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                                   Symmetry Under Transformation

                                   Cubical Maze Module

      Toward following the vision of Leonardo da Vinci to see Art and Science as one.


                 Let’s Not Underestimate the Mind of the Child   


                                                    CONSTRUCTION BLOCKS (Binary Processor

                                               Full-spectrum COGNITIVE TOOL

"The scientist is always working to discover the order and organization of the universe, and is thus playing a game against the arch enemy, disorganization."

                                                Norbert Wiener, The Human Use Of Human Beings, 1950

Let’s not underestimate the mind of the scientist ... the child.


"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 (April 10, 2008, cc m.s.)

This is the story of the Sommer Cube (S3) and leveraging binary and analogical reasoning in a hybrid toy / cognitive tool.  STEM:  in a low-tech / high-yield, cost-effective format.  

For anyone, across the age spectrum.

A story of, "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” / reachability test; think graph theory.)


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 self-reflection (not "mere facts", but principles); functional relations, particularly goals and feedback (What information is relevant?  What are my assumptions?  Are they justified?)

A simple premise, but powerful.   

Ask Albert Einstein, Frank Lloyd Wright, Buckminster Fuller, etc.

S3 is about adaptation to uncertainty from the perspective of multiple disciplines:  unleashing the instinct to think, with Hands and Mind; play, with freedom of means and ends, and why it is important, in scientific terms.  A challenge to prevailing assumptions, integrating concepts that have been carefully examined in radically different fields.

The question is:  What invisible forces are driving Play, and intuition?


                                                  We Are All Programmers

A story of Spatial Thinking  (space, representation, process:  multifaceted, interconnecting competencies, human and robot), the holistic approach -- manifested by a cubical maze module -- the rational in conflict with the intuitive, and why it is important. 

In particular, Wayfinding:  spatial problem solving critically examined as cognitive tool, in conjunction with an overdue operational definition of Play, a theoretical framework in Cybernetic regulatory terms:  the hidden architecture of play.

Screates an efficiency, as algebra (“al-jabr”:  reunion of broken parts) facilitates the solution of geometry problems by other means (Cartesian Plane) — illumination of diverse problem sets (classes) across a broad range of disciplines. CHALLENGES / GAMES 

The Sinitiative:  introducing S3 play to a wide audience; proposing Spatial Thinking and Learning Research / Children’s Museum creative partnership, as well as S3 availability in homes and schools. 



                                                                                         Rolling Ball “Tilt-Switch

Formal early childhood education needs a powerful learning / research tool (research on spatial cognition and documentation on the transfer of skills and logic acquisition from Sto STEM -- complex and real world stimuli).  

Thus, S3 addresses the need for more learning proficiency than problem solving competence: situation awareness, creativity, and the careful construction of logical arguments, and their limitations.  Scientifically verifiable Spatial Thinking  (spatial:  ability, reasoning, cognition, concepts, intelligence, environmental cognition, cognitive mapping, mental maps, etc.) tools and learning opportunities are needed, for pilots, astronauts, surgeons, etc., as well as teachers and students, diagnostic as well as development.

Sachieves this by leveraging full-spectrum cognitive / perceptual processing, advanced thinking and intuition (essential for  Science Technology Engineering Mathematics, STEM, and its inseparable partner the Arts -- therefore STEAMwith Hands and Mind.  (Slearning / development research is a first step toward rigorous, scientific demonstration that STEAM is as effecitve as STEM.)

“I was particularly taken with the possibilities the S3 offers for strengthening spatial, mathematical, and problem-solving skills in children, beginning in the preschool years.  The S3 readily engages and captivates children, sustains their attention, and challenges and stretches them cognitively.  Moreover, it enables children to calibrate their own complexity of play--making the S3 ideal for any age and age-mix of players, child-child or child-adult.  We have much scientific evidence to indicate that children benefit enormously from such collaborative endeavors, both cognitively and socially.” 

                                     Laura E. Berk, Ph.D.  Child Development, 8th Edition (May 13, 2008, cc m.s.)


Science Technology Engineering Mathematics -- the future of the U.S. is on the line.

Research "results suggest that spatially enriched education could pay substantial dividends….”

"At present, it seems that inadequate attention is being paid to the teaching of STEM in early childhood education. Less than 5 percent of time in formal early childhood education settings is devoted to STEM learning, according to research presented at a recent conference on fostering STEM trajectories organized by the Joan Ganz Cooney Center and the New America research institute.”  

                                                         Rafiq Dossani, "Integrating STEM in Early Childhood Education

Think of that black swan which disproves the contention that all swans are white (S3 is a physical hybrid as well as a hybrid of ideas), like the promise of Spatial Thinking, a long overdue consolidation of important scientific inquiries and educational initiatives.

STEM development should be about power -- first, the power of abstract reasoning and then its practical application -- right from the beginning (bottom-up, in addition to top-down).  Spatial processing -- (teaching and learning).

