Lucretian Geometry (a speculation)

Is an hoop just a dot by other means?

A dot and an enclosed circle are certainly more dissimilar than a dot and a connected line in space (i.e. a hoop). A circle is an outline, a trace, history’s most pragmatic abstraction, whereas hoops, loops, Möbius strips – these more closely resemble the paradox of the dot.

If you think about it, dots are simply impossible: they’re either tiny (thick) lines or filled-in circles (i.e.:“a hair or a hairball”). Dots are uncanny.

Dots are not the same as points.

In Euclidean geometry, points are the beginning realities (so called one-dimensional space), but in fact they are complex abstractions of the imagination. If you think about how lines are defined as the shortest distance between two points, this is phenomenologically incorrect since in fact the lines include – or subsume – the points themselves. In this formulation, “point” – or dot – means “end,” an intangible concept denoting the place where a thing ceases to be a thing and instead becomes a not-thing, separated, void. Following Lucretius, who posits that the world is made up only of “bodies” and “void” (and no “third things”), an end is not an abstraction but the observation of void. Consecutively, points (vis-à-vis Euclid) are ill-defined abstractions that seek to make sense of impossible dots, and in turn, impossible realities.

In three-dimensional space, ends become edges, giving further shape to the Lucretian geometry of void and furthering the groundlessness of points.

But knives do cut, you may say.

Importantly knifes do not cut on their own, independently; they cut into other things. We say a knife cuts because we see an apple or a cheese wedge bisection at the point of contact with the knife’s edge. [Furthermore, there is no knife that is not a tool (or a consumer product), hence a further abstraction.] A cut is equal parts cutting-thing and cuttable-thing.

Though it possesses more verisimilitude, the Euclidean third-dimension is in a way the most problematic because it gives body to shapes but leaves out the energy, the vitality, the presence of thing-ness. Disappointingly, three-dimensional space is less-than inert. If you think about a stationary wheel (the one on your bike in the shed), it is not a circle in the third dimension (albeit with the accidental imperfections necessarily added to bring abstract shapes into actuality, through welded metal and molded rubber, etc.). What do wheels do but spin? Balls but roll? Blocks but impede? Pyramids but erode? Three-dimensional space is still abstracted “space” and does not represent “live” space. Motion – kinematics, not time – is the fourth dimension because movement necessarily resembles living bodies – living bodies in active relationships with other living bodies. [It is Lucretius not Augustine who first insists that time is insubstantial: “Time also exists not of itself, but from things themselves is derived the sense of what has been done in the past, then what thing is present with us, further what is to follow after. Nor may we admit that anyone has a sense of time by itself separated from the movement of things and their quiet calm” (On the Nature of Things, 1. 459-463, Rouse & Smith).] The Euclidean spectrum of x, y, z, t is both abstract (as opposed to descriptive) and ideological (as opposed to universal) as it attempts to present the sense of things as a heap of sensible things extra-dependent of nothing, where in fact objects are extra dependent of precisely nothing, of void, that allow for the presence and interactivity of other objects.

A hoop then has two conceptions, one in the third-dimension and the other in the fourth-dimension. A hoop, like a circle, has no beginning or end, except when we remember that a circle does indeed have a beginning (the abstract x center) and an end (the circumference dictated by d distance from x in all directions). Hoops in the third dimension have this same finite quality to them, only with added complexities of gauge and further demarcations shaped by their bloated edgings with void.

In the fourth dimension of kinematic geometry, however, hoops begin to assemble their symbolic association with infinity and no longer resemble the Gordian knot whose complexity is merely a complicit illusion. This is because a moving hoop, a loop looping, a shaping of objects into a material blur of non-beginning and non-ending, mirrors the velocity of experience. [A model Mobius strip then is really just a three-dimensional representation of the fourth-dimensional hoop.]

When a bus darts past your field of vision, it moves away in a linear vector defined in essence by the surface of the earth’s crust. If you are on a bus darting past Overland Avenue, the earth is transformed into blurring lines of light and matter. The bus stops and the experiential hooping of earth-things comes to an end. Just as waking and sleeping are not fixed points of experience but only definite matters in the abstract (a fact also mirrored in our entire existence of being here and then eventually not), these liminal continuums mirror the experience of being hooped.

Returning to the failures of three-dimensional geometry versus actual “live” geometries, all three-dimensional objects imply a spectral dissection dictated by their coming into contact with experience (i.e. with sense and reflection) and perhaps none so illustratively as a hoop. The are two ways a subject can be presented to a hoop: as a thing-that-hoops and as a hooping-thing. As a thing-that-hoops, it is a child’s hula-hoop or a metal bracelets – but a hooping thing can be any encapsulating space of varying thickness: a belt, a car, a department store floor, the woods, etc. Life cocoons experience, but observers – others – are finite, specifically our widths, even just the space of a brain synapse, or an atom. A loop is a tube in the moment. A moment in a tube is a container. In life, the wild is counterintuitively a confined space. Our conceptions are always-already limited, yet simultaneously more complex than their abstraction attempts to conceal.

Back to the original question: Is a hoop just a dot by other means?

If you go fast enough, all shapes turn to lines, and if you are constantly moving then what’s the difference between the continual line connected in space (a hoop) and a dot? At top speeds, dots become indistinguishable from hoops, not to mention other dots. Dots – not points – are live space when they are blots, spots, smudges, spills, drops – all synonyms of human accident. The information age devalues uncanny dots and their vicissitudes because they avert the binary codes of mobility.

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Word #8: “phyllotaxis” (n.)

noun. [Botany]

  1. The arrangement of leaves or other lateral members (e.g. the scales of a pine cone, the florets of a flower of the family Asteraceae ( Compositae), etc.) on an axis or stem; the geometrical principles of such arrangement.

Below are mathematical functions for variant phyllotaxis shapes found in organic structures, and, even more fantastically, their nomenclature.

patterns