“Logic is for demonstration, intuition for invention.”  

                                                                                      H. Poincare, L’Enseignement Mathematique (1899)

                                                     The  Dialectic


This is the story of the Sommer Cube (S3).

Leveraging binary and analogical reasoning —  variations on a theme of square and circle -- flip-flop, a counterpoint of logic and intuition. 

A cubical maze module; it is at once fundamental and obscure — leveraging Binary and Analogical reasoning, as both classic toy (Tate Museums) and Cognitive Toolconcrete to abstract, (a switching of “true" and “false" logical arguments presented visually and tactilely — “sameness" and “differentness” at multiple levels of abstraction in terms of relational complexity.  Causal interaction within dynamic core. 

S3 is a combinational (technically “permutational”, like a combination lock) logic block which contains four tunnels, effectively an array of unconnected, gravity-dependent switches / routers to be programmed (with or without a ball) by the user as the block is rotated in space, which can be connected to other logic blocks to create multiple adaptive, simultaneous routes (network theory), depicted physically or logically.  An architecture of nested dichotomies. 

It’s about patterns — always patterns.


                                  CHALLENGE CONSTRUCTION BLOCKS (Analog-Binary Processor


It’s about tension of switchiness (compound cognitive, perceptual, mechanical flip-flop and schedule of reinforcement reversal learning":  analogical reasoning, instrumental learning):  switching of directed attention, relative motion and form, and navigational strategies.  (Gestalt Engine (analog)  x  Tilt Switch (binary)  =  "Directed Attention" (flip-flop)  =  "Cognitive Flexibility”)

Analytic and holistic thinking; a competition of rational (programming desired logic functions) and nonrational.

It's about simultaneous mental (rational and nonrational) and manual rotation.

Giving form to binary and analogical information, a coupling between physical objects and binary information where bits are directly manipulable and perceptible.  

In other words, the S3 physical state embodies the binary state of the system, as well as the analogical. (think Tangible User Interface -- TUI)

A "Distributed Adaptive Message Block Networkin Paul Baran’s Internet terms, in a low-tech format.  A fractal command and control, strategic and tactical planning challenge.


Think of the ball as “ping”:  a network test for reachability.; think graph theory.

A simple block and ball networking system which leverages full-spectrum cognitive flexibility / perceptual processing   :  learning algorithms, and the art of design, with an emphasis on advanced thinking and intuition -- and self-reflection (not "mere facts", but principles); functional relations, particularly goals and feedback (What information is relevant?  What are my assumptions?  Are they justified?)

It's about relative motion (flip-flop of manipulator frame of reference.  

Egocentric  >>  allocentric) and patterns of thought (higher levels of abstraction) in a paradoxical environment, and why and how to get the most out of them:  a partnership of feedback and symmetry.  Control under continuously varying context (among eccentrically rotating local" and expanding “absolute" coordinate systems) — multivariate, celestial mechanics.

It's about levels of abstraction, pattern.

The rule which governs a system or phenomenon, exactly like numeric, musical, or visual relationships; patterns of thought exceeding the parameters normally experienced in logical operations:  Apex reasoning (think Bayesian inference -- subjective probabiity as epistemic tool).  Where the strategies and tactics are evolutionary.

An holarchy, an architecture of nested uncertainty (uncertainty of environment, and cognitive flip-flop and perceptual flip-flop -- "unknown unknowns"), of constant and unexpected contradiction and recursion;  an Hegelian logic hierarchy.

In Hegel’s Science of Logic terms, S3 is about thinking which focuses on itself. 

"I liken my study of logic to the study of grammar. You only really see the rewards when you later come to observe language in use and you grasp what it is that makes the language of poetry so evocative".


dialectic engine of paradox, demanding logic and intuition; a cubical maze module (four tunnels = four binary (0/1) switches = gate array) offering a development of choices (control flow) to create linearly independent / dependent paths, using a ball, or symmetry in mathematics.

"… the logic of dichotomies ….  A container shall be provided with holes in such a way that they can be opened and closed.  They are to be open at those places that correspond to a 1 and remain closed at those that correspond to a 0.  Through the opened gates small cubes or marbles are to fall into tracks, through the others nothing.  It [the gate array] is to be shifted from column to column as required."

