimes said to be less than ten chunks of information – even leaving aside the various tricky boundary problems, knowing whether or not a given bit of image is to count as a new pebble or not.

There is no right answer

Figure 13

 

As with Figures 1 and 13 above, the problem here is knowing which pebbles, which parts of the beach are to be counted. Anything in the snap which can honestly be made out with the naked eye? Including all the pebbles which are underwater? And even supposing one made such a count, how replicable would it be? Would one person get the same result a second time? What about a second person? To which question, the emerging answer seems to be that counting pebbles is not going to be replicable. And given that probabilities are creeping into the answers given at reference 6, even doing it by computer may not give a replicable count.

Figure 14

In the left hand panel of the snap above we have clear boundaries, in the form of a box. And the pebbles have been selected, washed and cleaned. No marginal bits and pieces of dead crab, other debris, sand, slate or anything else.

But there is still room for doubt. What about what is lurking down below, in the zone highlighted in purple, bottom right? What about the two, possibly more, pebbles lurking down there? With the answer depending on where the eyes are, on the state of the pebbles – are they wet and/or shiny? – and on the lighting conditions. We are probably going to get a slightly different answer every time we do the count, every we time we make, we build the count set.

And even if we had a complete description in our computer of the sizes, shapes and positions of all the pebbles involved, deciding exactly which ones were to count as potentially visible would be reasonably complicated and would involve setting several, more or less arbitrary visibility-flavoured thresholds. There is no canonical answer to the question ‘how many pebbles can you see’. All we can be sure about it is that the number you see cannot be more than the number that there are, which can be determined in this case by tipping them out of their box and spreading them out.

At reference 6, we avoided these problems, to some extent, with the notion of a count set of regions. This what was available to be counted on the day, on this particular occasion. Accepting that on a different day, even with the same beach, with the identical configuration of pebbles (in practise rather unlikely), one might have a different count set. And even if one had the same count set, the rules for count path, the vagaries of count paths, do not deliver the same count on each occasion.

A novel counting strategy

Figure 15

 

In the linear, one dimensional case maintaining the cursor, maintaining one’s position, is manageable, certainly with practise. Two dimensional bricks are harder, but possible, as illustrated above. Two dimensional pebbles, of irregular size, shape and position, are much harder – but maybe there is something that can be done, short of memorising the whole picture and ticking off the pebbles, in the way of a computer, as sketched above.

Trying to count the pebbles by rows works after a fashion, in the way that we previously counted rows of bricks, provided the rows are reasonably short. But even then, one is soon defeated by the pebbles not being in tidy rows, by not being sure whether one has already counted a pebble or not.

Maybe a savant, one of these people with special skills, either could – or could train himself to – unconsciously maintain a record of his moves, the zig-zag red line in the right hand panel of the figure above, moving slowly up the beach, with the front end of the record growing piecemeal as pebbles are counted and marked off and with the back end of the record fading away. The savant holds his (these savants are more commonly men) attention on this front end and knows which pebbles he has counted recently and where he needs to count next. With the count being maintained as long as the record is rolling slowly but steadily forward. A device which means he only needs to hold the recent past in memory, not the all the past. A device which does not look too bad to a normal in the example shown above, where the width of the patch of pebbles to be counted is small – but which will rapidly get hard as the width gets large.

Maybe a savant could do it, could slowly pan up a bounded beach, unconsciously maintaining this sort of rolling image in memory, counting the pebbles and ticking them off as he went.

I note that some savants can train themselves to do very improbable mental feats and that some footballers (for example) can train themselves to do very improbable things with footballs. Zen archers who can shoot at a distant target with their eyes shut. Australian aborigines who can do improbably well at throwing stones, feats like throwing a pebble into a waste paper bin thirty metres away – a loose translation of the paper at reference 4. Plus I have seen YouTube clips of footballers doing much the same sort of thing. The brain does respond – in some mysterious way – to sustained effort at such things. Provided there is feedback, as there is in these cases, mostly from the eyes. 

