I’d like to make a contribution with regards to circularity/linearity, from a geometrical standpoint.  If you don’t like geometry, stop reading, or better yet, read with increased intensity.

The polarity between circle/line is one that is fundamental to many geometries – they are taken to be quite different logical entities.  Primarily this arises because of (in a move parallel to Russel’s need to introduce the Theory of Types to avoid paradox) a limited way of dealing with infinity.  Cybernetics shows us, by providing a wider view (attention to relations, recursions, and thus relations of relations, etc.) that the Theory of Types is unnecessary and points towards a more flexible and mysterious understanding of paradox.  In the same way, projective geometry provides a way of dealing with infinity that encompasses, expands, and re-frames the conventional geometric view.

Just as cybernetics acts as the field/ground out of which say, a physics can precipitate, so too many logical geometries fall out of the wider field of projective geometry; ex. Euclidean geometric relations ‘fall’ out of the more general set of possible relations when certain assumptions (starting conditions) are taken.

With respect to the line/circle polarity, projective geometry shows how these two seemingly incompatible geometric entities are in fact transformations of each other – they are recursively compatible geometric forms that are related coherently through the infinite.

If you are confused, don’t worry, because the beauty of projective geometry lies in its ability to deal clearly with process.  Try the following experiment:

Imagine a line a.
Imagine a point not on the line A.
Imagine a circle c whose radius r is made by drawing a new line between point A and line a such that they meet at a 90 degree angle. This means that line a is tangent to circle c.

Got it?  See the circle?  The tangent line a?  (Alchemically, the beginning facts are Earth.)

Now slowly, without any gaps in your imagination, watch how the situation changes as you move the point A away from line a.  That is, increase the length of the radius r. (Alchemically this is the Water element.)

Note what happens to the curvature of the circle as r increases.

Now keep lengthening the radius r.  Note that you will probably at some point start to lose your ability to ‘see’ the WHOLE situation at once, and your imagination will start to ‘skip’ from one possible viewpoint to another.  Don’t worry about this!  It is precisely what happens when we ask ourselves to stretch our point of view beyond its normal comfort zone (alchemically this is moving to the Air element).

Focus on what is happening at the point of intersection between the line a and the circle c.  Allow your imagination to see only a partial arc (attention all Brad Keeney fans) of the complete circle and its relationship with the line.

With this focus, move the center of the circle, point A, TO INFINITY.

Yes, that’s right, expand the circle so that it’s radius becomes infinite… and notice what happens to the curvature of the circle near where you are focusing your attention (where the circle meets line a).

Can you see it?  What has happened to the circle?  What has happened to the line?  Where is point A, and in which direction?

Now don’t stop imagining, but reverse the whole procedure so that point A comes back from infinity, approaches the line, and ends up comfortably near so that you can easily see the line and the complete circle all at once.

For those of you who want some additional thoughts (no need if you can do the exercise well), I offer the following (NOTE: don’t read this unless you have accomplished the above instructions, as it will otherwise potentially spoil some of your discovery).

You may also try the variation in which instead of focusing on the line, keep your view focused on the center of the circle, point A, while completing the same exercise as above…

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First of all, we can see that as the point A moves away from the line, the curvature of the circle decreases (i.e. becomes flatter).  When point A reaches infinity, the circle is coincident with the line; it BECOMES the line.

In other words, the line IS the circle, and the circle IS the line.  More specifically, lines as such are circles whose centers are infinitely far away.  Indeed, this argument about lines applies equally to planes; every plane is a sphere whose center is infinitely distant.

We see that every section of a line or bounded area in a plane is geometrically part of an infinite circle or sphere, whose center is in the direction orthogonally (at 90 degrees to) the line or plane.  In other words, if we are both standing on a line, me at point M and you at point Y, and if we both are directed to point to the center of the “circle” on which we are standing, I must point at 90 degrees to the line.  But you do the same… and we are both pointing to the SAME point.

Euclidean geometry fails at this point, and Euclid had to create a completely separate postulate that parallel lines do NOT meet.  In fact, this postulate is a defining feature of his geometry, but it is not necessary.

In projective geometry,  parallel lines meet at a point, and the point is said to be at infinity.  Additionally, when you have parallel lines, you see that they are parallel both in one direction and in the other.  In other words, to point to the point at infinity, you MUST point in BOTH directions.  If you were on train track, you would point one arm along the tracks in one direction and the other along the other direction, 180 degrees to the first.

Relating this to the line/circle polarity, we can see that the ‘inside’ of the circle fills the entire plane, above and below the line, when its radius is infinite.  Moreover, the center is equally far from the line on both sides, and is always the same distance no matter where you are on the line.

…there are many more ways that this geometrical relationship can be discussed, but I want to make sure that you are thinking metaphorically about this exercise.

That is: every circle is a line, and every line is a circle.  When we focus on my linear path from here to there, I usually do not have the perspective that I am walking along a circle, and that what lies in front of me is also behind me and vice versa.  To achieve this perspective requires a movement THROUGH infinity.  Walking the infinite circle, we realize that the distance between any two points is aribtrary – i.e. it is CONSTRUCTED by our perspective, and is not inherent in the geometry (relations are more fundamental than quantities).

The path I actually walk in the world is always a partial arc of an infinite circularity.  My uprightness itself indicates the direction to the infinitely distant center, below my feet and above my head.  My walking is always on the boundary between the inside and the outside, the exact middle between an infinitely distant point and itself.

This is just a taste; there are many more projective geometric imaginations which help recontextualize according to process rather than form.