The Holofractal Omniverse · The Map of Dimensions

Euclid defines a plane as a surface that lies evenly with the straight lines on itself — flat all the way through, with no privileged direction anywhere in it. He defines a circle as a figure bounded by one line, every straight line drawn to that boundary from a single interior point being equal to every other. That interior point is the centre. Worth noticing what Euclid is actually naming: for him the circle is the whole disc, and the curve around it is the circumference.

There is an older way of saying it that suits this page better. Thomas Heath, translating Euclid in 1908, notes a genetic definition that goes back to Heron: a circle is the figure described when a straight line, staying always in one plane, turns about one of its ends as a fixed point until it comes back to where it began. Not a shape you draw — a shape you generate, by sweeping. One end held still, the other end going round.

Out of that sweep falls the strangest number we have. In flat geometry every circle, without exception, has the same ratio of circumference to diameter: 3.14159…, and it never resolves. Johann Lambert proved in 1761 that π is irrational — no fraction will ever equal it. Ferdinand von Lindemann proved in 1882 that it is transcendental, the root of no polynomial with whole-number coefficients, and in doing so finally closed a question that had been open for two thousand years: you cannot square the circle with straightedge and compass. Not because nobody has been clever enough. Because the number forbids it. Archimedes, working with nothing but patience, had already trapped π between 223/71 and 22/7 by squeezing a circle between an inscribed and a circumscribed 96-sided polygon.

And then most of mathematics comes out of it. Trigonometry is named for the triangle, but its functions live on the circle — sine and cosine are simply where you are on a circle after turning through a given angle, which is why they repeat forever. That is Euler’s doing: in 1748 he set the radius to one and turned the sine from a line segment drawn inside a circle into a number, the first time anyone treated these as functions at all. The radian comes from the same place — the angle whose arc is exactly as long as the radius — so the circle even supplies its own unit for measuring turning. Calculus has a claim on it too: inscribe a polygon in a circle, double its sides, and double again, and you have the method of exhaustion. Eudoxus invented it, Euclid used it, Archimedes made it sing, and it is the ancestor of the integral — though its inventors would not have put it that way, since they argued by contradiction precisely so they would never have to speak of a limit. One honest correction to a famous story: Euler never wrote eⁿᵢπ + 1 = 0. What he wrote in 1748 was e to the power v√−1 = cos v + √−1 · sin v. He did not even have the symbol i yet.

One more, because it is the hinge into the next dimension. A sphere’s surface is finite — 4πr², a number you can write down. But puncture it. Remove a single point, and what remains is exactly the infinite plane: stereographic projection maps every remaining point of the sphere onto the plane and back again, one for one, angles preserved. Run it the other way and the plane plus one point at infinity is the sphere. It is the same move as the line closing into a circle one dimension down, done one dimension up. A caution to keep the claim honest: Gauss proved in 1827 that no patch of a sphere, however small, is ever truly flat — curvature survives any amount of bending. But the error shrinks with the square of the patch, which is why the earth looks flat from where you are standing, and why every flat map of it lies.

The circle is the most productive object in mathematics. Nearly everything below came out of asking what happens when a line turns about one of its ends.

Emerson opens his essay Circles by pointing out that you are already looking through one. The eye is the first circle, the horizon it draws is the second, and after that he says the figure simply repeats without end. He then credits Augustine with the famous line about God being a circle whose centre is everywhere and circumference nowhere — and here is a small, pleasing correction to make. Augustine never wrote it. It descends from the anonymous twelfth-century Liber XXIV philosophorum, through Alan of Lille, Nicholas of Cusa and Pascal, collecting a different famous name at almost every stop. A saying about a centre that is everywhere turns out to have no fixed origin of its own.

Edwin Abbott asked in 1884 what it would actually be like to be confined to two dimensions, and the answer he gave in Flatland is still the sharpest thing written on the subject. His narrator, a Square, lives on a surface where everything he meets is a line, because a line is all a flat eye can see. When a Sphere finally passes through his world, the Square perceives a circle that grows, shrinks and vanishes, and the phrase he cannot afterwards shake — upward, not northward — is the sound of a mind reaching for a direction its whole universe has no word for. That is worth sitting with on this page, since the reader is standing exactly one rung above it.

And a circle brings back what a line never could: the same place again. Nietzsche made recurrence the heaviest question he knew how to ask — what if you had to live this life over, unchanged, endlessly? A footnote for anyone who has met the line “all truth is crooked, time itself is a circle” and taken it for his conclusion: it is spoken in Zarathustra by a dwarf, and Zarathustra rounds on him for it — do not make it too easy on yourself. The cheap version of the circle is that everything comes round. The expensive version is being asked whether you would say yes to it.

The circle may be the most widely shared religious symbol there is, and it carries the same freight almost everywhere it appears: wholeness, because it has no gap; eternity, because it has no end; divinity, because it has no preferred direction and no edge to stand outside of; and the turning of things, because whatever goes round it comes back. Buddhism puts a wheel at the centre of its teaching and calls the first sermon the turning of it. Hinduism speaks of the Brahma-wheel, turned by the greatness of God, with every living thing riding on it. Ecclesiastes watches the wind go south and come round to the north and go on circling, and the rivers run to the sea and return to where they started, and reaches the flattest and most modern-sounding conclusion in scripture: there is no new thing under the sun.

And the passage below from Black Elk is the one people remember longest, so it deserves an honest note. Black Elk Speaks is John Neihardt’s English rendering of interviews given through Black Elk’s son as interpreter, and the surviving stenographic transcript shows Neihardt shaped the prose considerably — the images, the argument and the circle-against-the-square are all Black Elk’s, but these exact sentences are the writer’s. Worth knowing, and it takes nothing away from them.

From this dimension emerges the Strong Force. There is nothing to find by stretching farther — the line could reach forever and meet only more of the endless white. So it does something new: one self anchors, and the other begins to spin.

Picture two men holding the ends of a rope. One of them plants his feet and starts to turn. The other has no choice in the matter at all — he goes where the spin takes him. And the shape that travelling end draws as it goes is a circle. Nothing decided that. It is simply what a fixed length dragged around a fixed point does.

When the travelling end arrives back where it started, something has been finished. A plane has been created, and every point of space between the two of them has been visited on the way round. Not sampled — explored, all of it. The spin creates an infinite sweep, and the sweep traces the plane: all possible lines, held in a continuum. Length gains width. This is linear rotation — the line becoming harmonics, a string plucked into standing waves, drawing a perfect circle around its anchor.

And it is worth stopping to count what each rung has cost. The zero dimension needed one point in order to exist. The first dimension needed two. This one needs three. And to get to the third dimension, we are going to need a fourth.

The strong force is the vibration, the rope, and the spin together — the builder and the binder, the force that will close the next dimension shut and never let it go.

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