Quite different would be an approach which might be helpful if one was doing the job with statistical clerks. Cut the image into rectangular blocks of modest size. Phase 1: delete the smaller parts when a pebble appears in two or more blocks. Phase 2: dish out the blocks to be counted, one block to the clerk. Phase 3: pass the blocks around and count again, just to be on the safe side. Phase 4: add up the blocks. All of which seems a bit complicated for a general purpose brain to tackle.

Some experiments

In the light of the foregoing, I tried some experiments.

Experiment 1

Figure 16

 

In this experiment, I tried counting the pebbles – known to the trade as inch and a half shingle – outside our back door, where the beach, as it were, is neatly bounded by concrete and the problem of what pebbles to count in large part vanishes. The idea being to start at the bottom of left hand panel of the snap above and to work up. With the result that the counting was easy enough; it was easy enough to lock onto a pebble, to increment the count and then to move on, to left or right. What I found more or less impossible was knowing which pebbles I had already counted when I came back again the other way.

A variation was just to count, but without attempting to count all the stones, in the spirit of the count paths of reference 6. The task was to maintain the count, fixing and counting one pebble after another, with the only rule being that one did not count any one pebble more than once, with the process being illustrated in the snap above. The aim was to count a band of pebbles, and while the band might be a bit ragged at the edges, the idea was that in the middle of the band one counted everything, the count was exhaustive. This is sketched in the right hand panel of the snap above.

The problem here was that even the business of fixing on the next pebble, even when one was not too fussy about exactly which pebble, used up brain resources and it was quite easy for the count to go astray, to miss a number out or to use a number twice.

Experiment 2

Figure 17

 

Here the idea was, rather than counting real stones, to see how one got on with the computer where one was able to mark the image up as one went along. So the raw image was loaded into the Microsoft Snip & Sketch tool. I then used the highlighter tool therein to mark off stones as I counted them. It was not as easy as I had first thought and one needed to get the right colour and width of highlighter and to develop a convention for marking the stones. One needs to try various colours, it not being obvious which colour is going to work best given the strength and variety of colours in the raw image. I think the idea should be to mark right across the width of the stone at it widest point, in one stroke. Then, inter alia, the count should be the number of strokes, in the top right hand panel of the snap above, around sixty.

But in order to cope with the more difficult cases it needed to be done carefully and neatly, which required concentration – which meant in turn that one was apt to lose the count. Not enough brain cycles available to do both.

Bottom left, the idea was to draw around each stone, counting as I went. Which worked quite well, but which was quite tiring, both for hand and brain. Working in the rather organic way shown seemed easier than trying to do it in regular lines.

A weakness was a tendency to obscure the smaller pebbles in the gaps between the large pebbles, which resulted in some of them getting left out. In some cases there was doubt about whether one was seeing a new pebble at all. Was it just the shadow of one which had already been counted? Furthermore, after the event, one could not check the count as one could no longer always distinguish small stones from spaces which were intentionally left blank.

A slight refinement was to draw around the pebble when it was large and to fill it in when it was small. Which worked even better – except that it was more or less impossible to check after the event. The colouring in was a prop during the proceedings, not a record which would be worked on after the event.

Consciousness

Some sorts of simple counting, like counting the steps up a long staircase, like counting the down strokes on the pedals on a bicycle, can be done more or less unconsciously. If one is a habitual counter, one does not even need to turn the count on as a conscious decision, or (on another view of agency) as a conscious registration of a decision already taken by the unconscious.

Counting the storeys of a tower block does not come so easily and requires both the conscious decision to count and conscious attention to maintain the count. I have, for example, never found myself in front of a tall building, half way through the count, not having much idea how I got there. Something which can happen, for example, when one is driving, perhaps while composing the dinner menu for the day following. One suddenly realises one has got somewhere without any memory of the business of getting there. Nor, incidentally, can I remember ever having dreamed that I was counting either steps or storeys.

Note that driving monopolises the vision system, of which we have just one. And counting monopolises the speaking system, of which we have just one again, even if the speaking in question is silent. While computers are not constrained in either way.

