---
title: "content"
description: "Syntegrity, Beer wrote at the end of the passage discussed in chapter 20, “depends on structural closure” (BD 1994: 14). How can a group of people be closed…"
url: "https://wiki.cybernetic-futures.org/pure-vsm/part-04-team-syntegrity/ch21-the-icosahedron-as-organisational-form/ch21-the-icosahedron-as-organisational-form-content"
modified: "2026-11-12T15:26:26.085Z"
language: "en"
site: "Cybernetic Futures"
---

# content

## 21.1 Why a regular polyhedron
Syntegrity, Beer wrote at the end of the passage discussed in chapter 20, "depends on structural closure" (BD 1994: 14). How can a group of people be closed structurally, so that ideas reverberate among them and no member is privileged? Beer's answer starts from a criterion taken from the idea of democracy and ends with one geometric form that meets it.
The criterion comes straight from the demand for "perfect democracy" (BD 1994: 12). Such a structure "surely means that no individual, and initially no cause, should have ascendance over any other. Then in looking for polyhedra on which to construct democratic tensegrity models, we must consider only *regular* polyhedra: figures which have no top, no bottom, no sides—indeed no feature by which they may be specially oriented at all" (BD 1994: 14). A political requirement becomes a geometric property. If no position in the structure differs from any other, no person placed in it can hold a privileged position. The words "initially no cause" matter as well: topics, too, are to be equal at the start, so that the structure does not settle in advance which issues count most.
Five regular convex polyhedra exist: tetrahedron, cube, octahedron, dodecahedron and icosahedron. Beer compares them by Euler's law, "the number of faces plus the number of vertices is equal to the number of edges plus two" (BD 1994: 14), and tabulates faces, vertices, edges and edges per vertex for each. The icosahedron has 20 faces, 12 vertices, 30 edges and 5 edges meeting at each vertex. "Without pursuing all the relevant arguments here, I decided to base my major experiments on the icosahedron, and to consider the edges as representing Infoset members (namely 30) and the vertices as representing topics" (BD 1994: 14). Each vertex is a topic, and the five edges that meet there are the five people who work on it.
The choice was not deduced, and Beer says so: "There are no proofs to offer, but there is accumulated knowledge about interpersonal systems" (BD 1994: 15). He gives three reasons from experience. First, "five discussants per topic is a 'good' number whereas three is certainly low" (ibid.). Second, thirty members "guarantees a reasonably high variety of viewpoints, without reproducing too many clones or look-alikes" (ibid.), and thirty is also, "roughly speaking (a personal observation), the key number of influential members of a management group" (ibid.). Third, he first treats the fixed number of twelve topics as a possible "straitjacket", but concludes from the 1990 experiments "that the number is arbitrary. Any number of topics may be contracted or expanded to 12 by sensible drafting" (ibid.).
Chapter 11 of BD, at the start of Part Three, restates the original criterion: the five regular convex polyhedra were chosen "because each edge (a person) has exactly the same status in the geometry as every other edge. There is no top, bottom or sideways in any of the five structures" (BD 1994: 181). Beer recalls that the icosahedron had been selected informally in chapter 1, "'without pursuing all the relevant arguments here'" (ibid.). Part Three sets out to supply those arguments (see 21.3).
The idea predates BD. In the essay "On Suicidal Rabbits", reprinted in *Think Before You Think*, Beer sketches the same structure: "Let us typify an organization as an icosahedron. Its twenty sides are triangles. It has thirty edges. Let each edge represent a person" (TBYT 2009, "On Suicidal Rabbits": 217). The same passage draws the key combinatorial consequence: "Since each edge (equals a person) has two ends, each of thirty people belongs to two teams, and no two people belong to the same two teams" (ibid.).
## 21.2 Consequences of the geometry
With people as edges and topics as vertices, a series of properties follow from the figure itself. Beer draws them out one by one.
- **Roles.** Each person (edge) connects two vertices and is therefore a member of two teams (topics); each team has five members. Beer notes that "one third of the Infoset has direct responsibility in topic formulation and development for any two topics", which he calls "a high Centrality ratio, whilst not being stressful for the member concerned" (BD 1994: 15). Critics are assigned between teams as well. In the Syntegrity diagram of ch. 2 of BD, "Each team appoints one member as a critic of another team" (BD 1994: 29). In chapter 1 of BD the critic is appointed "to the next-but-one neighbour team", with the consequence "that not only is s/he a *member* of two teams, but also a *critic* of two near-neighbour teams. There are now 120 roles being played out in the Infoset" (BD 1994: 16). Later chapters revise the allocation of critic roles (e.g. for the tetrahedral variant, BD 1994: 74; for the icosahedron, BD 1994: 125–129). In the tetrahedral case, with only six players, "each player is a critic of the two topic groups of which s/he is not a member" (BD 1994: 74), which Beer finds has "a certain incestuous quality about it" (ibid.). For the icosahedron, chapter 8 of BD adds two exclusion rules. The Freesman Exclusion reads: "A person may choose to play a critic role in only one of a pair of teams that are themselves polar opposites" (BD 1994: 125). The Adjacency Exclusion rules out criticizing a team that shares a member with the person's other team (BD 1994: 126–127). Together they leave nothing to choose: "Now it is clear that the critical roles are fully determined by the dual team membership" (BD 1994: 129).