                                                                            Leibniz, G.W., "De Progressione Dyadica -- Pars 1", 1679

A Leibniz germinis "machinea rationcinatrix.”


                  Rolling Ball “Tilt-Switch”                                                Asynchronous Analog-Binary Processor (0/1)

           (Resets according to orientation.)

Each S3 reorientation simultaneously reprograms the four "gravity feed” tunnels differentially, nonlinearly; each acts as a binary (0/1) logic gate (rolling ball "tilt-switch") to impede (0) / allow (1) ball flow.  (Think Field Programmable Gate Array.)

Think of the ball as “ping” :  a network monitor / troubleshooting tool testing reachability.

In logic / mathematical terms, the S3 manipulator must personally (without symbolic / numerical tools) visualize, intuit, reason incommensurate concepts, like Gödel’s Paradox (“this statement is false” requires a broader logical perspective), like the S3, self-referential), like the Pythagorean triangle, 1,1, √2; 1,1,1, √3, and reinvent accordingly (Oops! all numbers are not integers, rational numbers -- let’s look at the bigger picture — think of it as irrational number along a line of possible numbers).

It's about focus on the collapse of our expectation of a logical universe (like the Frame Problem:  no boundary for what information is relevant for each action, what will change, what will remain the same), the problem of delimiting the consequences and non-consequences of actions.  Just as mathematician Lewis Carroll’s Alice was presented with numerous logical paradoxes in Wonderland -- a topology of paradoxWatzlawick’s "contradiction that follows correct deduction from consistent premises”.

In other words, S3 is about uncertainty of analogical / binary interplay. 

Uncertainty as tool.


Its about switching.  In Zen (cognitive flip-flop) terms, embracing the contradiction (instead of accepting the Law of Non-Contradiction) creates the necessary tension to escape from the underpinning concepts -- to see them as dynamic patterns, part of a process of transformation.

It's about regulatory switching (synchronizing concurrent communication / interference among rotating switching stations), a variety increasing (informal as well as formal reasoning), heuristic routing (ball or vector) system seeking best paths (optimization) in an unfriendly (blocking), gravity-dependent network of switchable links.

A Cybernetic switch, encompassing the Systems Thinking idea, the operation of wholeness, and the relation of whole and parts,


structure of systems and their behavior, laws of regulation, and recursive processes of self-observation and self-reference in the service of reasoning.

“The basic thesis of gestalt theory might be formulated thus:  there are contexts in which what is happening in the whole cannot be deduced from the characteristics of the separate pieces, but conversely; what happens to a part of the whole is, in clear-cut cases, determined by the laws of the inner structure of its whole.”

                                                        Max Wertheimer, "Gestalt Theory”, 1924  ("Social Research", 1944)

The subtlety here is that the whole (the Gestalt) has a reality of its own:  subjective contours are real, according to the brain, and therefore must be taken into account (illusion of whole affects interpretation of each part); illusion is operational fact.  Actions are taken, or not taken, accordingly.

The problem (more about this later) is that the Systems Thinking axiom / mantra, The Whole Is “Greater" Than The Sum Of its Parts, is a misquote, a misunderstanding, and represents thinking not robust enough for an environment of paradox or chaos (think Bayesian inference -- epistemic tool).  (Gestalt psychologist Kurt Koffka’s original statement was "the whole is other than the sum of its parts”; “this is not a principle of addition” he complained, a significant intellectual degeneration.)  

In other words, Sis a dynamic domain (Wiener Animal-Machine Control and Communication) of nested and recursive structures / functions / processes, a tangible, lucid demonstration of Synthesis of Cybernetics and Systems Thinking -- goal systems of circular, regulatory chains influencing each other.  A paradoxical environment, with hierarchies of different rules and changing order of operations (principles have to be unlearned).

A spiraling, recursive, nested, back and forth of cognitive and perceptual cycles within a dynamic logic hierarchy, an Hegelian “Totality”:  a mixing of levels of abstraction, self-reference.

