Lines

In this post, we introduce another structuring device, the line, linear objects likely to be interesting in their own right, and which can also be used to express lists. For example, the alphabet or a list of places on the way to work. Noting in passing the allegation at reference 3, that the ability to work with such lists is possibly part of what we are buying along with consciousness. With this being the third post in the ‘sra’ series.

Stepping back a little, we think of geometry, guessing how much of the geometry of the shapes, positions and relative positions of objects that neuronal processing might reasonably know about.
Some objects will be blobs, roughly circular in shape, possibly convex and certainly perforation-free.

But other objects, the objects we are presently concerned with, will take the form of simple lines, that is to say line segments with two terminals, with or without the distinction of start and end. And of loops, where the start and end more or less coincide, if they are deemed to exist at all. Lines will have synaptic connections compiled into them which allow arousal to travel along them, in one direction or the other, or both – although probably not at the same time.

[Figure I]

With figure I above (click to enlarge) illustrating the shape of some of the lines and curves to come. Maybe the ‘119’ and ‘179’ of the fat lines tell us something about those lines, beyond their trajectories: think of all the properties of lines in Powerpoint. The thin green line is a distinct object from the fat pink line, being defined on a different pattern. While the assemblies bottom left and bottom right do not count as lines in the sense used here at all.

We have some rules about lines.

In what follows we suppose that P1, P2 and P3 are successive patterns on some line. And that the touch of two consecutive patterns is the side, the part of the side or the corner at which they meet. In the figure that follows, working from the left, the touch starts at four points of the underlying grid, then three and ends at two.

[Figure II]

First rule, a line is a sequence of contiguous patterns.

[Figure III]

Second rule, P2 must not touch P3 on the same side or corner as it touches P1. Our lines cannot turn too sharply. This exclusion is illustrated in figure III above.

Third rule, a pattern in a line must touch at most two other patterns. A terminal pattern will touch one other pattern, while an intermediate pattern will touch two.

Fourth rule, a line will either be simple and have two terminal patterns, be a simple line which one can traverse from one terminal to the other, from one end to the other, or be a loop with no terminals.

Fifth, most of the time, with most in some sense to be defined, the touch of P1 and P2 will be opposite the touch of P2 and P3. Opposite but not necessarily identical. As is illustrated in figure II above. We call a sequence of patterns on a line with opposite touches a segment, with the line as whole being composed of a number of segments, typically quite a small number, say less than 10.

Two successive segments share one touch, but cannot share two, this last being a consequence of the second rule.

[Figure IV]

Very roughly speaking, a line segment moves north, east, south or west on the page. To turn, the segment must be broken with a right or left hand turn. So, on the line in Figure IV above we need four turns to complete the loop. We can traverse a loop in either a clockwise direction or an anti-clockwise direction – but whether this distinction will be important remains to be seen. With the same arrangement in Figure V below, but with the line stretched out a bit, to be more like a circle than a square.

[Figure V]

Sixth rule, a line can only be a simple loop, in the way of Figure V above. We do not allow segments of a line to cross, or even to touch. Which is a variation on the third rule.

Given that a eastward line, for example, can vary between close to north to close to south, the third rule suggests that a loop needs at least three turning points, one between each of three segments, as shown in Figure VI below.

[Figure VI]

Our lines are not directed, in the sense that there is nothing to distinguish travelling in one direction rather than the other, although looking forward to layers, it would be possible to add something to mark start and end.

In which case, one could travel along a line from start to finish, travel which could generate a sequence, perhaps a sequence or numbers or the alphabet. Travel which would need some sense of direction of travel, there being two available, except at the start and finish.

Which last would slightly complicate travelling on a loop as at the start one would need to travel away from the end next door. And one would probably want to know when one gets back to one’s starting point.

Lines as containers for sequences

This requires the notion of layers, to be developed in the next post.

Lines as enclosures or borders

A loop will always have an interior, an interior which comes in one piece and without holes.
It may be possible to define this interior as an object in its own right, although this would not be possible in the case of the triangle above and we might need to allow filling in: we define as much of the interior as we can as an object, but then deem the bits and bobs left over as being part of that object, as least in some sense.

The loop plus the interior might also be an object, but this requires the concept of layers.

Or we might just raise another sort of object, an object defined by a loop, which might or might not be considered to include the loop itself and which would not be bothered by any awkwardness of shape that the line or its interior might have.

Going at it from the other side, as it were, the boundary of some objects, the better behaved objects, will be a loop in the present sense. But such an object does not need to be convex. It could be a blob with projecting arms and legs.

Processing

Looking ahead, it may well also be that our probing impulses will go a lot faster along simple lines than along complicated lines, with these last involving a lot more brain cycles. There might also be a preference for straight lines going up and down, then other sorts of straight line, then gently curving lines. I associate to the notion of momentum, a vector quantity, and wonder whether we can have some similar mechanism here, implementing arousal travelling across a sheet of neurons. Bending good, but straight ahead better.

Conclusion

We have two more structuring devices, the line and the loop, to be used in posts to come.

Reference 1: http://psmv3.blogspot.co.uk/search?q=sra.

Reference 2: http://psmv3.blogspot.co.uk/2016/12/from-grids-to-objects.html.

Reference 3: http://psmv3.blogspot.co.uk/2016/09/what-is-consciousness-for.html.

Special words: line, loop, segment, touch.

Group search key: sra. For which you can click on reference 1 above.