- **Polar opposites.** Pairs of vertices that are furthest apart (BD 1994: 16, 29). The icosahedron has six axes, "which can be made to spin serenely between each pair of poles" (BD 1994: 16). The twelve topics therefore form six pairs of opposites, and that pairing becomes the basis of the Hexadic Reduction in the revised protocol (see chapter 22).
- **Tension.** Beyond the 30 edges there are internal relations: from each vertex five vertices "that are neither neighbours nor polar opposites. That makes 60 internal *tensile* relationships, represented by 30 internal struts" (BD 1994: 16). Beer asks what these struts could represent in human terms, and answers that the teams at the vertices provide compression and the critics provide tension. When each team appoints critics to next-but-one neighbours, "the whole icosahedral space is interlaced with tensile componentry" (ibid.). The critic role is thus the social counterpart of the struts. It holds the structure together by pulling across it; nothing is added as a further layer above.
- **Limited variety.** A group of 30 has 870 directional relationships, Beer notes, and "An Infoset meeting that tried to accommodate this number of direct personal interactions would take forever. But if each person has only four roles, we reduce proliferating variety to 120—and this, it seems, can be handled" (BD 1994: 17). In the terms of chapter 2 of this book, the geometry works as a variety attenuator that everyone shares equally.
- **No marginalisation.** Beer applies the Bavelas measures from his early work (see chapter 20). A player who is a member of two teams and a critic of two others "may contact 20 people (including his *nodal* self) in ONE step", while the ten members of the two polar-opposite teams are reached in three steps (BD 1994: 17). Since every player is in the same position, all centralities are equal: "Peripherality is 30 – 30 = zero. This is exactly what our presuppositions about democracy and symmetry led us to expect from adopting such a model as this: zero marginalization of any one person" (BD 1994: 17). A friend, Donald Burrill, arrived at different numbers, and Beer notes that the invariant matters more than the count. He quotes Burrill that syntegrity requires identical values for every member, "so that Peripherality for each member is identically zero whatever the value of Relative Centrality" (ibid.). In one respect this resolves the old Bavelas paradox (chapter 20): symmetry removes peripheral isolation without making anyone central.
## 21.3 Why this form
Experience with the protocol led Beer to reconsider how teams at opposite poles of the icosahedron should relate. He restates the purpose of the whole design: "To replace hierarchic and authoritarian notions of management with a polyhedral envelope that maintains its equilibrium through the geodesic principles of tensegrity was the very basis for adopting this model" (BD 1994: 114). From this he concludes that "strong internal connexions, pole to pole, were barred. Such arrangements would inevitably lead to centralization (see Chapter One, and compare the Star Net), and destroy the 'holographic' equality of perception and potency at the periphery that was sought" (ibid.). The geometry constrains the design as well as picturing it. A direct channel through the centre would bring back a hub and, with it, hierarchy.
In Part Three (chs. 11–13) Beer examines the mathematics of "icosahedral space". Much of the formal work is presented by collaborators in Part Five of BD and is not attributed to Beer here; Beer reports its outcome in his own chapters (for the matrix algebra he sends the reader to "Part Five, Surplus One", BD 1994: 222). One result concerns which structures meet the conditions for a "syntegral graph". His conclusion: "The syntegral graph conditions evinced by the icosahedral model are not met by most of the Platonic Solids with which we started in Chapter One. The tetrahedron, octahedron, and dodecahedron are out" (BD 1994: 220). The other result concerns the icosahedron's inner geometry, including its relation to the golden rectangle. He summarizes: the model, first chosen "for its convenience in describing a 'democratic' societary structure, using numbers for infosets and teams that looked appropriate, turns out to be a fundamental structure, with a potency derived from a minimalist logic of relations" (BD 1994: 187).
The sequence matters for reading that sentence. Beer chose the icosahedron first, on grounds of symmetry and practical experience (BD 1994: 14–15). Its mathematical distinction was found afterwards, and Beer himself calls the finding "a surprise" (BD 1994: 187). He does not claim that the mathematics proves the social claims. He claims that the form chosen for democratic reasons turned out to have well-defined formal properties that account for how information spreads in it. Chapter 22 takes up that spreading as reverberation.
## 21.4 The mathematics of Part Three
Beer's own part of the mathematics fills three chapters: "The Structure of Icosahedral Space" (ch. 11), "The Dynamics of Icosahedral Space" (ch. 12) and "Self-Reference in Icosahedral Space" (ch. 13) (BD 1994: 179). He sets the level himself: "The discussion works at the high-school level of difficulty" (BD 1994: 182). Each chapter adds one layer: a constant, a closed pathway, and a graph with its eigenvalues.
