PART I
Contents of this part
Introduction
- Playing with dimensions
- One dimensional views
- Blobs
- Not much data
- References
Contents of next part (https://psmv4.blogspot.com/2021/03/a-one-dimensional-structure-part-ii.html)
- Some processing
- Cellular automata
- Some structure
- Rather more data
- Some examples of moves in one dimension
- Additional information
- Conclusions
- References
Introduction
This post is, inter alia, a development of that at reference 1 (getting from a real world to an array of pixels on a computer), with reference 8 (occlusion, gravity and spatial clues in a painting) having been a diversion on the way. A post which perhaps falls between two stools in that it takes an interest both in what a one dimensional world might be like and in how a one dimensional model of a two or three dimensional world might work – which is not quite the same thing.
So while we can be thought to live in time in a three dimensional world, a lot of what we think about, a lot of what we look at, is essentially two dimensional. And the brain can be thought of as a two dimensional sheet of cortex. Against this background we have been looking at the sort of things that can be done in three dimensions which do not work in two. Here we go the next step and look at what can be done in one dimension.
So a lot of the modelling of the visual world that humans do takes places in two dimensions, and needs a lot of neurons and energy. Would one dimension give better value for the money?
Playing with dimensions
But first we review some of the things that carry on working as one moves down from three dimensions to two dimensions, or even to one dimension, and some of those that stop working. With some of these last perhaps being of more interest to the biologist who takes an interest in the interior of things, than to the man in the street.
First, the concept of size. If we stick to reasonably straightforward, connected objects (that is to say objects which are not made up of a number of separated parts), size and volume both work in one, two or three dimensions, although in one dimension size and volume coincide with length. In two dimensions we have diameter and area; in three we have diameter and volume. Again, in one, two or three dimensions we can say how far apart two objects are.
So far so good. However, objects in three dimensions cannot generally move through each other, but, geometry permitting, they can move around each other. The same is true in two dimensions, but not in one. There is no possibility of one object moving around another in one dimension. You are either stuck with the arrangement of objects you start with, or you have to get up to tricks.
Vision degrades quite fast too. So while we can see the world, we can see a two dimensional picture and we can see a one dimensional line – things are not so good if we are actually inside those diminished worlds. In two dimensions, assuming that vision works along the same lines as it does in three, one would see a one dimensional line, not nearly as informative as the two dimensional image we actually get. And in one dimension one just gets a couple of dots, one to the left and one to the right. A worm’s eye view of the world.
Then there is the question of colour, which might be characterised here as a property of an open disc on a reasonably smooth, two dimensional surface, possibly a plane, mathematically {x : |x-α| < β} where α is a point on the surface and β is some small constant, but not arbitrarily small as we must have β > ε, where ε is some small positive number, the lower limit for the perception of colour. A sensible closed loop in a plane will enclose something which can have colour. Colour – whether just black and white or the full gamut of colours – is an essential part of any two dimensional image. But being pedantic, nothing of the sort is possible in one dimension. A line must have some substance if it is to have colour.
A work around, at least for modelling purposes, is to depict the line as a thin band. Such a thin band can have colour, colour which can be used to distinguish one line from another or one part of the line from another part.
But there is at least one property which works well in one dimension but which does not work in higher dimensions; and that is the property of order. Once we agree on direction, for any two distinct things on a line, we can this one is before that one, or this one is after that one. Simple statements of this sort are not possible in two or three dimensions.
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| Figure 1 |
Then moving from things in spaces to biology, and starting with A & B in the figure above, we see that while we cannot get from A to B directly, be they ever so close, at least in two dimensions we can go around the green blob to get from A to B, the pale blue being some medium in which objects of interest can move reasonably freely. But this may not be convenient and what we can’t do is go through the green line without cutting it in half. While in three dimensions, two one dimensional lines can cross over, close as you like, but not actually interfering with each other, as at C. A difficulty which makes the sort of pipework needed to run, for example, a three dimensional human, more or less impossible in two dimensions.
And as already noted above, going round is not an option at all in one dimension: you need the additional dimension to make way for the move. All your objects are stuck in whatever relative positions they start with.
