"Transire suum pectus mundoque potiri”    (Archimedes)


The S3 is a hybrid:  combined puzzle and maze and rolling ball device, where intuition is a prerequisite -- logic is necessary but not sufficient.

S3 is about mental and manual rotation in the face of contradiction of action and environment:  an illogical universe, and its implications.

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

                                                                       The Dialectic 


                        Operation and Activity of Abstraction


"[The universe] cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word." 

                                                                       Galileo Galilei, Opere Il Saggiatore, 1623


Q.  What is a number?

A.  An abstract idea, used in measurement and counting.

Q.  What is a numeral?

A.  A symbol, representing a number.

Q.  Where to begin?

A.   Albert Einstein's “Combinatory Play”, the genesis of mathematics.


“The psychical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be 'voluntarily' reproduced and combined... this combinatory play [emphasis mine] seems to be the essential feature in productive thought before there is any connection with logical construction in words or other kinds of signs which can be communicated to others."

                                           Albert Einstein:  letter to Jacques Hadamard,

                                                     The Psychology of Invention in the Mathematical Field, 1945 

S3 is about continuous synchronous 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.

                       S3:  Programming Binary Logic Gates

             Discovering and Representing A System Of Rules


                                                                                                                                 Binary Processor (0/1)






Michael’s little idea:

Symmetry under transformation.

Juxtaposition of square and circle across three dimensions: 



The rule which governs a system or phenomenon, exactly like numeric, musical, or visual relationships, and its simultaneous mental and manual rotation.


Buckminster Fuller's "coordination of thought and physical action, the genesis of geometry, system, and structure.


While S3 demands recognition of sequence / pattern of repeating events formed in accordance with a definite rule(s), different individuals can perceive the same pattern differently, reach different generalizations.  

It’s mysterious.

In Cybernetic terms, S3 is about demonstration of functional relations, particularly goals and feedback, of Norbert Wiener’s ANIMAL  /  MACHINE cooperation in DYNAMIC SITUATIONS -- fields which can change (or seem to change) autonomously.

S3 fields are "pattern blocks", which create porus polyhedron "chains" formed by distinct collections of figurative elements:  circles and squares.

Thus, Sis a conspiracy of pattern, a dynamic router, a system of programmable binary (0/1) switches, a “flying junction” (an over-and-under junction of tunnels which weave past each other) housed within a cubical exoskeleton with multiple axes of rotation and alternative, manipulator-controlled ball trajectories, offering exit in all orientations.  

A distributed processor (S3) programming network.



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


"… 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


With each additional S3 (transparent or opaque) linked tunnels offer paths of exponentially 

a)  more combinations (complex binary sets and patterns and groups), but 

b)  fewer permutations (possible successful paths).

That is, the closer to the desired end, the more high level reasoning is required to proceed without disrupting the entire Snetwork.  And with each additional path the other path(s) / algorithm(s) must be redone in concert with the new interrelationships to optimize multiple simultaneous paths:  (ike the Frame Problem:  no boundary for what information is relevant, what will change, what will remain the same), and how to adapt (they don't know the rules until the game evolves):

"'solutions to the narrowly conceived problem have the effect of driving a (deeper) difficulty into some other quarter of the broad problem ” 

                                                   Daniel Dennett  (Hayes):  Cognitive Wheels: The Frame Problem Of AI

      Approaching Complexity (Something You Cannot Predict)

                                      Unconscious Inference

"From the perspective of a user, the crucial factor is not whether the problem is intrinsically complex (suitable software can make even complex problems seem simple), but rather the cognitive resources that the user is prepared to devote to solving the problem."

                                                          Alan F. Blackwell, ˆWhat is Programming?” (2002)

The developing S3 network requires constant revision of operational sets and simultaneous consideration of many possibilities from many perspectives: the order of steps is not fixed, there is no algorithm.

From this perspective, Sis an engine of paradox which generates continuous ambiguous, bi-stable stimuli, cognitive and perceptual, such that the brain cannot commit to just one interpretation, but must switch back and forth, providing best guesses, unconscious inference ("unbewusster Schlussin Helmholtz's 1867 terms.   

It demands the manipulator understand and assimilate evolving information about patterns and functional relationships and analyze change in both concrete and hypothetical contexts: a meeting of abstract ideas with the spirit of mathematical rigor.

In other words, the Sis about awareness of environment, modifying self-behavior on the basis of experience (feedback), through the optimization of multiple simultaneous paths, in a constant stream of evolving problems -- within a topology of paradox (contradictory cognitive and perceptual signals / messages at different levels, with a negative injunction forbidding acknowledgement of that contradiction:  Double Bind):

A dialectic engine, developing logic and intuition; a tangible, lucid demonstration of basic behaviors (consciousness) of the brain through mechanical concepts (switches); 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.


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


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 ever expanding universe of S3 (overarching whole / system of systems).  Then go back down the hierarchy of hierarchies and repeat the process (while coping with contradiction of action and environment).  A dynamic cycle of recalibration and reflection.

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

The exploration of symmetry, influences among parts and whole, square and circle --  


adaptation to accelerating rates of change and complexity of system and environment: a cognitive development tool.  While Rubik’s Cube, without that paradox, exercises logic, Sdemands logic and intuition.