**Structure: the golden section.** Chapter 11 derives phi from the only proportion that can be written with two terms, a/b = b/(a + b), whose positive solution is 1.618034 (BD 1994: 182–184). It passes through the Fibonacci series to the golden rectangle, which divides into a square and a smaller golden rectangle without end (BD 1994: 184–185). Three equal golden rectangles set orthogonally have twelve corners; joined by thirty edges, they give the icosahedron (BD 1994: 187; the "surprise" of 21.3). In the pentagon under each vertex the diagonal measures phi when the edges measure one, and that diagonal "is an internal brace—a 'critic' trajectory in the dynamic structure" (BD 1994: 188–190). Of the icosahedron's three lengths (edge, polar axis, brace), the axis has no structural role, so "in a unit icosahedron, the *only* significant measure is phi" (ibid.). Each of the fifteen pairs of parallel edges, with the braces joining their ends, forms a golden rectangle, and the whole space is "an interpenetration of phi-ness" (BD 1994: 191). The crossing braces enclose a regular dodecahedron at the centre. Beer kept this space free of polar axes because "strength comes from elastic tension, and not from brittle rigidity" (BD 1994: 192); from his meditation practice he calls it the "sacred space" (ibid.). The chapter ends with a practical correction. The "clocks" of BD ch. 2 "are really entailment charts rather than projections" (BD 1994: 193), which is why they served badly for allocating topics. Don Burrill supplied proper projections centred on an edge, a vertex or a face (BD 1994: 193–195).
**Dynamics: the enneagram.** Chapter 12 asks whether reverberation has a basis in the model itself. Beer thinks "that there is a dynamic implicit in the icosahedral model that pre-exists the hum-and-buzz activity" of the protocol (BD 1994: 197). He stands the model on a vertex, sights along a polar axis and removes three teams: the two at the poles and the one at the vertex it stands on. Nine vertices and 16 of the 30 players remain (BD 1994: 198–199). On these nine points a closed triangle 3–6–9 appears, and a second closed path is traced by the digits of one seventh, 0.142857: "It is only the fraction of sevenths that generates a closed path" (BD 1994: 200). The figure is the enneagram (BD 1994: 202). Joe Truss found it in the model. Because it can be found along any axis, the icosahedron "enfolds *30* three-dimensional, four-planar, enneagrams" (BD 1994: 207). Beer marks the status of the history that follows: "This book is a story and an illumination rather than a text book, and the account of the enneagram that follows is a personal account" (BD 1994: 202). The chapter ends with the hypothesis of an "infosettic" consciousness (BD 1994: 208), taken up in chapter 23 (23.2).
**Self-reference: graph and eigenvalues.** Chapter 13 starts from closure and recursion. After von Foerster's recursive operators, Beer shows a "recursive square-rooting machine" that takes 93 down to 1.00 and stays there; 1.00 is "the eigenvalue of *the operation of a square root*", and the starting value disappears (BD 1994: 214–216). He then turns to graph theory, following Assad Jalali. A graph that conforms to "the syntegral logic" satisfies kn = 2e, n(n − 2) = 4e and n = 2(k + 1), with k the valency, n the vertices and e the edges (BD 1994: 218–219). "Summing up: the syntegral graph is symmetric, connected, of diameter 3, and exhibits involutive isomorphism around all six polar axes" (BD 1994: 219). Jalali's list by valency gives the hexagon (k = 2), the cube (k = 3), an unfamiliar figure with twenty edges and ten vertices (k = 4) and the icosahedron (k = 5); the other Platonic solids fail (BD 1994: 220; see 21.3). The octahedron's diameter is two, which makes it "trivial", and Beer connects this with the atypical course of the California experiment (ibid.). Paley graphs continue the series to 90, 182 and 306 members; whether protocols could work at that size "is a matter for conjecture" (BD 1994: 221). Last, the syntegral graph has, besides the eigenvalue 1.00, the eigenvalues plus and minus one over the square root of five. The positive one sets how fast information evens out over the iterations (BD 1994: 222–223). What Beer draws from it for the number of iterations is reported in chapter 22 (22.5).
On my reading, the three chapters carry different weight in Beer's own presentation. The golden-section geometry and the graph conditions are stated as mathematics. The enneagram's history is a personal account, in his words. The group mind is a hypothesis "for which there is so far no scientific support" (BD 1994: 208).

## References

- **BD** — Beer, Stafford (1994). *Beyond Dispute: The Invention of Team Syntegrity*. Chichester: John Wiley & Sons.
- **TBYT** — Beer, Stafford (2009). *Think Before You Think: Social Complexity and Knowledge of Knowing*. Edited by David Whittaker. Charlbury: Wavestone Press.