Note also the curious image processing artefacts at D. First, the holes in the boundary are made with small squares with patterned interior, intended to match that outside, but no boundary. In Powerpoint, these squares are invisible, but here in Word, where they are rather smaller, they are visible. Second, one gets horizontal, flashing pale stripes in the pale blue background when one pages Word up and down. Three of them, evenly spread down through the open black rectangle.
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| Figure 2 |
Elaborating on C we might have the above in two dimensions. Which to the human eye is a reasonably clear depiction of a three dimensional situation, with the vertical pipe running above the horizontal pipe. So we can depict something in two dimensions which we cannot actually do in two dimensions – with the brain joining up the dots.
Then in three dimensions, we can have closed surfaces, like the surface of a sphere or a torus (otherwise the sort of doughnut which has a hole in the middle), where outside, boundary and inside are all well defined and the outside does not touch or otherwise interfere with the inside. Furthermore, the boundary can define an interesting shape, perhaps something much more complicated than a sphere or a torus, perhaps some plant or animal. This still works in two dimensions, where we can have closed loops which do something of the sort, with G segregated from H by a boundary fence, more or less impassable, depending. And while such a closed loop cannot delineate the space occupied by some plant or animal, it can delineate a recognisable, two-dimensional, version of same.
In one dimension, nothing of the sort is possible, one just has points along a line.
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| Figure 3 |
In three dimensions again, many animals have a tube running from one end to the other called the alimentary canal, with the entrance and exit to the tube both being small relative to the surface area of the animal. Its overall structure is not much disturbed, at least to outward appearances. While in two dimensions such a canal cuts the animal in half, as illustrated at E & F in the first figure, Figure 1, above. Something like a lung is however possible, as the air goes out the same say as it comes in, rather than a one way traffic in one end and out at the other. The host object might have been rather cut about by the branching structure making its way in, but it is still in one piece.
Even just the hole is not so good in two dimensions, with a gap in the one dimensional boundary of the blob at D being much more drastic than a small hole in the surface of, say, a sphere. And two such gaps breaks the boundary into two parts. How are those two parts going to be held together, how is the integrity of the two dimensional blob as a whole to be maintained? How do we stop stuff leaking in or leaking out in an uncontrolled way?
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| Figure 4 |
We talk above of mediums in which objects can move about. As will be developed below, object may be simple, as the smaller red object lower left, or complex, as the larger pink object upper left. Complex objects have a boundary and an interior, an interior which is likely to contain other objects. We can speak of outer and inner mediums. These mediums are often fluids, that is say a medium can flow around things, things can flow through a medium, more or less easily, and this is what happens when objects move about, perhaps in the left hand panel of the figure above.
We distinguish compressible fluids, gases like air, from incompressible fluids, liquids like water. So in the middle panel, when sliding the pink object up the left hand side, there will be plenty of resistance. With more resistance as the pink object grows, get closer to the right hand side, and more resistance from water than from air. While in the right hand panel, where the blue object is more like a piston, sliding is not going to be an option at all in the case that the medium is water. In this, three dimensions and two dimensions have a lot in common. But once we drop down to one dimension where the boundary of an object drops down from a closed loop to two points, we only have what amounts to the right hand option in the figure above: the dark blue object can jiggle around in its light blue medium, provided that medium is air-like rather than water-like.
So even if we stop here, it is clear that a lot of the options and devices available to a three dimensional blob, particularly those to do with communications and interactions, are not available to a two dimensional blob. Maybe all this really does has something to do with the fact that most vaguely two dimensional life forms are fairly elementary in both structure and function, even when extensive in space. And very few of these options and devices are available to a one dimensional object.
Not relevant here, but we also often suppose that our fluids are well mixed. That they might be a mixture of chemicals, but they are well mixed chemicals and the mixture is the same, pretty much everywhere. An assumption often made, and not always justified, about human blood.
One dimensional views
So life as we know it is not going to get on very well in one dimension. But what about a one dimensional view of a three dimensional world? A one dimensional view which can be informed by the full power of the eyes and brain? Which can be labelled with stuff from supporting layers, in the way of LWS-R of reference 5?
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| Figure 5 |
So in lower part of the figure above we have a sketch, perhaps more a diagram, of a one dimensional structure, made up of seven objects (A thru G) and two gaps (X and Y). Objects are distinguished one from another by being coloured – we suppose a narrow band rather than a full-on one dimensional line which is hard to colour. Gaps are marked with the null colour, here white. If one object is behind another, as A is behind B, with B occluding the right hand edge of A, then the right hand edge of A is broken using the convention of a narrow gap followed by a narrow band of colour. In the same way, B is behind C, D is behind C and F is behind E. In this way our one dimensional visualisation of a three dimensional scene is able to carry more than information about direction.
