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.