Determining the requisite number of patterns, partners and balls in governance

Governance as "juggling" -- Juggling as "governance" (Part #13)

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Clearly mathematics has a major role to play in clarifying what amounts to a pattern language usefully encoded in the process of juggling, as discussed separately (Category juggling reframed through visualization dynamics, 2016). As noted above, Wikimedia provides access to 46 patterns (as gif animations) -- distinguishing the number of "balls" and jugglers.

In his study Burkard Polster explores the question of how many ways there are to juggle -- as being the question most frequently asked of jugglers. His response is "infinite" when unconstrained, usefully raising the question whether such jugging possibilities are comparable to infinite games (James P. Carse, Finite and Infinite Games: A Vision of Life as Play and Possibility, 1986; Niki Harré, The Infinite Game: How to Live Well Together, 2018).

Polster does however offer a preliminary answer to the effect that:

However, it still makes sense to ask for the number of juggling sequences that are distinguished in some way. The three most natural parameters used to define distinguished classes of juggling are:

• the number of balls used to juggle a juggling sequence
• the period of a juggling sequence
• the maximum height of a throw in a juggling sequence

If we only fix the number of balls, or the period, or a maximum throw height, the resulting class of juggling sequences will still be infinite, except for some trivial exceptions. Fixing the period p and a maximum throw height h yields a finite class of juggling sequences. Clearly, there are no more than (h+1)p such sequences. (p. 37)

He also offers a more complex indication in the following terms:

Numbers of juggling sequences

(Burkard Polster, The Mathematics of Juggling, 2006, p. 40) See explanation of MÖbius function

These formulae are also discussed by Steve Butler, et al (Juggling Card Sequences, 6 April 2015). The issue could be framed otherwise through the extensive study of passing patterns in other ball sports, most notably football and basketball (Howie Long and John Czarnecki, American Football Passing Patterns).

Fruitful questions might be framed in the following terms:

• If one expert juggler can "manage" 7 balls (say), how many balls can a team of 12 expert jugglers "manage" together?
• What is the mimimum number of "balls" and partnes reauired to sustain a viable pattern, with the maximum number of "balls" similtaneously "in the air"?
• What constraints become evident when the jugging capacity of the parners is significantly different?

Of relevance here is the apparent absence of consideration of constraint on the number of balls which can be effectively juggled, notably as these might relate to the number of participants between which they are passed. Some constraints are evident from the details listed by Wikipedia with respect to juggling world records. It is curious that the number for an individual is consistent with the psychological constraint famously highlighted by George Miller (The Magical Number Seven, Plus or Minus Two: some limits on our capacity for processing information, Psychological Review, 1956). How this constraint might relate to a limited group of individuals (or a group of limited individuals) is another matter.

Given the frequently cited difficulties for global governance of addressing issue complexity, also of interest -- if the "balls" are strategic issues -- is the question of how many issues can be effectively juggled in the light of the number of juggling partners. How many "balls" can be successfully "kept in the air" without being "dropped" given the relative juggling incompetence of some partners? Is this the challenge of sustainable governance?

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