We seem to have here another example of the tricky link between attention and consciousness: they do seem to be linked – but one can attend to something (for example, driving the car) while not being conscious of it – while we do seem to need to attend to something in order to be conscious. One has to be conscious of something, even if one is trying to reach, to touch the void – something which I believe Buddhists aspire to. Or, slightly off-message here, some high altitude mountaineers. While for some activities, like taking a tricky shot at golf, one really does try to empty one’s mind, while maintaining one’s visual attention on the ball below. To try to stop consciousness getting in the way of unconscious performance. Rather oxymoronically, to consciously try not to be conscious.

Counting the bricks on a wall or the pebbles on a beach is much harder again than counting the floors of a tower block and I doubt whether either count could proceed without a conscious start or while doing anything else. And either count would be fatally disturbed by almost any kind of interruption, almost any kind of stimulus.

Which is not to say that the heavy lifting involved in counting pebbles is conscious. What is conscious is maintaining visual attention on the bit of beach being counted. But there is no need to be aware of how that visual stimulation is converted into a count – although it may well be that this conversion can be learned, trained or improved with practise and feedback. 

Conclusions

We have poked around in various aspect of the beach counting problem.

 

Figure 18

 

The bad news is that, given the nature of the beast, there is no right answer, even if we had some way of checking the counts. The concept of ‘the number of pebbles on the beach’ is not well defined. One just can’t count the pebbles on the beach. 

The good news is that one can certainly do some counting, using the pebbles as a prop, but one cannot be sure that one has not missed some pebbles out and counted other pebbles more than once. And if the count goes on for a long time, the count itself may become a bit ragged, one cannot be sure than one has not missed out some numbers and used other numbers more than once.

Maybe my recollection of a pebble counter is a little off the mark, in that maybe what the chap in question was doing was estimating rather than counting, perhaps using some of the unusual mathematical skills pointed up in reference 2 – for example the ability to do large sums, to calculate day of the week of a date and to calculate intervals in days from two dates – in both these last two cases taking proper account of leap years – with reference 3 being about a lady who can do some of this. In her case, more of a problem than a gift, at least for the first half of her life. 

Given that neither Bing nor Google turned up much that was relevant for the search key ‘counting pebbles beach’, maybe the best one can do is hope that someone comes forward who can count the pebbles on a beach.

One better, I am encouraged by a report in chapter 3 of reference 5, of a pair of twins, George and Charles, severely handicapped in other ways, who, when a box of matches was dropped on the floor, were both able to count the matches, without doing much more than seem to glance at them, and shout out the number – which turned out to be correct.

Figure 19

And then there are the people, again severely handicapped in other ways, who, by dint of studying something like the perpetual calendar (from Wikipedia) snapped above, seem to be able to train their brains to say the day of the week dates for great chunks of years – with a weakness being the rather odd bounds that these chunks tend to have, perhaps 50 years, perhaps 500. And I don’t think any of them can cope with the switch from Julian to Gregorian calendars in 1582. Puzzling out the calendar above is left as an exercise for the reader.

So maybe a relatively easy next step – for someone with the right equipment – would be to try to train a  neural network to count the pebbles in something like the left hand panel of Figure 15 above. If a brain can train itself to do far-fetched things of the savant variety, a modern neural network ought to manage to count pebbles. We could worry about exactly how it was doing it afterwards.

Figure 20

PS: both Bing and Google turned up lots of hits about aids to teaching children to count and about collecting interesting pebbles. Slightly nearer the mark, Google turned up something about sampling pebbles on the beach at Robin Hood’s Bay in order to produce an analysis of roundness. A place we last visited perhaps thirty years ago. Top right here. A place perhaps better known for the distinctive white domes of the nearby RAF Fylingdales, on the right as to head up to Whitby. Bottom left here.. A place which I was able to run down as Wikipedia provided me with the coordinates to feed into gmaps. Wikipedia also alleges that Serco are mixed up in this bit of critical national infrastructure, the people who try to do tracking & tracing, although you would not guess that from the write up on the RAF web site. But I have learned that the station commander is a Wing Commander, Commander in senior service terms, one down from a captain. Very much a hot seat should the cold war start to run hot again.

