The Concept of the Circle

Where would we be without circles? Well, I guess we’d have to make do with polygons. But that’s ok since a circle is simply a polygon (an n-gon) with n = ∞. So plane (Euclidean) geometry, from the lowly triangle (n = 3), the minimal building block of the material world (Plato, et al.), to the infinitely sided circle (n = ∞), is consistent. And solid geometry is just the projection of 2-dimensional plane geometry (d = 2) into 3-dimensional space (d = 3)…and so on (d’ = d+1). 

So what do all n-gons have in common? They divide a plane into two distinct regions: inside and outside. What is in is in and what is out is out and never the twain shall meet. Well, that’s not exactly true. 

Every ‘normal’ n-gon (including a circle) has an (n-1) dimensional border region consisting of points that can be considered both in and out (or neither in nor out), but those points are fixed. A point inside a circle is inside forever; a point outside is forever out. 

Imagine what it would be like to live in a world where circles (n-gons) actually existed! Perhaps that’s what Hell is. Since events often occur in the context of broader events (e.g. many battles in one war), our universe would consist of a nearly infinite progression of Russian dolls. Every ‘inside world’ would be a universe unto itself, entirely sealed off from, and unaware of, whatever might be outside it. 

On the other hand, an outside world could potentially experience the existence of an inside world, but only as an interruption, as a hole in the whole. But would a point in the outside world really experience the existence of an inside world or would the hole be imperceptible…like the ‘blind spot’ in humans’ visual field? Or the experience of a patient under general anesthesia who awakes unaware of any ‘hole in time’? Would we imagine a continuous plane? Does Swiss cheese even know it’s full of holes?

If we lived in a world where circles (n-gons) were real, every definable entity (every object, event, experience, etc.) would be the functional equivalent of a ‘super black hole’:

– a black hole because its event horizon, the circle, is a point of no return; what happens in Vegas stays in Vegas. Sin City is not part of the USA any more than Vatican City is part of Rome.

– a super black hole because the circle would function as a ‘double event horizon’. In the case of a black hole, the outside world is unaware of anything inside the event horizon…but the inside world is still aware of whatever’s outside it. In the case of an n-gon, however, neither the inside nor the outside is aware of the other (except perhaps as a hole). 

Who’s afraid of a big black hole? Literally, everyone! And if you’re not, you darn well should be. Your body is ripped apart (spaghettification) and you are forever unable to communicate anything to anyone in the outside world. Hmm, sounds a lot like death!

Now imagine a ‘super black hole’. You cannot communicate with anyone and no one can communicate with you. The outside world behaves like its own black hole from the perspective of the inside world. 

The universe would then consist of two black holes with a single, shared horizon which functions as a double-sided mirror; it precludes cross border communication, it precludes cross border awareness. Everything is reflected back on itself. 

Yup, this is Hell alright! Or call it Narcissus’ Paradise? To see nothing but one’s own reflection for all eternity…I’ll pass on that, thank you very much.

Alleluia, Alleluia, thanks be to God, we do not live in a world with n-gons. Ok, that sounds crazy: I can see 100 polygonal objects from my desk chair. Of course, n-gons exist in our world!

Except they don’t. We’ve developed a massive series of hacks that allow us to model the world using n-gons while escaping the implications of total isolation. These ‘hacks’ allow us to imagine that we live in a polygonal world when in fact we do not.

Take the biological cell for example; it’s the basic structure of all life on Earth. Every cell is ‘separated’ from its environment by a cell wall resulting in a closed geometric shape. 

But cell walls are not ‘walls’ at all; they are membranes that allow molecules inside the cell to migrate out and molecules outside the cell to migrate in. Ok, but is this cross border traffic anything more than a trickle? Not unless you call a billion molecules per second a ‘trickle’! 

If the walls of cells were not permeable, no cell could live more than a few seconds and all life on Earth would vanish in a matter of minutes. Who needs nuclear annihilation when sclerotic cell walls could do the trick so much quicker?

But that’s organic chemistry. Flunked that class anyway. What about the inorganic world? Surely functional n-gons exist there, right? Sorry, no! 

First, “things fall apart.” (Yeats) Even inorganic matter ‘decays’. The polygonal books on my shelves are deteriorating, albeit imperceptibly; they are obeying the Second Law of Thermodynamics (entropy).