Not so very different from what we might do in a two dimensional visualisation of the same scene, as shown in the upper part.
But there are problems. If our one dimensional ‘real’ view is taken from a line across the scene, as at P-Q above, things are apt to be missed out. Exclusion and inclusion are far more capricious than they would be in the larger, more capacious two dimensional view. And if one does some kind of aggregation, the view gives a reliable indication of direction but no longer give a reliable indication of altitude and the relative positions of objects are apt to be confused.
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| Figure 6 |
Another problem is the absence of shape. In the figure above we have reduced a zebra to the upper black and white dashed line. How are we supposed to know that we have a zebra rather than a line of fence posts? What happens if we take our line at leg level rather than at body level, as in the lower line, where what we get might vary wildly with the precise position of our line? We can apply labels, after the fashion of LWS-R, but the direct, declarative experience of one dimension is rather impoverished compared with that of two.
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| Figure 7 |
Nevertheless, there are some scenes where a one dimensional view works, where the focus is naturally more or less one dimensional. So if we have a queue of cars or a queue of people, the view is essentially one dimensional. Or perhaps we a an East India Company man of the late 18th century anxiously scanning the horizon for French privateers.
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| Figure 8 |
And if we move from things which do well visually, to things which are more conceptual, like the age structure of the population, one dimension does just fine. Just think of the ubiquity of line graphs, where position on the vertical y-axis is used instead of the colour we have used above.
Blobs
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| Figure 9 |
We think of the two-dimensional figure above as a hierarchical collection of objects, with objects free to grow, to shrink and to move around in their host media, but generally speaking without much disturbing others. Simple objects are just undifferentiated blobs, in the figure above, recycled from reference 1, the small blobs of red. Complex objects have boundaries – otherwise skins or integuments – and inner mediums, inner mediums which may host other objects.
Note that an empty complex object is not the same as a simple object.
The world as a whole, delineated in the figure above by the open black rectangle, consists of the world medium, stippled pale blue in the figure above, with objects floating around in it. Roughly speaking, media are see-through fluids while objects are solid. Elementary objects are opaque solids.
Intimate interactions between peer objects are tricky, involving, as they must, high-risk breaches of boundaries. But our world is unlikely to be very interesting without them, with just the waxing and waning of the various objects within a fixed, hierarchic structure. We need some mergers and acquisitions.
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| Figure 10 |
Furthermore, the things in our two dimensional world live in two dimensional Euclidean space. There are sizes, distances and directions; sizes, distances and directions which matter. We are talking about something a bit more complicated than the simple, rather flat, tree structure at Figure 10 to which Figure 9 can be reduced – simple except for the bit where we have to distinguish terminal nodes which are complex (pale interior) from those which are simple (solid interior). The real life position of the eleven top level nodes is, for example, poorly caught by their ordering in Figure 10, necessarily rather arbitrary.
Perhaps more or less two dimensional blobs of bugs – be they animal, vegetable or something else – moving around on the surface of a growing medium in a Petri dish.
Then is there any kind of one dimensional equivalent? Can we do one object containing another in one dimension? Can we do one object moving around another? Or through another? Are there biologically plausible workarounds to the various difficulties mentioned at the outset?
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| Figure 11 |
Another thing which is easier with a simple one dimensional structure, is the visualisation of change over time. So in the figure above, the simple one dimensional structure left, with just five elements, can readily be extended to the right to show change over time. Indeed, one might think that such one dimensional structures only become interesting when there is change in time. Why the big blue bulge in the middle? What about the split in the red to the right of the bulge, where it seems to bounce off the ochre? The static view left does not compel in the same way at all.
Not much data
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| Figure 12 |
We make a start in our one dimensional world with a strictly ascending, finite sequence of points on the real line; that is to say P(1) … P(N+1), where P(i) is always less than P(i+1) for all i from 1 to N.
Let the P(i) be bounded below at zero and bounded above at B.
This can be thought of as a touching sequence of N intervals on the real line, I(1) … I(N). We call such a sequence of intervals a segment.