Thus, S3 is a topology of paradox:

"… a kind of Strange Loop, an interaction between levels in which the top level reaches back down towards the bottom level and influences it, while at the same time being itself determined by the bottom level.”    Gödel, Escher, Bach

A necessary redefinition / transcending of field (contradictory cognitive and perceptual signals / messages at different levels, where acknowledgement of that contradiction is forbidden:  Double Bind).  

An exploration of symmetry, transformation of parts and whole, square and circle -- adaptation to accelerating rates of change and complexity of system and environment: a cognitive development tool; a topology of paradox.  While Rubik’s Cube (Machine-think:  about problem solving competence), without that paradox, exercises logic (according to MIT / Rubik’s Cube"Solving the cube becomes almost trivial once a certain core set of algorithms, called macros, are learned.”)S(about the need for more human learning proficiency) demands logic and intuition: Open Systems Thinkinga working epistemological premise, within a cubical field of “forbiddingly complex environmental interactions.” (F. Emery, Systems Thinking, 1969)  

Here, S3 demands, gently at first, the manipulator simultaneously plan and direct multiple lines of thought, mentally and physically rotate, differentiate and process visual and logical parts and wholes, 


separate evolving tasks into manageable, self-referential subsequences, all within evolving layers of higher abstraction -- a self-reinforcing collaboration of left and right cerebral hemispheres — logic and intuition, naive commonsense reasoning.


Like the Tower of Hanoi classic recursive programming problem, 


(transfer the entire tower to another peg, moving one disk at a time, never a larger onto a smaller) 

each step is specified in terms of previous steps that the same function has already calculated (recursive functions and recurrence relations).  

But, unlike the Tower of Hanoi, S3 has simultaneous multiple problem 


paths where each successive simultaneous path requires reinventing the previous path(s) / (algorithm(s).


Here, each Stunnel is a two-state rolling ball "tilt-switch”, a binary (0/1) logic gate with forty-eight states (twenty-four possible attitude combinations:  six faces; times four points of the compass; times impede (0) / allow (1) ball flow).  Each S3 orthogonal attitude change  simultaneously resets the four "gravity feed" chiral tunnels — differently.

Sis simply a Claude Shannon (Information / Switching Theory) / Minsky “Ultimate Machine" whose job is to turn itself off / on … differentially … with four tunnels (four binary switches) with each rotation 



In Field Programmable Gate Array terms S3 is a programmable logic component; think of bus architecture, plug-ins, simple logic circuits.  

To repeat, S3 is a combinational (technically “permutational”, like a combination lock) logic block which contains four tunnels, effectively an array of unconnected, gravity-dependent switches / routers to be programmed (with or without a ball) by the user as the block is rotated in space, which can be connected to other logic blocks to create multiple adaptive, simultaneous routes (network theory), depicted physically or logically.  An architecture of nested dichotomies.

In other words, within each combinational logic block, each Stunnel, has twenty-four possible attitude combinations (six faces; times four points of the compass; times impede (0) / allow (1) ball flow). 

While each set of four tunnels (one hundred and ninety-two states) acts as a dynamic logic block, a low-tech logic operator (dynamic router) of programmable interconnects. 

With each additional S3 there is a corresponding exponential increase in the number of possible tunnel attitude combinations (two S3's = 24 attitudes; three S3's = 24attitudes; etc.) and states (two S3's = 1922  states; three S3's = 192states; etc.).

Thus, Sis a universe of “orbiting" possibilities, an integrated circuit where all users have the same basic platform and can map the logic gates to their own design, developing sequences of dynamic, recursive and conditional binary change of state. 


                                four bit (0/1) -- nibble                     Asynchronous Analog-Binary Processor (0/1)

"Turbulent  Field ...  the dynamic properties arise not simply from the interaction of the component organizations, but also from the field itself. The "ground" is in motion.” 

                                                    Emery and Trist, "The Causal Texture of Organizational Environments”, 1965.

To optimize multiple simultaneous paths (make most paths begin / cross within a single S3) the manipulator must simulate the future, conceptually "drill down” through nested coordinate systems of dynamic information.  But what is the manipulator’s reference frame?  Particularly in the face of an expanding universe of S3s.

In Michio Kaku’s, The Future of the Mind terms, “self awareness; the ability to put one’s self inside a model of the environment.” 

