Tori of Emergence

Each quadrant here is a level that emerged from the one before: a single cell that senses, a many-celled plant that perceives, an animal that thinks and wills, and a human who represents. The page draws the three arrangements you described.

• Folded. One torus. The four levels are the four quadrants of its small circle, and a lower level lives only part of the circle.
• Growing. One torus whose small circle gains a quadrant at each level. The circle grows in step: 1, 2, 3 and then 4 quarter-turns of the human circle.
• Nested. Four tori, one per level, each a whole cycle of its own, carried one inside the next.

The Layout switch keeps every torus on one shared equator or stacks them up the spindle, smallest at the bottom. The shape slider closes each torus into its sphere, as on the first page.

A single cycle is a circle: one phase angle θ, or the pair (cos θ, sin θ). A torus is a cycle carried round another cycle, so the torus you have been using is already one cycle inside another: the small circle carried round the large one. Each further independent cycle adds another circle. Four levels, each with its own phase, make a four-dimensional torus, T⁴ = S¹ × S¹ × S¹ × S¹. That cannot be drawn whole in three dimensions, which is why the page shows it as four tori.

On each small circle the cosine is the in–out position, from origination (+1) to the threshold (−1). The sine is the height, from pre-conscious to post-conscious. The spindle records only the sine, and the lower chart traces those sines over time.

Nested circles are also how waves add. A point carried on a circle that is itself carried on a circle traces z(t) = r₁e^iω₁t + r₂e^iω₂t + …, which is Ptolemy’s epicycles and Fourier’s series. When the frequencies stand in a whole-number ratio the combined motion closes up and repeats. When the ratio is irrational it never repeats, and winds over the whole torus.

Every level lives on the human’s small circle. A lower level runs through its own quadrants and then has to get back to origination, and the shortest way back is a straight chord across the circle.

• The cell senses and returns from the bottom of the circle, below the horizon.
• The plant senses and perceives, then returns along the horizon itself, from the threshold straight back to origination without crossing.
• The animal crosses the threshold into thought and comes down from the top, straight back to origination.
• The human carries on through representation and comes round. No chord is needed.

The regions these loops enclose sit one inside the next. On the cut faces they show as four coloured pieces of the circle, bounded by chords rather than radii. Swept round the spindle, the chords become a cone under the tube, a flat lid on the horizon and a cone above it (switch on Glass to see them).

The chords are √2, 2 and √2 long while the arcs are multiples of π/2, so at an even pace the loops take 1.90, 3.27, 3.90 and 4 quarters. Those numbers have no common multiple, and the folded levels drift apart and never fall back into step.

Here each level adds a quadrant of fixed length to the same circle. A circle made of k quadrants has k times the circumference, so its radius grows in step: 1, 2, 3, 4 quarters of the human radius. Each quadrant then spans 360°, 180°, 120° and finally 90° of its circle. Sense is the whole of the cell’s cycle, half the plant’s, a third of the animal’s and a quarter of the human’s.

The new quadrant opens at the seam, where the cycle returns to origination, and the earlier quadrants are carried round without shrinking. That moves the threshold between perception and thought. With equal quadrants it sits at 240° on the animal’s circle, above the horizon, and reaches the inner equator at 180° only on the human’s. On the plant’s circle, perception fills the whole upper half. In this model the horizon has a fixed meaning only for the human.

Every larger circle contains every smaller one. Keep the earlier stages (the Traces chip) and the growing model becomes the nested one.

Each level is a whole cycle in its own right, a torus of its own, and the four share one spindle without touching. The sizes are my choice, made to match the growing model. They come from one family of tori, those with R² − r² = 1, which are the surfaces of toroidal coordinates and also the shadows of the Hopf fibration of the 3-sphere. With r = ¼, ½, ¾ and 1:

• Each torus is conformally a flat rectangle with sides in the ratio k : 4. The cell’s is long and thin. The human’s is square, the square torus of the first page, where R/r = √2.
• On every torus the inner and outer equators multiply to 1. The plant’s are 1/φ and φ, the golden ratio. The human’s are √2 − 1 and √2 + 1.
• Below the cell the family shrinks onto a single circle of radius 1, the core. Above the human it swells toward the spindle, which closes into a circle through infinity. So the nested family runs from one circle to another, with every level in between.

At an even pace every quadrant takes the same time, so the cell goes round four times for each turn of the human: periods of 1, 2, 3 and 4 quarters. Neighbouring levels stand in the ratios 2:1, 3:2 and 4:3. These are the octave, fifth and fourth of Pythagorean tuning, built from the same numbers 1, 2, 3, 4. All four cycles realign every 12 quarters, which is three human moments.