We leave aside the complication of the point boundary between two intervals. And we might have it that a point cannot be coloured, that only an interval, albeit a short interval, can be coloured. We might set a lower bound on the length of intervals that can be coloured – with that lower bound on a computer screen being the width of a pixel, perhaps a hundredth of an inch.
Let C be a set of not more than 256 distinct colours, probably a lot less than that, colours which can be coded in one eight bit byte and one of which is called the null colour, usually in practise white, the physics of the matter notwithstanding. We colour each of our intervals with one of these colours.
By associating each colour with a number in the range [0..255], null with zero, we have an order on colours and we have a distance between colours - both, sadly, spurious. Colours do not arrange on a line interval very well, with Microsoft using three numbers in the range [0..255] to do the job and other people like colour wheels, wheels which do not usually include black or white. So we cannot do numerical things with these colours, like taking an average or saying that this colour is near that colour. But we do suppose that the eyes are quite good at discriminating between two different colours when place side by side. And because we only have a finite number of colours we do not have to deal with one colour merging into another.
An alternative, attractive from a numerical point of view, would have been to use a grey scale, rather than a decent repertoire of colours. But we preferred the more attractive visualisation offered by such a repertoire.
Null intervals are likely to attract special treatment in the rules table, for which see below.
I(1) and I(N) must have a non-null colour, possibly but not necessarily the same colour, in the figure above blue and black respectively. The line before I(1) and the line after I(N) takes the null colour. That is to say far left and far right in the figure above.
I(1) and I(N) are called the left and right terminal intervals respectively. The other intervals are called interior intervals. In the figure above, the two interior intervals taking the null colour are shown as open blue rectangles. They are called null intervals. In terms of Figure 9 above, null can be thought of as the world medium.
We wondered about whether to allow adjacent intervals to take the same colour, with series of identical intervals seeming to crop up in a reasonably natural way. However, our world is supposed to be one dimensional, and how can one interval be distinguished from another in the absence of a distinguishing feature? We opted for the rule that adjacent intervals must take different colours, with a series of identical intervals, each separated one from another by a short interval taking the null colour, being one way of doing the series first mentioned.
Note that this is not a problem for the cellular automata which we discuss below. There, the one dimensional structure on the real line is replaced by a one dimensional array of individual cells, and adjacent cells taking the same colour is not an issue, any more than it is with the pixels on a computer screen. But, for the present, we just have the real line, not conveniently parcelled up into cells or pixels, so adjacent intervals taking the same colour is a problem.
Note also that while, for the purposes of presentation, we show white gaps between the intervals in these figures, they are not really there. When we include the null intervals, the interior intervals cover the space between the two terminal intervals exactly once. That is to say, no gaps and no overlaps.
As noted above, if we were in such a world, rather than looking at it from the outside, we could not see much at all, even supposing the null segments to be transparent and that there was suitable illumination. A visual image would be limited to two coloured dots, one to the left and one to the right: not much use at all. Sound might give us something, but for the most part we would be reduced to whatever we might get from touching our two neighbours, right and left. Which, we suppose, is about where the earthworm is.
These segments can be readily translated into a one dimensional array on a computer or on a computer screen, say 2,000 elements or pixels, with our intervals being constrained to be well away from the beginning or end of this array. A translation which approximates to the map: M(x) = x × 2,000/B – with the value of M being constrained to be a positive integer which is less than or equal to 2,000; that is to say a position in our array rather than a position on the real line. We will return to these pixels in due course, but for the moment we put them aside.
References
Reference 1: http://psmv4.blogspot.com/2021/01/layers-of-approximation.html.
Reference 2: https://en.wikipedia.org/wiki/Turing_machine.
Reference 3: A new kind of science – Stephen Wolfram – 2002.
Reference 4: https://en.wikipedia.org/wiki/Peano_axioms.
Reference 5: https://psmv4.blogspot.com/2020/09/an-updated-introduction-to-lws-r.html.
Reference 6: https://www.vectorstock.com/.
Reference 7: Painters centre one eye in portraits - Tyler, C. W. – 1998.
Reference 8: https://psmv4.blogspot.com/2021/03/relationship-cues-in-painting.html.
Reference 9: [next post/last post] https://psmv4.blogspot.com/2021/03/a-one-dimensional-structure-part-ii.html.
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