Holding Topology

I began by looking for a strict definition of juggling and its difference from object manipulation. The search narrowed to a physical question: what changes when bodily retention gives way to release?

Definitions and scope

In this article, object manipulation is the generic class of practices in which a performer intentionally changes or maintains the position, motion, or configuration of physical objects through bodily contact or a physical linkage. The comparison class is continuous-retention manipulation, where every active object remains directly retained by the body throughout the episode. The form of juggling studied here is object manipulation whose continuation repeatedly includes release: for some interval at least one active object is no longer directly retained by the body, while its trajectory remains part of an intended capture, strike, pass, bounce, or subsequent movement.

Within this generic class, juggling transfers part of control from continuous contact to initial conditions, prediction, and renewed constraint. Under continuous retention, the body directly constrains motion at every moment; after release, the object state and environmental dynamics carry the motion. This difference changes the model’s state space: a retention snapshot describes an instant, while juggling is identified by a trajectory of releases and captures together with a continuation relation. A complete hold may occur as a pause and complete release as a flash.

The article is limited to the physical geometry of retention, release, and renewed constraint. Performer psychology, social relations, subcultures, identity, artistic meaning, and problems of digitally representing performance remain outside its scope. The simulator below serves as an illustration of the formal model. Intention appears only as an external continuation condition supplied by a task, pattern, performer, or observer; the model assigns no psychological variables.

The system has a body, a finite set of objects, and an environment. The body is whatever supplies direct physical retention: a palm, a pinch, a press against the chest, a ball supported on a foot, several people treated together, or a robot. Once released, an object follows continuous dynamics set by its state at release and by gravity, air, collisions, and later interventions. Release therefore removes bodily retention; it does not by itself remove predictability, influence, or control.

Retention state

Let be the finite active object domain chosen for the description. Membership comes from the performance, task, or interval under study. Attention and an intended continuation may help an observer choose , but neither belongs to the physical retention state.

For each labelled object , let when the body directly retains it and otherwise. The held subset and its occupancy are

The complete discrete microstate is , equivalently a bit vector in . Occupancy forgets identity. A further projection gives three macrostates for every nonempty domain:

Thus separates three resolutions: which objects are retained, how many are retained, and whether retention is absent, mixed, or total. The sign denotes , where there is no active system. It is a useful neutral sign when domains are joined, but it is outside the three-state algebra of a nonempty domain. “Some held” and “some unheld” are complementary aggregates derived from the same object states; for every nonempty domain they give exactly the three active regimes above.

If an observer does not know the value of , the physical state still belongs to one of the three regimes. Partial observation is defined over , for example as a set of compatible microstates or a probability distribution over them.

The letter is borrowed from akrateia. Historically the Greek word names want of power or self-control; here it is declared as a formal label for absence of direct bodily retention, with no claim about will, ethics, or effective control (LSJ). The sign recalls kratos. The mathematics depends only on .

For disjoint domains, adjoining the null sign gives a convenient composition. If a sign is represented by the pair “some object unheld, some object held,” then

Here is neutral and absorbs. This join records the coarse state of the union with three active regimes.

Paths through the cube

Let be the Boolean hypercube whose vertices are the held subsets . Join two vertices when one object changes retention:

A release changes to , and a capture changes to . A performance then writes an event word and traces a path through .

On an interval with a fixed domain , let and count elementary releases and captures. In a simultaneous event, each object change is counted separately. Then

Equal initial and final occupancies give . In particular, an episode returning to its initial contains equal numbers of captures and releases, while a transition has balance and contains at least releases.

For one object, the throw path is . For two, the mixed microstates and differ in two bits. Any path between them made of single-object events touches either total release or total retention .

Remove those two homogeneous vertices and call the remaining induced graph . For it contains microstates; the removed vertices represent and , while all of projects to . For , is connected exactly when . To see this, shrink any nonempty proper subset to one retained object, move between singleton subsets through a two-object subset, then reverse the shrinking process toward the target. The two-object intermediate remains proper only when a third object exists. For three objects the whole mixed region is already a cycle:

Three is therefore the first object count with internal mobility inside mixed retention under single-object events. A three-object cascade can follow this cycle while its macrostate remains .