References

Reference 1: https://psmv4.blogspot.com/2020/01/on-as-aspect-of-attention-and.html.

Reference 2: The savant syndrome: an extraordinary condition. A synopsis: past, present, future - Darold A. Treffert – 2009.

Reference 3: https://psmv2.blogspot.com/2013/04/remainder-shelf.html. 

Reference 4: A Dynamical Analysis of the Suitability of Prehistoric Spheroids from the Cave of Hearths as Thrown Projectiles - Andrew D. Wilson, Qin Zhu, Lawrence Barham, Ian Stanistreet, Geoffrey P. Bingham – 2016.

Reference 5: Extraordinary People: Understanding savant syndrome – Darold A. Treffert – 1989.

Reference 6: http://psmv4.blogspot.com/2020/06/some-preliminaries.html. 

Reference 7: https://www.auctionschools.com/. 

Reference 8: https://www.youtube.com/watch?v=IBEGo3QpwT0. 

Some preliminaries

Figure 1

This by way of building some infrastructure to support the counting of pebbles on beaches, the subject a post to come. In three parts.

First, to help with saying which pebbles are to be counted. Our starting point here is what can be seen by someone standing on the beach – or what might be seen in a large format photograph of same, taken from above. We get to count sets.

Second, some thoughts about how the count might proceed and how it might be verified. We get to count paths.

Third, some thoughts about the various count lists – other than the usual 1, 2 and 3 one – which might be used for the count.

The counting story has been ongoing for a while now, and had been noticed, for example, at references 1 and 2. Counting pebbles is a move from the one-dimensional count to the much harder two-dimensional count. Perhaps the preserve of savants – the sort of people who have improbable skills with numbers and who are the subject of reference 3 – although that remains to be seen.

Count sets

Looking at the snap of a beach above, it is not at all clear what it is that we are trying to count. Where does it start and where does it end? What is in and what is out?

So we simplify and we start here with a circular virtual slab (S) in three-dimensional Euclidean, a slab which will serve as the container for our pebbles. A slab which is flat and does not slope down to the sea or anywhere else. No water, no grass, no sunbathers and no cliffs in our slab. But a world which one could create in real life, for experimental purposes. A for the present, a world which can be used to clarify what it is exactly that we want to count.

Figure 2

This slab is defined by: {<x,y,z> | x^2 + y^2 < SR^2, 0 < z < SH}, where SR (slab radius) and SH (slab height) are positive real constants, with SR a good deal bigger than SH. So a relatively thin slab, defined here as an open, convex subset of three-dimensional Euclidean space. The plane below defined by {<x,y,z> | z=0} is called the base. Gravity pulls things above the base towards the base and our pebbles sit on the base. The plane above defined by {<x,y,z> | z=SH} is called the canvas, which is used to describe what we actually see in the way of pebbles. A projection, of sorts, of the heap of pebbles onto that canvas.

This slab can be arbitrarily large, with the zone of interest being well inside. There are no interactions with the side boundaries or with the canvas. Our pebbles are piled up on the base of that interior zone, well away from the sides, well below the canvas above, and are assumed to be at rest, stationary, in equilibrium.

Let X be a small, open, convex subset of S. Let the surface of X be smooth, without corners or sharp edges. At the same time that surface is rough, so that there is plenty of friction between touching stones. They do not slide around on each other too much. We further suppose that they solid, rigid and opaque – and that they are the same all the way through, that they have uniform density, this last to simplify considerations of stability. Quite like the mostly flint pebbles to be found on our beaches.

Then we say that the inner diameter is the diameter of the largest sphere than can be placed inside X and the outer diameter is the diameter of the smallest sphere which encloses X. The roundness of X is the ratio of inner to outer diameter, that is to say the quotient of inner and outer diameter, a number less than or equal to one. Equal to one in the case that X is an open sphere.