Which level goes outside is a choice. Human outside makes the human a whole composed of the earlier levels, Stanley Salthe’s scale hierarchy, [whole [[parts]]]. Human inside makes it the most particular case of the levels around it, his specification hierarchy, {physical {material {biological {sociocultural}}}}. On the inside the human is also the smallest and fastest cycle.

My addition. Every ring torus carries tilted circles, its Villarceau circles, that go once round the spindle and once round the tube. On the nested tori they tilt by 14°, 26.6°, 36.9° and 45°. While the tori are nested, each level’s circle links every other level’s exactly once, like the links of a chain. I checked this numerically with the Gauss linking integral: each of the six pairs links once. Stack the tori up the spindle and every pair comes apart.

The 45° circle on the human torus is the spiral from the first page, arg z = π + log |z|. On the square torus that spiral is a perfect circle.

Vertical time can mean two different things.

• Shared equator. Height is the phase within each cycle. Every level passes through the same present at once.
• Stepped. Height is the order of emergence within a moment. The cell’s cycle sits at the bottom, then the plant’s, the animal’s and the human’s, each torus resting on the summit of the one below. I put the horizon at the junction between the plant’s torus and the animal’s, so it falls between perception and thought, where your threshold is.

In the stepped tower the levels can be taken apart. In the shared, nested arrangement their circles interlink.

Each torus throws a shadow on the spindle: the height of its small circle. Close the torus into its sphere and the shadow becomes the sphere’s axis, from 0 at the bottom pole to ∞ at the top. In the growing and nested models the shadows are 1, 2, 3 and 4 units tall, because circumference and height grow together. Together they make 10, the Pythagorean tetractys.

Stacked spheres touch pole to pole, so each level’s ∞ is the next level’s 0. What was the unbounded outside for the cell becomes the ground of the plant, and so on up the tower.

Read upward, the tower is the smoke ring’s spindle with each segment blown up into its own sphere: sense at the bottom, perception up to the horizon, thought and representation above it. Join the human’s ∞ back to the cell’s 0, through infinity, and the chain closes. Its four junctions are then the four quadrant boundaries: the bottom, the threshold, the top and origination.

A ring of four spheres joined like this is what algebraic geometers call the Néron 4-gon, type I₄ in Kodaira’s list of degenerate elliptic curves. It is a torus pinched along four parallel circles, and opening the pinches gives a torus back. Pinch the folded torus along its four quadrant boundaries and you get exactly this ring. The first page’s sphere with 0 and ∞ joined is the 1-gon, I₁: a torus pinched along one circle, the threshold.

Each arrangement makes a different claim about how the levels live together in one person.

• Folded reads every form from the human moment. The lower forms are partial cycles that short-circuit. The cell’s sensing goes straight back into action. The plant perceives and turns, skimming the horizon without crossing it. The animal’s thought discharges directly into action. Only the human returns to origination through representation, so only the human loop closes through language. The risk is reading the cell as an unfinished human.
• Growing is about proportion. Nothing earlier shrinks, but every addition makes it a smaller share, and each new capacity opens where the cycle returns to its origin. Most of a human moment is not sensing.
• Nested is a holarchy in Arthur Koestler’s sense: each level is whole in itself and a part of the next. A person still contains cells sensing salt, a body whose daily clock is set by the sun, and an animal that wills, each with its own faster rhythm. At an even pace those rhythms lock into simple ratios and come back into step every three human moments. Nested on one equator their cycles interlink and cannot be pulled apart. Stepped, they become stages that can.
• The stepped tower also points somewhere. On the first page ∞ was the social. Here the human’s ∞ sits at the top of the spindle, where the 0 of a fifth level would go.

• Which model. The three may be views of one structure. Growing with its earlier stages kept is the nested model, and the folded torus pinched at its quadrant boundaries is the stepped tower closed into a ring.
• The sizes. Each level here grows by one equal quadrant, 1 : 2 : 3 : 4. Doubling, 1 : 2 : 4 : 8, would make every level as large as all the earlier ones together.
• The pace. Even pace lets the lower levels turn faster. In unison gives every level the same moment, one turn in four quarters.
• The order. Is the human the outermost whole or the innermost particular?
• The examples. An amoeba already carries mitochondria; that merger came before both amoebae and many-celled life. The merger that belongs to plants is the chloroplast, a captured cyanobacterium. A bacterium swimming up a chemical gradient is the cleanest case of sense on its own, and a young sunflower turning with the sun each day is perception run on a daily clock.
• A fifth torus. Should the social be the next level, grounded in the human’s ∞?