"Spatial ability and STEM:  A sleeping giant for talent identification and development” for the U.S. and the developing world.

Thus Early Spatial Thinking ("gateway or barrier?) should be incorporated into existing standards-based K-12 instruction acrosss the school curriculum.  


                            The Smart Block.

                                                             Binary and Analogical Reasoning

                                                             (switch from fixed to changing values).

                                                             The Upgrade. 

S3.  What it is.  What it does.  

The Answers take a little explaining:  follow multiple perspectives -- mental and manual rotation:  the sciences, arts ( SPACE AS ART), and music -- with Links. 


                                                                                                                         Insightful Elephant & work


Adventure By Topology   

(technical hints)

Or go directly to OVERVIEW 

 An architecture of complexity.  (Remember, link multiple simultaneous paths.)



                                                    (checkered paths allow exit)

Sis 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)


                                                    (VIRTUAL REALITY paths confirmed)

Back to basics.

S3 is about exploration of situation awareness, transformation of parts and whole, square and circle, feedback and symmetry -- adaptation to accelerating rates of change and complexity of system and environment:  a cognitive development tool (manual / mental rotation studies).  Network topology, physical and logical -- with hands and Mind.


                                                                         Problem Universe

                              "Control and Communication in the Animal and the Machine

Thus, S3 is transformative, a kick-start, a self-generating, self-sustaining escalating causal loop:  an exploration of deep causal structure (in "hard science”, beyond behaviorist psychology descriptive terms -- think endogeneity); an evolutionary cycle of cognitive development where the manipulator is both agent and object of change. 

To be precise, S3 is a technological mechanism, a dynamic router, a Binary Processor which employs corrective feedback (non-linear feedback controlled "purpose”) which generates oscillating, intermittent restriction of cognition and perception within a dynamic logic hierarchy, a topology of paradox:  the cognitive mechanics of Gregory Bateson's Double Bind paradox (Zen koans employ the same flip-flop mechanics): contradictory communication at different levels, where acknowledgement of that contradiction is forbidden); and Wieners’s oscillatory response, "pathological oscillations”, to "set-theoretical" paradox). 

Thus, Sis predicated on the tension of switchiness (compound cognitive, perceptual, mechanical flip-flop and schedule of reinforcement):  switching of relative motion and formand navigational strategies.  

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.


Thus, Sis about Apex reasoning, patterns of thought exceeding the parameters normally experienced in logical operations -- where all the rules change; discovering order in what is seemingly random, in starkly contrasting conceptual frameworks, different problems sharing a common medium.


Variations on a theme of square and circle -- flip-flop, juxtaposition across three dimensions.

A counterpoint of logic and intuition.


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


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.

(Note that Snonlinearity -- change in one variable which does not produce a directly proportional change in the result -- even in the single S3, is effectively a nonlinear expression / experience which allows one to intuitively graph the output as a curve -- very exciting stuff for the manipulator with a questioning mind.   And that’s just the first S3.)

Thus, S3 is an eminently measurable (think signal-to-noise ratio: mental opposed to manual rotation / neurophysiological) interface (invites interaction and responds to it) in a paradoxical environment -- hierarchies of different rules and changing order of operations, principles have to be unlearned) versus logical algorithms, "naive physics”.  

A tangible 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.

A programmable logic component (In Field Programmable Gate Array terms), a combinational logic block which contains four tunnels, effectively an array of unconnected, gravity-dependent switches to be programmed by the user as the block is rotated in space, which can be connected to other logic blocks to create multiple adaptive, simultaneous, routes by reconfigurable interconnects. 

S3 turns four switches on / off differentially, unpredictably, with each rotation:  sort of purposeful, having apparent agency, independent intentions and decision making; reconfigures / “randomizes” routing with each rotation (note compound reinforcement of manipulator).  

In summary, to repeat, S3 is 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. 

With Hands and Mind.)  

And it’s fun.  (CHALLENGES / GAMES)


Toy Industry.

Consumer Knows Best 

Scientifically verifiable Spatial Thinking toys are needed -- today.

Unless toy manufacturing reflects what people want, it can win ROI, but lose the customer base.

STEM should not be just a toy "niche market" issue (a "money loser") -- it should be strategic policy, responsibility to consumer issue.  STEM WITHOUT BOUNDARIES (STEAM)

Putting today’s "education-minded consumer" into the strategic planning / intelligence driver's seat.

Spatial Thinking.  Classic Toy.  Full-spectrum cognitive engagement across market segments.

SBusiness.  (PITCH DECK)




How to cite this material:

Sommer, M. S., (2017).  Sommer Cube:  Symmetry Under Transformation.  Retrieved from

"North Haven:   A State of Mind




                                                                                 "l'espoir vis en moi

© Michael S. Sommer, Ph.D, 2017