We further specify the minimum outer diameter DI, the maximum outer diameter DO and the minimum roundness DR. DI, DO and DR are positive real constants, with DO > DI and DR < 1. DI might be one centimetre and DO might be ten centimetres. DR might be one half, or as a percentage 50%.

We constrain our pebbles to be subsets of this sort.

Figure 3

We have a finite population P of pebbles. Pebbles are solid and so may not intersect but they may touch, theoretically at a point or points, but practically speaking over a small area. They must be in a physically plausible configuration, with the population as a whole resting and at rest on the base. This is illustrated in the snap above, of a small part of our slab, taken from the side. The configuration left is implausible because one pebble is suspended in mid-air and another is in an unstable relationship with the one it is resting on. The slightly larger configuration right suffers from neither defect.

We now think about looking down on our pebbles from above. What can we see? What is there to be counted? 

Figure 4

We say that pebble A occludes pebble B, if, roughly speaking, pebble A covers all or part of pebble B, as viewed from vertically above. So, in the snap above, the blue pebble, shown for clarity at 20% transparency, partially occludes the red pebble below. Note that while our two pebbles might both be convex and smooth, what it left of the red pebble after occlusion is neither convex nor smooth.

Sometimes regions will touch tangentially, without there being any occlusion. This can happen, for example when there two identical, spherical pebbles resting on the base. It will seem to happen when the two pebbles are both more or less spherical and both of more or less of the same size. 

Figure 5

Note that occlusion is not a very comfortable relationship. It is not transitive, as it does not follow that if G(reen) occludes B and B occludes R, then A occludes R – even in the perfectly ordinary scenario sketched above.

Figure 6

Indeed, it is uncomfortable to the extent of allowing G to occlude B, B to occlude R and R to occlude G – although Powerpoint is not keen, and one has to resort to trickery to get the snap above. Although to be fair, one has to resort to trickery, to unusual configurations of real pebbles to break transitivity.

The result of this is that the occlusion relationship does not define a (partial) order. We cannot talk about highest and lowest elements.

Other relationships we might use to describe the relation between pebbles, like ‘touching’ or ‘near’ are not transitive either, in fact much less transitive than occlusion.

Figure 7

We suppose that what we can see can be represented as a finite population C of two dimensional, open regions on the canvas, derived by projection up from some of the pebbles below. C is the count set and, as will be seen below, the objective is not to count the number of regions in C, rather to count from C. Such a count should be exclusive, any one region is counted just once, but it will rarely be exhaustive.

The idea is that C stands for what can be seen from above on some particular occasion, what gets projected onto the retina. A member of C might be the projection of an entire pebble onto the canvas (Y), or there may be occlusion (X and Z). And such occlusion may result in one pebble being represented by more than one region. Note that in the sketch above the pebble below Y is completely occluded and can never appear on the canvas. And the two small pebbles right have not made it to the canvas either: they are too small to be seen, to be noticed, given the prevailing lighting conditions.

The regions of C will be open and connected. They will not necessarily be convex - but they are still reasonably sensible shapes and they will not intersect each other.

Figure 8

Figure 9

Some more of the possible permutations are sketched above. Although, if one was being picky, one might say that the blue pebbles in the top half of the second sketch do not conform to the roundness rule we started out with.

Every count set will contain a root region, approximating to the position of the viewer, above the canvas. We suppose the root to be roughly in the middle, in some sense or other, of the count set.

Generally speaking, where we have a large pebble X, which is not occluded and which is reasonably near the root, there will be exactly one region in C for X.

We expect C to be complete in the sense that if a region X* of C is derived from pebble X of P and pebble Y partially occludes pebble X, then there is a region Y* in C which is derived from Y and which touches X*. However, C is incomplete in the sense that the converse is not necessarily true. We do not necessarily see stuff down in the depths.

 

Figure 10

In this slightly more complicated example, the blue pebbles are completely occluded and will never make it to a count set.

