For white backgrounds (without audio / video) click Safari Reader View. 


                                   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 (Analog-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

This is the story of the Sommer Cube (S3), a cubical maze module — leveraging Binary and Analogical reasoning in a hybrid toy:  a cognitive tool.  Analytic and Holistic thinking; a competition of Rational (programming desired logic functions) and Nonrational.

An exoskeleton, with form and space consistent with Gestalt Illusion and Architecture.  Thus, it is analogical in nature.  However, the exoskeleton also circumscribes a system of linkable tunnels which depend on the Law of Gravity.  Thus, it is also binary in nature.


In other words, S3 is a systematic confusion of whole(s) and part(s), a topology of paradox (different sorts of paradoxes) requiring a back and forth of cognitive and perceptual faculties where the manipulator must rise to successive higher levels of abstraction:  Hegel’s dialectic.

Thus, S3 is about the physical and mental universe, a dynamic, precise structure of inner connected ideas, their context, and their logical expression:  variations on a theme of square and circle   -- flip-flop, a counterpoint of logic and intuition with myriad “permutations” of dead-ends and thruways (where particular order matters, like a combination lock.)

In operational terms, Swith its nested reference frames creates an interdependence of analogical and binary reasoning, transformation to higher and higher levels of abstraction, much as algebra (“al-jabr”:  reunion of broken parts) converts geometry into algebra (and vice versa) with the Cartesian Plane.    CHALLENGES / GAMES 

(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.)


“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)

One of the most powerful forces in the universe:  the human mind as Gestalt Engine — mind as machine — a dynamic field where the parts are at war with each other.   The furious evolutionary competition of patterns, structure, and logic — parts to their contexts:  the foundation of cognitive process and Artificial Intelligence.

With S3, that human mind (context), with its evolutionary field effects (parts), confronts an external Gestalt confusion, which demands an evolutionary cycle of cognitive development.  That is, the S3 is both agent and object of change; the mind is both agent and object of change.  In an eminently researchable partnership.

Thus, Science Technology Engineering Mathematics in a low-tech / high-yield, cost-effective format, where questions of function, role and position determine meaning.  What seems to be not understood is that science is a question, not an answer — any child will understand that.  


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


“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.)

A contribution to Child Development and Gestalt Theory, and a little more. 

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).

It's about low-tech in a high-tech world and understanding, and therefore balancing, the benefits of each.  

It’s not just philosophy — it’s about reinventing us:  carefully.

The question is, Machine-think or Human-think?  Remember, we can switch how we think  re-program ourselves.  We are all programmers  but not limited to programmatic thinking

Rubik’s Cube "cooperates” — there is a winning algorithm. 


S3 won’t “cooperate”.

There is no winning algorithm (think like child, or Cybernetician — like Shannon's Ultimate Machine).  


Enter S3. 

The new reasoning is that if it is not electronic, it can't be significant.  Which is too bad  and dumb.  A subtle form of tunnel vision.

Thus, the Sinitiative:  introducing SPlay (within an operational definition of Play as evolutionary regulatory arena) to a wide audience; proposing Spatial Thinking and Learning Research / Children’s Museum creative partnership, as well as S3 availability in homes and schools. 

As Strauss put it “… collecting data on human learning based on children’s behavior in school is like collecting data on killer whales based on their behavior at Sea World.”


In other words, the world of cognitive studies / learning development (not to mention the toy companies) is often too comfortable making new inquiry into old developmental precepts, without Gestalt (descriptive and explanatory) benchmarks.  (Think endogeneity:  the solution is married to the error.)

The new video-computer / Machine-think learning precepts have become a problem in their own right:  the fix  the silver bullet"  the big “silo”.

It's all about supporting and strengthening the disposition to go on learning throughout life, while affirming the quality of human life. 

S3 is waiting. 


Let’s begin.

The challenge:  understand the dynamic, structural features of navigation in topology of paradox, where the strategies and tactics are evolutionary.  Key is the difference between objective and subjective reality, and factoring that in.  


The Gestalt engine (de Kreyser’s “we are furious pattern-matchers”), resolving dissonance by escalating perception:  bird’s-eye view — like Zen.



It’s about survival of the creativist.

S3 is a combinational (technically “permutational”, like a combination lock) 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.   

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

A dialectic.  


                                                                                         Rolling Ball “Tilt-Switch

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.  

While the S3 exterior has six faces (three sets of opposing faces), each "quartered" face with one or two holes (mismatched hole quadrants between modules are dead-ends, and proliferate erratically), the S3 interior has four intertwined chiral tunnels (each tunnel with independent logic gate orientations) with eight entrance / exits.


                                                           Asynchronous Analog-Binary Processor (0/1)

In other words, Sis a programmable logic component (in Field Programmable Gate Array terms) where the manipulator can program the desired logic functions (physical logic arrays) — actually see and touch and test them.  Algebraic analysis by other means.



Thus, S3 is a physical hybrid as well as a hybrid of ideas.

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?)

For anyone, across the age spectrum.

A simple premise, but powerful.   

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


                                                  We Are All Programmers

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).

"Recognizing Spatial Intelligence", SCIENTIFIC AMERICAN (2010)  

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


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.  

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.)


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

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.  


Toward a Science of Play (here defined as evolutionary regulatory arena) — cognitive and educational assessment — STEM:  in a low-tech, high-yield, cost effective format.


                            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)

To optimize multiple simultaneous paths (make most paths begin / cross within a single S3) the manipulator must simulate the future, conceptually "drill down” a hierarchy of nested coordinate systems.  

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).

To recap, Swith its nested reference frames creates an interdependence of analogical and binary reasoning, transformation to higher and higher levels of abstraction, much as algebra (“al-jabr”:  reunion of broken parts) converts geometry into algebra (and vice versa) with the Cartesian Plane.   CHALLENGES / GAMES 

Combinatory play.

The hypothesis:  Play as regulatory arena (survival of the fittest) for neural housecleaning (rearranging / “'loosening’ rigidly held intellectual content” — “pruningunused synapses), a harmony-seeking oscillatory activity; regulation of neural binary and analogical computation and plasticity.  


         (Analogical x Binary reasoning  >>  Probabilistic reasoning  >>  Intuition.)

Thus, S3 is about the essence of Artificial Intelligence:  information, computer and human in a situation where logic alone will fail.

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)

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. 

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.



Each Sreorientation simultaneously reprograms the four "gravity feed” tunnels 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.

In other words, 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).  


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

A counterpoint of logic and intuition.





                                                                          four bit (0/1) — nibble




                                                      Asynchronous Analog-Binary Processor (0/1)             



Hello, 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.

Let’s raise the bar.

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