(From child to super-mathematician, the Smanipulator has only one path:  reason / intuit up the original "evolution of mathematics” ladderfrom things, to abstractions of things, to binaryness, to geometryness, to algebraness, to setness -- create more sophisticated problem models, know what the laws are.)

To discover order in what is seemingly random.

In other words, navigation of nested levels of abstraction (sets: binary and analog), mental discrimination / integration of different problem models, starkly contrasting conceptual frameworks (each with its own set of rules):  

CUBE exoskeleton (six faces)  >>  

HOLES quadrant (eight holes)  >>  

TUNNELS / HOLES (four couplingsgravity gate array; (four binary switches)  >> 

TUNNEL / HOLES (one binary switch). 


                            Analog-Binary Processor (0/1)  /  three axis rotation problem space    

Each of these nested frames of reference establishes conditions / opportunities as the focal point of each frame is moved in any of three perpendicular directions; a dynamic cycle of abstractionnested recalibration and reflection.

With each rotation (precession of the rotational axis of each tunnel), and each additional S3, the manipulator must focus on the binary issues (0/1) of a given tunnel (part), proceed up the logic hierarchy to the S3 (whole), then to the expanding universe of S3 (overarching whole / system of systems).  Then go back down the hierarchy of nested coordinate systems, and repeat the process (while coping with contradiction of action and environment).  

In group theory terms, S3 changes state with each action but remains the same in fundamental ways, in each of six canonical positions:  a symmetry transformation.

An evolution of compound statements of control flow, similar to block-structured programming where control structures are formed from blocks, including building blocks nested within other blocks, minimizing range of functions, variables, procedures.


In programming terms, Sis about sequences of dynamic, conditional and recursive binary change of state (each step must be specified in terms of previous steps that the same function has already calculated); streams of operations which operate concurrently:  concurrent programming, computations which must be executed simultaneously, potentially interacting with each other on physically seperated processors (S3).

Thus, the manipulation of each S3 and the evolving S3 logic universe requires programming a nonlinear overarching control structure -- dynamic, nested, self-referential -- function calls, instructions, and statements with increasingly less predictable flow and changing order of operations.

An evolving asymmetric topology of nested and recursive, interdependent logical frames, cascading, two-valued logic operations demanding the manipulator exercise forward / backward chaining (trace backward or forward, from result to cause, etc.)

A topology of reciprocal influence of Hands and Mind.  

Each rotation also generates nested and interdependent illusory frames of reference; all frames, illusory or not, remembered but adaptable, creating significant and conflicting changes of viewpoint (contradictory goals within changing contexts).

"The matching process which decides whether a proposed frame is suitable is controlled partly by one's current goals and partly by information attached to the frame; the frames carry terminal markers and other constraints [S3 confusing distinctive-feature information:  squares / cubes, circles / tunnels, illusory / real symmetries], while the goals are used to decide which of these constraints are currently relevant."

                                                                                        Marvin Minsky, "Frames Systems Theory", 1975

In other words, the manipulator is forced to make plans / take actions while facing the dichotomy of the relevant from the irrelevant -- what stays the same within a dynamic environment  -- a nascent Frame Problem (Philosophy / Artificial Intelligence).


                                            (four checkered paths cross and exit within a single S3)

In Dennett's terms, the Smanipulator must 

"recognize, not only …  intended implications of its acts, but also … the implications about their side-effects … the difference between relevant implications and irrelevant implications …"  and then ignore "thousands of implications ... deduced to be irrelevant.


       Approaching Complexity (Something You Cannot Predict)

The only recourse (causal inference) is intuition, naive commonsense reasoning.

In Poincare’s terms: “Logic is for demonstration, intuition for invention.”

No small task.

"Try to acquire the weird practice of savoring your mistakes, delighting in uncovering the strange quirks that led you astray. Then, once you have sucked out all the goodness to be gained from having made them, you can cheerfully set them behind you, and go on to the next big opportunity. But that is not enough: you should actively seek out opportunities to make grand mistakes, just so you can then recover from them.

                                                               Daniel C. DennettIntuition Pumps and Other Tools for Thinking

Easy task?  With each reorientation, what’s on the other side of the cubical form / S3?