Figure 11

While in this version of much the same configuration, the blue pebbles are not completely occluded, but it is unlikely that they will make it to many count sets.

Note that things will get more complicated in three dimensions that they are in two.

Note that in coming up with our count sets, we have defined what might be seen in an exact, geometric way. Whatever it is that a brain does may not be exact in this way at all. It might, for example, tidy things up a bit to stop the image getting too complicated.

Count paths

We now turn to count paths. One-dimensional structures drawn in a two-dimensional world.

Figure 12

Let us suppose that we have a population, count list C which is a reasonable representation of the stones snapped above.

Let the large, grey stone, bottom left be our root, our start point. We then define a path from that start point. A path which might go in any direction, but which in this example will run roughly east north east from the start point, that is to say it heads roughly for the opposite corner.

A path can be expressed as a sequence: (P(i) | i=1..N) where each P(i) is a region in C and N is the number of regions in the sequence. For each i greater than one, we must have it that: first, P(i) touches or nearly touches P(i-1); and, second, P(i) has not been used before. The idea being that the eyes fix on one region, then shift to some neighbouring region and then the brain increments the count – possibly audibly – with the brain also keeping track of which regions in the vicinity have already been used. So rather more for the brain to do than when counting the floors of a tower block.

It is the activity which we are interested in here – the fix, shift and count – rather than determining how many pebbles there might be. Rather how many pebbles have we counted.

Note that as things stand, a path might easily get itself into a cul-de-sac, where all the regions to hand have been used.

Experiment suggests that this works best when the eyes are more or less smoothly and evenly tracking across the pebbles, with the trace of one such experiment illustrated below at Figure 14 below. And as there is a direction for any one path, by default maintained for the duration. An arrangement which makes cul-de-sacs quite unlikely.

Figure 13

In the snap above, an enlarged version of the bottom left hand corner of the previous snap, we have superimposed part of the current definition of regions, C. We are at the root region, region 0, north is up and the task is to decide which region to move to. Given that we are tracking roughly ENE, this suggests regions No.1 to No.5. Distant regions like No.6 tend to be avoided, without being excluded. We propose a probabilistic algorithm which, roughly speaking, chooses among the regions in the east to north east sector, about one eighth of whole gamut of directions, with decreasing probabilities as follows: No.2=No.4, No.1=No.5=No.3 and last No.6. No.2 and No.4 are preferred because they are contiguous and roughly in the right direction. Then we have the regions either off the preferred direction or a little further away. Then last No.6, good on direction but bad on distance. A sort of guided random walk, for which we dare say there is a technical term.

Figure 14

We carry on, perhaps yielding something like the path snapped above. Which is all well and good, we have maintained a count. But while we have length, we do not have area. We have not covered much ground. What we really want is a broad ribbon or band rather than a line. Something more two-dimensional than one-dimensional. Doing the whole expanse is not likely to be possible, but we ought to be able to manage a ribbon.

Figure 15

We think the way forward is a sort of zig-zag, with the zigs and zags being at right angles to the intended direction of travel and with the zigs and zags not being so long that the brain loses track of the regions which have been counted at the other end. So in the snap above, when we get back to the point β, we will not have forgotten which regions we counted when we were at α. Clearly the difficulty of this will, other things being equal, increase with the length of the zigs and zags. The ground we cover, the more the brain has to do, the more it has to be able to hold in working memory, or wherever it holds this sort of information.

Now we want some measure of how much or little has been missed out from our ribbon. How many regions our zig-zagging have missed out, particularly from the centre of the ribbon. Then how many that we have counted twice – or even more times. One could perhaps do something by drawing the axis and then weighting errors inversely with distance from that axis. A least squares fit to some low order polynomial?

Some sort of measure of how close together the points on a path are? With tight zigzags being good, that is to say close.

Noting that while we might want such a measure, we wonder whether the brain bothers with such a thing.