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

                          Symmetry Under Transformation

From another perspective, Sis a self-generating, self-sustaining, escalating causal loop, where objects / ideas are taken to pieces, analyzed, and reassembled (a cycle of cognitive development where the manipulator / observer is both agent and object of change).

Thus, Sis transformative,  demonstrating:

Open Systems Thinking as a working epistemological premise: a rotating, cubical field of “forbiddingly complex environmental interactions” (F. Emery, Systems Thinking, 1969). 

Second-order Cybernetics:  "the cybernetician, by entering his own domain, has to account for his or her own activity. (Heinz von Foerster, Cybernetics of Cybernetics)

Artificial Intelligence / Mathematics (the science of patterns) Play (without symbols), Symmetry and Consciousness -- “strange loops”, feedback on higher levels. (Gödel, Escher, Bach / Norbert Wiener / Ross Ashby)

Play of causal inference / adaptation.

A cubical maze module.  Symmetry under transformation.



From yet another perspective, the Sis about active control of thinking while learning -- not just goal-attainment;   partnership of motor and mental rotation (unlike "mouse learning" /  computer learning), essential in the very early years, which facilitates robust exploration of abstract concepts in visuospatial terms. 

        “Logic is for demonstration, intuition for invention.”  

                                                                                      H. Poincare, L’Enseignement Mathematique (1899)


Simple "puzzle solving" logic is not enough -- there is no algorithm with S3.  As combinatorial binary issues (0/1) rapidly become overwhelming a robust flexibility of logic and intuition,"intelligent trial and error", must be exploited to continuously compose / decompose the problem from / into sub-problems and multiple constellations, by continuous analogical mapping and perceptual processing to program dynamic patterns and causality of structure, function, and process of square and circle:  the manipulation of abstractions without symbolic and numeric tools -- core ideas spanning all disciplines.

Thus, the S3 is a low-tech, high-yield, cognitive tool which can play a critical role for the individual as well as formal education and science disciplines' (STEM) efforts to recognize and develop spatial reasoning -- key to intelligence.

And it's fun.

"Recognizing Spatial Intelligence", SCIENTIFIC AMERICAN (2010) 


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

                                          Play of Ideas

From another perspective, S3 is the Cybernetics equivalent of a model train set.

"The first rule of discovery is to have brains and good luck. The second rule of discovery is to sit tight and wait till you get a bright idea."

"Analogy pervades all our thinking, our everyday speech and our trivial conclusions as well as artistic ways of expression and the highest scientific achievements."

"… the main achievement in the solution of a problem is to conceive the idea of a plan.” 

                                                                       G. Polya, How to Solve It (1945)

"You can turn ideas around in your mind to examine them from different perspectives until you find one that works for you.  And that's what we mean by thinking." 

                                                        Marvin Minsky, "Commonsense-based Interfaces", CACM (2000)

"Logic is a poor model of cause and effect.

                                                                        Gregory Bateson, Mind and Nature: A Necessary Unity, 1979

“The most beautiful thing we can experience is the mysterious. It is the source of all true art and all sciences.”

                                                                       Albert Einstein, Living Philosophies (1931)

“Painting isn’t an aesthetic operation; it’s a form of magic designed as mediator between this strange hostile world and us.”

                                                                       Picasso, Quoted in Mario Livio, The Golden Ratio (2002)

"A set is a collection of definite, well-distinguished objects of our intuition or thought.”

                                                                       Georg Cantor, Math. Am., 46 (1895)


                                           Why S3?

"It [Cybernetics] takes as its subject-matter the domain of "all possible machines" .... What Cybernetics offers is the framework on which all  individual machines may be ordered, related, or understood.  

                                                       Ross Ashby, An Introduction To Cybernetics, 1956


S3 Talks / Papers / Demos

“What Ever Happened To Systems Thinking?” 


8 / 3 / 2014, Washington D.C. 

American Society for Cybernetics Conference


8 / 9 / 2014, Washington D.C.

International Society for the Systems Sciences Conference


Answers to ASC President Stuart Umpleby’s Questions (6 / 30 / 14)


"What is S3?

“Game?"  >>  CHALLENGES

"Puzzle?"  >>  IDEA  >>  PUZZLEMENT

"Metaphor?"  >>  ECOLOGY OF MIND

"Three dimensional math, etc.?"  (bad term / right idea)  >>  ECOLOGY OF MIND  >>  LOGIC

"How did you come up with the idea?” (this is my favorite question)  >>  ARCHITECTURE / GEOMETRY

"How can the S3 be used?"  >>  HOME

"You seem to see them as a metaphor for many phenomena." 

Yes, von Bertalanfy / Norbert Wiener / Ashby did too.

"Please explain the matching between the S3s and the real world situation."  >>  ECOLOGY OF MIND  >>  COGNITIVE TOOL  >>  ECOLOGY OF MIND  >>  MATH / RATCHET  >>  ECOLOGY OF MIND  >>  LOGIC  >>  ECOLOGY OF MIND  >>  RESEARCH  >>  IDEA  >>  BLOCKS / HANDS AND MIND  >>  IDEA  >>  SPACE AS ART  >>  IDEA  >>  QUAD PROBLEM  >>  IDEA  >>  CHALLENGES        


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© Michael S. Sommer, Ph.D. 2015