But in the common way of taking the view of any opake object [S3], that part of its surface, which fronts the eye, is apt to occupy the mind alone, and the opposite, nay even every other part of it whatever, is left unthought of at that time:  and the least motion we make to reconnoitre any other side of the object, confounds our first idea, for want of the connexion of the two ideas, which the complete knowledge of the whole world would naturally have given us, if we had considered it the other way before."

                                                          W. Hogarth, 1753, The Analysis of Beauty, (Oxford, 1955) 

The manipulator is immersed in a tug of war between local optimization / organic strategies and global purpose.

In C.S. Peirce’s Pragmatism terms, A) flip-flops of abduction (intuition) / deduction / induction, a dynamic cycle (or spectrum) of inquiry only understandable as a whole, B) systematically impeded by gestalt figure / ground illusion flip-flop (active resistance, error).

The subtlety here is that the whole (the Gestalt) has a reality of its own:  subjective contours are real, according to the brain, and therefore must be taken into account (illusion of whole affects interpretation of each part -- illusion is functional fact).

In programming terms, a tug of war between the need for:

Top-down programming (deductive reasoning),

Specifying complex pieces, then dividing them into successively smaller pieces; and

Bottom-up programming, inductive and abductive (intuition) reasoning.

A geometry of oscillating paradox -- asymmetric, nested self-referencing and contradictory.

"Wiener had described a computer's oscillatory response ["'pathalogical' osciliations"] to being presented with a Russsellian paradox … learning about learning could lead to something analogous to a Russellian paradox (e.g., if one learns that whatever one learns is nonsense, one has a paradoxical bit of knowledge.)" 

                                                     Steve P. Heimes:  Journal of the History of the Behavioral Sciences

"Gregory Bateson and The Mathematicians: From Interdisciplinary Interaction 
To Societal Functions", 1977  


Thus, S3 is about continuous synchronous and asynchronous switching:  nested, dynamic, switching of cognitive and perceptual and mathematical dichotomies, where the manipulator must constantly re-examine reigning assumptions, transcend and control nested and evolving recursion and contradiction, by inventing new propositions (Boolean algebra -- a way of seeing new structures -- fundamental to the design of binary computer circuits and programming language).                

Like Cantor’s paradoxKant’s transcending spheres, an antinomy, Srotation creates an effect that is not in direct proportion to cause, with sensitive dependence on initial conditions; the dichotomy of classical Boolean two-valued logic (every proposition is either true or false) does not apply to the S3 (it’s a holon -- simultaneously whole and part).  Note that individual S3 logic elements do not necessarily have a discrete true / false state at any given time; simple Boolean, bivalent logic is inadequate for this, thus extensions are required.

In Aristotle’s terms, with each S3 rotation there is a back-and-forth, a switching between "efficient cause" (guiding principle / linear assumptions with prior condition as cause) and "final cause" (aim or purpose / nonlinear assumptions of future condition as cause).

But Aristotle's logic is concerned with discrete elements in a deductive pattern, like the Rubik’s Cube:  find the answer (instead of the question, the concepts); accept the formulation of the problem as a given (instead of calibrating the permutations and difficulty).

Hegel, to the contrary, dissolves this classical static view and presents a dynamic, spiraling movement toward a whole, which embraces each of the ideas or stages which it has subsumed (contradiction of overcoming and preserving: Aufhebung, “sublation”):  a totality, a sort of fractal architecture. 

Thus, a demonstration of Hegelian dialectics of pattern and of process, a mixing of different levels of abstraction, self-reference.

In Open Systems Thinking terms,

“… analysis of a system, which begins by disassembling it, can reveal only its structure and how it works, but not its essential properties or why it works the way it does.  Synthesis or synthetic thinking, on the other hand, treats the thing to be explained as part of a containing whole and focuses on function rather than on structure.  It reveals why things operate as they do.

                   M. S. Sommer, "Open Systems Listening: Conversation As A Research Tool", Ph.D. Diss.,1988 

Thus, in a nutshell, whether from the perspective of programming, or logic, or Open System Thinking, etc., the Spresents the manipulator with interesting operational issues,  Fuzzy issues.  