These snaps have been made using one of the marking tools in Microsoft Snip & Sketch while counting pebbles on the screen, which requires careful work with the (usually) right hand and which is at some remove from what we are really trying to do. So we would also like some less intrusive way to make a record of our path from the root, without that record being visible as we go, all to apt to cover up that which we are trying to focus on. For the red trace in the snaps above to be created off-stage, for inspection afterwards rather than in real time, as we go. Inspection afterwards which would enable us to work out how well we had done, how many pebbles we had missed and how many pebbles we had counted more than once.

Figure 16

But how do we do that, without leaving the comfortable world of Microsoft Office? Could one get the right hand to track movement of the eyes across the pebbles and trace a polygon, what Powerpoint calls a ‘Freeform: shape’, something like that snapped above, but with Powerpoint transparency set to 100% so that you couldn’t see it? We have not yet worked out how to set the defaults to do that.

Bearing in mind that the business of getting the hand to track the eyes in this way is taking a lot of brain cycles away from the eyes and from the count, interfering with the very processes which one is trying to record, although the interference would probably reduce with practise. Would modern eye tracking tools, the sort of thing sold by the people at references 4 or 5 be accurate and unintrusive enough for present purposes?

Count lists

The count list is the list of names for numbers. The count list is what you recite, silently or out loud, as you focus on successive elements of the count path.

Figure 17

Most of us, having been so taught when quite young, have got so used to using the Arabic numbers for these purposes, that we forget that there are other ways of doing things.

Most people find it convenient to use the names for these Arabic numbers. The first twenty one positive numbers being given at the top row of the snap above. This sort of number has a lot of advantages when it comes to doing sums, still a good thing even now that we do a lot fewer sums by hand that we did fifty years ago. But it is not so good when it comes to counting out loud, as the numbers rapidly get quite long and it is hard to maintain, say, a rhythm of one number per second, once one gets past a hundred or so. There are too many syllables.

One way to deal with this is to name the numbers in the same way as they are written, so ‘one two three four’ rather than ‘one thousand two hundreds three tens and four’ or ‘one thousand two hundred and thirty four’ for short. Not much used in ordinary, everyday life, maybe because the loss of redundancy increases the error rate, but something else to be experimented with in the present context.

Figure 18

The second row is how Excel does columns. There it is convenient to use Arabic numbers for rows and upper case letters for columns. So B23 is the second element across in the 23rd row, simpler and easier to work with than something like ‘2-23’. With the snap above showing what happens when you get to the 27th and the 703rd columns. So more or less unlimited, just like Arabic numbers – but one would need to experiment to find out whether they worked well in the present context. In any event, fine for the smaller numbers.

A variation on Excel is the 26 code words of the NATO phonetic alphabet: Alfa, Bravo, Charlie, Delta, Echo, Foxtrot, Golf, Hotel, India, Juliett, Kilo, Lima, Mike, November, Oscar, Papa, Quebec, Romeo, Sierra, Tango, Uniform, Victor, Whiskey, X-ray, Yankee, Zulu. The words have chosen for their clarity and exclusivity – and the misspellings are entirely deliberate. Clarity which might also help with keeping one’s place in the count – provided one does not want to count beyond 26 that is.

The fourth row is Latin Numbers, represented by short sequences of upper case letters. Not much to recommend them in the present context.

The fifth row is binary numbers, the ultimate fate of numbers in computers. Not much to recommend them in the present context either.

Figure 19

 

The sixth row is a bit more promising, but suffers from the important defect that, not being in regular usage, it has to be learned. If we were starting over, it might be a good candidate. It might even be convenient for a counting hobbyist. A structured number, with the letter giving the tens and the digit giving the units. We use the digits 0 thru 9 to represent the numbers 1 thru 10 for the convenience of the name always being two characters – perhaps not a convenience here, given that we are saying them rather than writing them. Furthermore, the spoken versions mostly have 2 syllables and never more than 5 – unlike, for example, ‘one hundred and seventy six’ with eight syllables, common enough in the Arabic world. A system which works quite well for numbers up to 260 – enough for the counting hobbyist but probably not enough for the counting savant. The sort of person who crops up in the book at reference 3 and who can very quickly count the number of peas that have been thrown on the table, seemingly just by looking at them.

