A Reconciliation of the Linear and Circular

Posted on 19 October 2008


The concepts of linear and circular are root metaphors in many worldviews. For example, linear and circular provide root metaphors for the conception of time. The linear and circular are fundamentally different in several respects. The circular, for example, has the quality of recurrence while the linear has the quality of infinite extension without repetition. The presence of these contradictory root metaphors in different worldviews raises the question of which one is correct, or, if both are somehow correct, how they can be reconciled.

What appears paradoxical from limited perspectives often has a straightforward reconciliation from a more comprehensive perspective. The linear and circular, I will argue, can be integrated through the mathematics of projective geometry. The circle can be seen, in a precise mathematical way, to be identical to the line. This model provides a way to see, without contradiction, how a circular conception of time (eternal recurrence) is equivalent to a linear conception of time.

For the following discussion, the reader is referred to the figure to the right which shows a blue circle horizontally bisected by a red line. A point on the bottom of the circle serves as a pivot point for a black line that freely rotates through the pivot point. At any particular non-horizontal angle of orientation, the black line intersects exactly one point on the horizontal red line and exactly one other point on the circle. This creates a one-to-one correspondence between points on the circle and points on the line. The one exceptional case occurs when the black line is exactly horizontal. It then intersects neither the circle nor the horizontal red line. The pivot point, therefore, corresponds to no point on the horizontal red line. The pivot point is the singular point of projection that is not itself projected.

In mathematical terms, this model illustrates how, despite their apparent differences, the line and circle are fundamentally isomorphic to each other. Thus, this isomorphism provides a reconciliation of circular and linear conceptions of time by showing how they are merely two ways of viewing the same fundamental structure.

As a metaphor, this model also provides insight into the relationship between circular and linear perspectives. For example, instead of drawing the red line horizontally through the blue circle, instead draw the red line vertically through the circle and select the rightmost point on the circle as the point of projection (in other words, rotate the red line and pivot point in the original figure 90 degrees counter-clockwise). The same points on the circle now correspond to points projected along the vertical red line. Similarly, any other choice of pivot point will map the points of the circle to points along a different red line. These different red lines can be interpreted as symbols of different worlds projected from the same circle from the perspective of a particular pivot point on the circle. This provides a metaphor for how different worlds may be viewed as projections from a single reality, while each world is at the same time isomorphic to that reality, and hence to all the other worlds. If the circle and line are given a temporal interpretation, the model provides a way of reconciling time conceived as infinitely extended in a line with time conceived as cyclic and recurring. The two are different ways of representing a single common fundamental structure, which we might call with Plato a moving image of eternity.

Finally, we may also observe that the circle has an underlying rotational symmetry. That is, a circle can be rotated without changing any essential property of the circle. To put it in somewhat paradoxical terms, a rotation of a circle is a change that results in no change. We can imagine rotating a circle, but in doing so we do not in fact do anything. Thus, in a sense, each point on the circle is equivalent and indistinguishable from every other point. The symmetry of the circle thus provides a symbol for the identity-in-difference of all the points on the circle. Each pivot point is both the same and different from all other pivot points.

Note: The ideas in this post are largely based upon a 2004 article by the author entitled “The Integral Sphere: A Mathematical Mandala of Reality” located at http://www.integralscience.org/sphere.html. The article provides a more detailed discussion of the model and generalizes it to higher dimensions.

Posted in: Philosophy, Science