Sis a topology of paradox (contradictory cognitive and perceptual signals / messages at different levels, with a negative injunction forbidding acknowledgement of that contradiction).

In other words, the manipulator must continuously focus on the binary issue (part) of a given tunnel, proceed upward through the logic hierarchy to the S3(s) (whole), and then go back down the hierarchy and repeat the process, again and again, as all the issues compound.

(Note the interesting Sproblem spaces, Boolean Construction, and Atomic Warfare -- surgical elimination of command / control node-links -- at Challenges / Games.) 

                                       To Repeat 

                                         SOpen Systems Thinking

"Turbulent  Field ...  the dynamic properties arise not simply from the interaction of the component organizations, but also from the field itself. The "ground" is in motion.” 

                                                    Emery and Trist, "The Causal Texture of Organizational Environments”, 1965.

             Discovering and Representing A System Of Rules

                      S3:  Programming Binary Logic Gates

Simply put, the manipulator must continually switch between deductive reasoning (rule driven) and inductive & abductive reasoning (discovery driven) skills in an evolving spiral of abstract reasoning.

A cycle of concrete to abstract    

                            Thesis                     Antithesis                     Synthesis  


where each S3 rotation / reorientation of a logic gate (single tunnel) / logic block (four tunnels in concert) impacts goals and information creating dynamic decision environments, uncertainty over means and ends: 

the same action, or order of operations, can be right one moment and wrong the next. Current goals and information are ambiguous.


                                                                     Analog-Binary Processor (0/1)

S3s is a holarchy, an architecture of constant and unexpected contradiction and recursion -- the manipulator must "drill down" through these dichotomies -- an Hegelian logic hierarchy:

cubical form logic frame (six faces) > holes quadrant logic frame (eight holes) > tunnel (array) logic frame > tunnel (binary switch) logic frame); 

The manipulator must continuously focus on the binary issues (part) of a given tunnel, proceed upward through the logic hierarchy to the S3(s) (whole), and then go back down the hierarchy and repeat the process, again and again, as all the issues compound.  And continuously ask, Why? 

In idiomatic terms, with the S3 (which is self-referencing to the core) “you can’t get there from here” (unless you know where you are going), so you have to start there (the map) to begin from here (the terrain).  In ironic programming parlance -- "to understand recursion, you must first understand recursion.”

A cyclical, dynamic process of analysis (separation into constituent parts) and synthesis (fusion of parts into whole), verification and correction, expressed as programming steps, from formulation to execution. 

The S3s is a a Cybernetic switch, encompassing the Systems Thinking idea and operation of wholeness and the relation of whole and parts, structure of systems and their behavior, laws of regulation, and recursive processes of self-observation and self-reference in the service of reasoning. 

In other words, S3 is about adaptation to uncertainty:  unleashing the instinct to think, with Hands and Mind; play, with freedom of means and ends, and why it is important.

Thus, in Systems Thinking parlance:

Saccommodates nested and recursive structures / functions / processes.

S3 is a holon, at once whole and part, 

S3 is other than the sum of its parts.

Part of the S3 affects all others.

Small change has big impact.

Evolving network resists central design or coordination.

Global system emerges from local agent interaction.

No objective physical space obeys a single set of laws.

Each S3 rotation changes the ground rules, locally and globally.

Each S3 rotation changes the navigation constraints / opportunities.

Competing (necessary-but-not-sufficient) Sstructures / functions / processes are not necessarly similar. 

S3 is not within a field; it is the field.

Field is not a passive arena (simple game, puzzle and maze): it is effectively “purposeful” (telology)-- it has a “will” of its own.

Field is turbulent: dynamic processes arise from that field.

Environment causality is "richly textured":  S3 cause and effect is nonlinear, dynamic.

Actions, concurrent and reactive, must compensate for processes with no forewarning.

Paths from same starting points can end at different places.

Paths from different starting points can end at the same place.

System breaks down (AIR BALL can require trajectory adjustments).

It “works” / “does not work” thinking is inadequate; there are other contextual possibilities.    


                              "Science and Complexity



© Michael S. Sommer, Ph.D, 2018