Psychosocial Torus

The shape and its two circles are the same in every tab. Only the names of the points change.

• The moment turns around the hole. Its four phases run counterclockwise through the quadrants from +1: sense, perception, thought, representation. Sense and perception are the preconscious half, above the real axis; thought and representation are the post-conscious half, below it. The line through −1 is the threshold of consciousness, between perception and thought. The line through +1, the body’s source of sense, is where representation gives way to new sense.
• Scale runs through the hole, from 0 out to ∞. Each reading names 0, ∞ and the circle |z| = 1 between them.
• The now moves around (one circle of the moment at a level set by the slider), through (one phase carried from 0 to ∞), or on the spiral (both at once at 45°, arg z = π + log |z|). Moving at a steady speed, it takes a finite time to reach a pole even though the spiral winds round it infinitely often.
• Others are the now multiplied by the twelfth roots of unity: twelve participants sharing one circle, each at a different phase.
• Turn over is the half-turn about the real axis, through +1 and −1. It is the map z ↦ 1/z, the same map that swaps 0 and ∞ on the Riemann sphere. It swaps sense with representation and perception with thought, and on the circle |z| = 1 it leaves exactly two points where they are: the body’s source at +1 and the threshold of consciousness at −1. In all it fixes two points on the sphere, three at the horn (adding the node) and four on the ring (adding two on the throat), and it carries the spiral onto itself, run backwards.

On the square torus, and only there, the spiral closes after exactly one turn and one loop. Sense leaves the throat on the 0 side at +1, the threshold of consciousness falls at −1 on the circle |z| = 1, and representation reaches the throat from the ∞ side and re-enters as new sense. That closed spiral passes through two of the four fixed points of the turn-over: the threshold point on |z| = 1 and the re-entry point on the throat.

0 is the individual’s source, unknowable, in the moving moment of now. ∞ is everyone modelled together. The circle |z| = 1 is the meeting, one person with another.

On the sphere, every rotation and rescaling z ↦ az leaves 0 and ∞ where they are, so the two poles are the only privileged points, and every line out of the source ends at the social pole. Bringing the poles together through the apple changes distances but not this, which is why the map stays put.

At the horn the lines close into loops, but only through the node. Near the node the surface is two discs touching at a point, and nothing joins a direction in one disc to a direction in the other. A moment passing through loses its phase.

On the ring the node has opened into the throat. The local equation uv = q joins each direction into 0 to the same direction out of ∞, so a moment comes back to the source with its phase. The scale loop is the circuit Berger and Luckmann set out in The Social Construction of Reality (1966): society is a human product, society is an objective reality, man is a social product. On the torus it is one loop with no first step.

On the square torus the spiral crosses the threshold of consciousness exactly at the meeting circle, between the inner half of the surface and the shared half. In this reading, conscious thought begins where one person meets another.

0 is where an observation starts, ∞ is what it reaches, and the circle |z| = 1 is the distinction between them.

The turn-over maps the surface onto itself at every stage, so nothing in the shape says which end is the observer. Whether 0 is the individual and ∞ the social, or the other way round, is not settled by the structure. It depends on where the observing is done from.

The stages line up with the orders of cybernetics. On the sphere the observer looks at the observed from outside: first order. At the horn they meet in one point and the observer is part of what it observes: second order, von Foerster’s cybernetics of observing systems. On the ring the meeting has become a circulation through the throat in both directions, with no position outside it: third order, with no privileged observer.

The points the turn-over leaves fixed are eigenforms in von Foerster’s sense, values that a recursive operation returns unchanged. There are two on the sphere (+1 and −1), three at the horn (+1, −1 and the node) and four on the ring (+1 and −1 on the distinction, and two on the throat).

0 is the instant, the vanishing edge of now. ∞ is all of time. Distance from 0 is duration on a log scale, and the circle |z| = 1 is a generation. Taking a second for the instant and the age of the universe for all time, the halfway point on a log scale is about 21 years.

On the sphere time is a line: everything runs out from the instant to all time and stays there. At the horn the instant and all time are one point, an eternal now with no duration, through which no phase passes. On the ring history comes back into every instant and every instant feeds history. Giddens called this the duality of structure: structure is both the medium and the outcome of the practices it organises.

Each circle |z| = constant is one timescale, and the moment turns at all of them. Choose Around and move the slider to take the moment from short timescales to long ones.

Each member at 0, authority at ∞, the group on the circle |z| = 1. Twelve participants are shown in this reading.

• C, imposed control: the sphere. Two fixed poles on one axis of opposition. Every rotation and rescaling of the group leaves both poles in place, so the centre is fixed, not derived, and every member’s line runs up to it.
• B: the apple and the horn. Authority comes closer. At the horn the opposition has gone, but everything still passes through one point.
• A, aligned movement: the ring. No point is fixed. Every rotation or rescaling moves every point of the torus, and the centre is the hole, an absence the movement turns around. It is derived from the movement, not imposed on it.

Couzin and colleagues (2002) found this milling state in models of fish schools and bird flocks: individuals circle an empty core, and as alignment strengthens the group switches to travelling in parallel. As the ring thins, each stretch of tube looks straighter, which is the turn from murmuration to migration.

The animation runs from C to A and back again. Read the stages as conditions a group passes between, not rungs to climb: treating A as a goal collapses it back into C.

One open question: your B is the multi-cluster relational state. It may sit better on the surfaces with more than one hole, in the fork, than on the horn.

• Which end is the individual. The turn-over shows the shape cannot decide this. These readings put the individual, the observer, the instant and the member at 0. Each can be turned over.
• The quadrant layout. This page runs sense, perception, thought and representation counterclockwise from +1, one quadrant each, so the threshold of consciousness is at −1 (θ = π) and a moment closes at +1. If you place the threshold elsewhere, the spiral’s starting phase moves with it. The turn-over stays z ↦ 1/z as long as the threshold sits on the real axis.
• What the circle |z| = 1 is. The meeting, the distinction, a generation, the group. It is where the turn-over trades the two ends, so whatever it is, it is the place where the two ends exchange roles.
• Where the moment lives. Around the hole, through it, or on the spiral.
• A twist on return. A complex q would turn the phase by a fixed angle at each pass through the throat, for example bringing a moment back one quadrant on. A torus of revolution cannot show this.
• A reversed return. If the throat sent a moment back with its phase running the other way, the surface would be a Klein bottle, which cannot be built in 3D without crossing itself. This is the Möbius-type version of the same move.