Second, inorganic matter consists of subatomic particles but these so-called ‘particles’ have no hardwired location. They enclose nothing and they are enclosed by nothing. In fact, it is impossible to determine the precise location of any particle at any moment. Every particle is a field and every field is cosmo-spanning. Position is a function of probability, not locale.

Well, black holes then. Surely they have impermeable event horizons! I mean, we just said so (above). But we were ‘speaking loosely’, as my grandfather used to say. In fact, event horizons are never impermeable. Stephen Hawking proved that all black holes radiate mass (Hawking Radiation)!

Does this mean that we must give up the dream of a geometry that approximates real world phenomena? By no means! We just need a different geometry, one that eliminates the bifurcation of the world into insides and outs. But how do we accomplish that? 

Step #1: Add a Mobius Strip to our model. Any space that includes a Mobius Strip is ‘non-orientable’: (1) surfaces are ‘one sided’, (2) objects have no inside or out. 

Step #2: Remove the Archimedean Property from our model. This property limits us to the set of Real + Complex Numbers. It specifically excludes infinite and infinitesimal quantities. 

Without the Archimedean Property, regions cannot overlap. C can be embedded in B (or vice versa)…or B and C can be disjoint. B and C can, in turn be embedded in A. But when B and C are disjoint subsets of A, a common metric across all three regions is no longer required for coherence. 

Each space is measured by its own metric. The size of a region measured internally need not be the same as the size of the same region measured from an external vantage point. In the long running BBC series, Doctor Who, there is a certain phone box/booth, the TARDIS, whose interior is vastly larger than its exterior. 

So, the volume of B (and/or C) and/or the volume of (B + C) can be greater than the volume of A...even though B and C are subsets of A. There is a region of A called X that is not part of B or C; in some cases the volume of X can actually be negative. If (B+C) > A, then X < 0.

The Archimedean world imagined in our traditional geometries is a mere shadow, or surface, of the non-Archimedean world that underlies it. Think of the shadows in Plato’s cave (The Republic). Or think of the ‘skin’ that sometimes forms atop a liquid.

If B and C are disjoint subsets of A, then B and C can impact A and A in turn can impact B and C. Influences flow freely and reciprocally between parts and wholes, but never directly between parts. B and C retain their autonomy at all times, but that does not mean they are locked in separate supermax cells. 

Step #3: Remove the Axiom of Foundation (FA) from set theory. FA prohibits any set from being a member of itself. 

Step #1 eliminates the concept of inside/out. Step #2 allows entities to be embedded and to influence one another without compromising their independence and integrity and without lapsing into solipsism. Step #3 makes it possible for us to acknowledge that every set is a subset of itself.  

Combining these tweaks allows us to build a model that better resonates with the phenomenal world. It enables us to account for the permeability of the cell wall, the entropic decay of ordered systems, and the existence of cosmos-spanning fields (e.g. EM and Gravity). It accounts for Karma – “What goes around comes around”, for the Great Commandment – “Love your neighbor as yourself’, and for Newton’s Third Law of Motion - “Every action has an equal and opposite reaction.”

Language is Logic’s fraternal twin. Orientable, Archimedean Geometry and the Axiom of Foundation are well suited to a language of active/passive verbs and subject/object nouns…like ours, for example.

Of course, this language is just as unrepresentative of reality as its logical counterpart. But is it even possible for a language to model a non-orientable, non-Archimedean universe with no Axiom of Foundation? You bet it is! In fact, Indo-European used to be just such a language.

Our active/passive voices are corruptions of a single reciprocating voice, the Middle Voice. The Middle Voice redefines ‘subjects and objects’. Now every subject is its own object (reflection) and every object its own subject (recursion). 

Bonus: Middle Voice allows us to talk about ‘what we talk about when we talk about love’.  (Raymond Carver) And as we all know, Love is what makes the world go round. (Carnival) No one knows better than lovers the gulf that separates a loved one’s objective exterior from their subjective interior. 

“Lord, we don’t need another (Archimedean) mountain…what the world needs now is love sweet love.” (Jackie DeShannon) But to realize that vision, we first need to overcome our addiction to mythical enclosed spaces, like circles. 

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William Blake’s The Great Red Dragon and the Woman Clothed with the Sun (1805–10) stages a vision of beings locked inside their own enclosing arcs of wings and shadow, a painterly hell of circles within circles that mirrors the isolation a true n-gon world would impose.