Four marks a different threshold. The distance from a mixed microstate to a homogeneous boundary is

Every mixed state for three objects lies one event from a boundary. With four objects, a state holding two has and remains mixed after any single-object event. In general the greatest boundary distance is . This distance measures topological redundancy. Juggling difficulty lies on other axes.

The thresholds depend on the event alphabet. Simultaneous actions can be represented by packets such as or . If a two-object exchange packet is primitive, can occur without an observed intermediate macrostate. Synchronous throws, multiplexes, handoffs, and zero-duration catch–release events therefore need packet semantics that preserves simultaneity.

Mixed retention describes one regime of juggling. A full account also contains excursions to the homogeneous boundaries: a cascade may persist in , a flash visits , and a complete hold visits . A three-object flash from rest traces

The same signs also occur in other manipulation. The same configuration or trajectory may belong to an ongoing performance, a pause, or a completed action. An external continuation relation distinguishes a drop from a recovery and a pass from the end of a phrase.

Siteswap beside retention

Siteswap records which hand an object is thrown to and how many beats pass before it is thrown again. Juggling Lab allows a to be interpreted as a hold, uses to force a normally held throw, and uses for no throw (Juggling Lab). Chung and Graham’s siteswap state is a landing schedule: a binary sequence marking the future beats on which airborne balls will land (Chung and Graham, 2008).

These are two projections of a richer physical performance :

describes scheduled throws and future landings; describes present bodily retention. Neither projection is generally injective. Different physical performances can share a siteswap, and different siteswaps can share a coarse retention trajectory. A numerical siteswap alone also leaves dwell and the physical reading of a open. A grip-aware pattern search would therefore ask for a legal siteswap, a legal retention path, and compatibility between them.

Simulator

The court below keeps these projections together. Under the hold convention, is a one-object pattern that stays in . Occupancy in the finite flash walks , while the synchronous cycle releases all three, passes through empty hands, and retains all three again. Throughout, the lamps expose occupancy and the macrostate while the siteswap supplies the landing schedule.

Measurements

For an interval of duration , mixed-state persistence is

Average normalized retention is

These quantities answer different questions. A system spending half its time in and half in has and ; a two-object system retaining exactly one object throughout has and . More detail lives in the occupancy signature

An entropy becomes meaningful only after choosing a distribution over occupancies or microstates, for example . It would measure diversity of visited retention configurations. Environmental uncertainty requires a separate stochastic model.

For a uniform juggle, Shannon related flight , dwell , vacant time , balls , and hands by

If is the dwell ratio, the mean number retained is . Under the constant-timing assumptions of an alternating two-hand three-object cascade, the two hand cycles are half a cycle apart. At least one hand is occupied throughout when ; together with at least one airborne ball, this keeps the macrostate in . Below that threshold, vacant intervals can open visits to .

Beek and Turvey reported , , and as prominent dwell ratios for three-ball cascade juggling and about for five and seven balls (Beek and Turvey, 1992). A 2024 robot-planning paper cites human preference near while choosing for its simulated hardware constraints (Gomez Andreu, Ploeger, and Peters, 2024). These timing results qualify a retention path; they do not determine its topology on their own.

Beyond a binary grip

The pair , where contains positions, velocities, rotations, and other continuous variables, is a small hybrid dynamical model. Retention changes discretely while physical motion flows continuously. Robotics already models juggling systems with continuous flows and impact jumps, and nonprehensile manipulation studies planning and control without stable grasping (Sanfelice, Teel, and Sepulchre, 2007; Ruggiero, Lippiello, and Siciliano, 2018; Ramadani et al., 2026).

Binary bodily retention remains a quotient of a richer relation. In a basic poi configuration, a hand retains a handle while a tether constrains the head. Bounce juggling lets the floor constrain an object. Passing distinguishes release by one body from capture by another, while a handoff may include joint retention. A time-varying constraint relation between agents, components, environment, and objects can represent these cases. The set keeps only the direct body–object retention edge. When several people are combined into one body, this projection erases who released or captured an object, how many hands participated, and whether retention was joint.

This quotient keeps release separate from loss of control, state separate from intention, and siteswap separate from physical realization. Under single-object retention events, it gives two exact thresholds: three objects are the first count whose mixed microstates form one connected region, while four are the first count with a mixed state one event away from either homogeneous boundary.

Note

I wanted a three-object flash from rest, then a hold of all three, then the flash again, written into . I looked a long time for a valid siteswap for that path. Direct does not close.