For the moment we stick with Arabic numbers, base 10, although both saying the digits and the scheme just described have merits and might be worth a try.

Conclusions

We have defined count sets, sets of pebbles – represented by what we have called regions – which can be the subject of counts. Sets with one member designated the root.

We have defined count paths from the roots of those sets.

We have looked at count lists. So what we are doing is reciting a count list as we travel along a count path. With the recognition that the counting might be as important as the answer.

Noting in passing that children learn to count before they learn the point of counting, that the idea is to know how many apples there are. You can read all about this side of things at reference 6.

We think we have done enough to be able to get back to the beach.

References

Reference 1: https://psmv4.blogspot.com/2020/01/on-as-aspect-of-attention-and.html.

Reference 2: https://psmv4.blogspot.com/2020/02/kuss-quartet.html.

Reference 3: Extraordinary People: Understanding savant syndrome – Darold A. Treffert – 1989.

Reference 4: https://eyeware.tech/.

Reference 5: https://imotions.com/.

Reference 6: Re-visiting the competence/performance debate in the acquisition of the counting principles – Le Corre, M., Van de Walle, G., Brannon, E. M., Carey, S. – 2006.

Counting

More counting in the margins of a talk at the Royal Institution last night, to be reported substantively in due course.

Usual drill, with me sat in the right hand part of the middle block, maybe two thirds of the way up. The idea being to count the people in the left hand block. A block which is a trapezium in shape, with thin edge at the bottom. The two sides clearly marked by stairways, the back clearly marked by the back wall. So easy enough for the brain to know whether any given person qualifies or not.

Also usual drill, in that I counted the numbers in the dozen or so rows a few times, working up, until I judged that I knew them well enough.

Then, still working up, I compute the cumulative count, saying to myself, for example, '41 plus 11 makes 52', where 41 was the cumulative count so far and 11 was the count for the next row. Articulating the sum in this way seemed to hold the current cumulative count in working memory for long enough for me to recover the count for the next row, pretty much lodged in memory somewhere by the first part of the operation, and to do the necessary sum. At least partly a question of getting into the rhythm of it.

It did require concentration for the minute or so it took to complete the count, but it did seem to work, rapidly converging on a stable count of 74 or 75. Only slightly confused by the arrival of a few late comers, it getting quite near to the time for the off.

Don't know why working up should work better than working down. But I do know that counting this number of people placed randomly, rather than in orderly rows, would be more or less impossible, at least for me. Maybe one needs to be autistic to be able to do that, the sort of person that goes in for counting the pebbles on the beach.

Group search key: jka.

Series 2, Episode VIII

Polly woke up feeling a bit bossy the other day and thought that she and her friends, Pedro and Yuri, would play school. She would be the teacher and they would be the pupils, this despite Pedro being of mature years, not to put too fine a point on it, rather older than she was.

Polly asked Grandpa if she could borrow the washers he had collected, augmented by a very large washer from the Ewell by-pass since the cache noticed at reference 2. The very large washer was for her. She then arranged fifteen of the washers like the desks in a class room and told Pedro and Yuri to pretend to be all the pupils by standing on two of them. And once they were all standing comfortably, Polly could get going, get into her stride.

Baby Bear kept out of the way and tried to snooze.

The project for the day was to be to arrange the remaining washers into piles of five and then to work out how many there were altogether, not counting the one she was standing on and not counting the ones standing for desks. Pedro and Yuri looked a bit blank when Grandpa peeped through the hatch, but he never got to know what happened next. He was too busy worrying about how you might go about counting the stones on a beach, on which more in due course.

Reference 1: https://psmv4.blogspot.com/2020/06/series-2-episode-vii.html.

Reference 2: https://psmv4.blogspot.com/2020/06/roadside-rape.html.

Group search key: wwwy.