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                                   Symmetry Under Transformation

                                   Cubical Maze Module

                                              a necessity

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:  functional relations, particularly goals and feedback.  Synergy.


To Support and Strengthen the Disposition to Go On Learning Throughout Life --

To enable the children to teach us -- In the face of paradox, the collapse of our expectation of a logical universe.


The Cybernetics equivalent of a model train set.

The Zen koan mechanics of enlightenment.

Mind and Machine as fugue.



“A too restricted view of human nature … even though only briefly ascendent, can significantly alter the expectations and, hence, the behavior of men and societies and may thus provide its own bogus validation.”

                                                                                             Geoffrey Vickers, The Art of Judgement, 1965



the new reasoning is that if it is not electronic, it can't be significant.  Which is too bad -- and dumb.

"Steve Jobs:  The Next  Insanely Great Thing." 

"I used to think that technology could help education.  I've probably spearheaded giving away more computer equipment to schools than anybody else on the planet.  But I've had to come to the inevitable conclusion that the problem is not one that technology can hope to solve.  What's wrong with education cannot be fixed with technology.  

"No amount of technology will make a dent."

                                      Steve Jobs (as quoted by Gary Wolf, The Wired Interview, 1996)

The sense of touch overrides the human interaction with complex cognitive and perceptual content in paradoxical ways.  The robust integration of haptics / Hands and Mind is missing, from early childhood computer learning to joint pilot / computer management of aircraft (increasing absence of hand-flying).  Dynamic interaction of the brain and its environment trumps brain alone.

"Nature appears to have built the apparatus of rationality not just on top of the apparatus of biological regulation, but also from it and with it."

                                                            (Daniel Dennett’s ReviewAntonio Domasio, "Descarte’s Error”, 1995


                                                                           TABLE OF CONTENTS    

S3:  What it is.  What it does.  The Answers take a little explaining:  follow multiple perspectives --  the sciences, art, and music -- with Links.  

REVIEWS                                                                                 ECOLOGY OF MIND

PROGRAMMERS                                                                                COGNITIVE TOOL (low-tech)

IDEA                                                                                                     INQUIRY / DISCOVERY

          ARCHITECTURE / GEOMETRY                                                MATH / RATCHET

          BLOCKS  / HANDS & MIND                                                       LOGIC

          PUZZLEMENT                                                                           PROBLEM SPACE / TOPOLOGY

          SPACE  AS ART                                                                        RESEARCH

          QUAD PROBLEM                                                           PLAY  AS THINKING 

          PITCH DECK                                                                             STEM WITHOUT BOUNDARIES 

          CHALLENGES                                                                           ALGEBRAIC PLAY 

                    COMBINATIONS / GAMES                                             “CREATIVITY” / INNOVATION 

                    COURTYARD                                                        CYBERNETICS / SYSTEMS THINKING

                    AIR BALL TOWER                                                 PROFILE

                    3x3x3                                                                      APPENDIX - PHOTOS     


                                              the dialectic -- open your mind


What does the Sommer Cube (S3have to do with our new world?  It exercises the cognitive resources necessary to program a computer (0/1) -- but without the computer.  There is too much computer, not enough peopleness.  That is, unlike the iPhone, etc., the S3 allows us to hold on to some of the things we are losing … (see also Norbert Wiener’s, The Human Use Of Human Beings)

The S3 is about pattern, the rule which governs a system or phenomenon, exactly like numeric, musical, or visual relationships -- and its simultaneous mental and manual rotation.

  The Smart Block.        The Upgrade      

S3 is about Play, the absence of its scientific definition and cognitive payoff.

S3 is about playfulness, imagination and rigor -- in the face of real and abstract variables.

Sis about the original Synthesis of Cybernetics and Systems Thinking, and its tangible, lucid demonstration:

Awareness of environment, modifying self-behavior on the basis of experience, through the optimization of multiple simultaneous paths, in a constant stream of evolving problems, within a topology of paradox.  (“How complex environments push brain evolution”)

In Cybernetic terms, Saccomplishes this by regulatory switching (synchronizing concurrent communication / interference among rotating S3s), a variety increasing, heuristic routing (ball or vector) system seeking best paths (optimization) in an unfriendly (blocking) network / environment of switchable links:  

A distributed processor (S3) programming network.  



Thus, the S3 is 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)

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.

A robust, low-tech, high-yield cognitive tool: a topology of Hands and Mind (note manual / mental rotation studies -- unlike "mouse learning" / computer learning).  "Brain Games are Bogus.

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 continuous switching of square and circle.


 The embodiment of Albert Einstein’s “Combinatory Play"            

(visuospatial:  analogical mapping and processing);

Mathematics (science of patterns) Play -- without symbols); a dynamic cycle of concrete and abstract reasoning.


From developing mind to powerful intellect,

infant, child or world-class mathematician, 


Transformative challenges and games.



From manipulation (manual and mental rotation) to research, across the age spectrum:  (Autism, Alzheimer's, visuospatial, etc.),


The S3 offers a true "parallelism of general cognitive fields", 


"I'd like to see the S3 in every preschool and elementary school classroom, and in the home of every family with young children.”  

                                                                             Laura E. Berk, Ph.D.  Child Development, 8th Edition (cc mss)


Crucial to the development / exercise of visuospatial reasoning in

Science Technology Engineering Mathematics (& Art), 

(particularly in the formative years).  

Discovery through paradox:  Freedom to think.

And it's fun.



                  Cognitive Tool / Classic Toy / Virtual Reality


                Marriage of Constructivism and Computer Learning

                           Full-Spectrum Cognitive Tool Kit

                                       S3 As Smart Block


    (S3s in every pre-K “block corner” / elementary school: explore abstract concepts)

PROGRAMABLE LOGIC BLOCKS (binary gate arrays / switches)

    (“Levels:  High School (9-12), College, Research …  

     Math Topics:  Higher-Dimensional Geometry, Topology”)







The S3 is about pattern, the rule which governs a situation or a phenomenon, exactly like visual, musical, or numeric patterns.

                                          S3 As Fugue

“... a work of instruction and enlightenment, a kit to be assembled ... that looks backward and forward at the same time ...architectural, rotatable, invertible.”   

                                                                                                                    Paul Epstein on J.S. Bach


continuous interweaving, user created, network of tunnel paths (inner  voices -- relating to bass line -- cantus firmus), imitative / developed contrapuntally into an orientation-independent and network-interdependent system of harmony.

S3 as Albert Einstein’s "musicality in the sphere of thought.



                                                                                          J.S. Bach, Die Kunst Der Fugue

In a joyful way  

S3.  Play.



Support The New World of Child-Designed Education!






wall flower 2 crop copy crop 2

                                                                           "l'espoir vis en moi'


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© Michael S. Sommer, Ph.D. 2014