Symmetry Under Transformation

                                   Cubical Maze Module

Toward following the vision of Leonardo da Vinci:  Arts and Science as one.  A hunger for experimentation, abstraction, a manipulation of concepts.  Enlightenment

"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


                                                   We Are All Programmers

This is the story of the Sommer Cube (S3), a cubical maze module:  classic toy (Tate Museums) and cognitive tool (developmental and research).  It is at once fundamental and obscure — leveraging Binary and Analogical reasoning, from concrete to abstract.

It’s about patterns — always patterns. 

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.  


              Children formulate a Spatial Systems Problem  (checkered paths allow exit)

                                          Mathematician too:  QUAD / FRAME / ROBOTICS PROBLEM

A system of logic blocks 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.  Spatial proficiency.

black stem red 2 STEAM xx- copy

It’s about the need to eliminate STEM / STEAM bias against "girls and minorities” by illuminating a side of the manipulator often not otherwise seen.


                                     S3 Day at Pennsylvania Academy of the Fine Arts -- April 9, 2002

It’s about abstract idea.  (“What is a number?"  —  "An abstract idea”.)

“What turned me on was this mysterious relationship between things.  There is this wonderful world of logic and connections that is very difficult to see.  I can see trees and cats and people, but there is this other world and it’s very, very powerful.”

                                                                                          John H. Conway, (NYT, 1993). The Book of Numbers


Science as the Arts by other means.  "Come play with me".


Topology -- from planar to spatial environment.  

(Note Descartes’ insight that "smooth surface can be approximated with a polyhedron”.)

Let’s discover how it all began.


                                                   Tangram morphs from 2D to 3D.

Stasis  vs. Rotational Motion.(squareness  vs. circleness)   

   Root (what it is) -- squareness.  

(extrude cube)

   Process (what it does) -- planar rotation of parts — circleness.  

(extrude tunnels, sphere)

It’s about patterns -- always patterns. 



                                        CONSTRUCTION LOGIC BLOCKS (Analog-Binary Processor)

                                                                      CHALLENGES / GAMES 

It’s about thinking skills and evolution.

Optimization of multiple simultaneous paths (make most paths begin / cross within a single S3), in a constant stream of evolving problems, within a topology of paradox: Watzlawick’s "contradiction that follows correct deduction from consistent premises”.  “How complex environments push brain evolution”  and Bohm and Peat's Science, Order, and Creativity.  

Survival of the creativist.

 "Constructivist developmental theory is needed in developmental neuroscience.

Problem space as evolving construct:  a furious evolutionary  competition of patterns, structure and logic at multiple levels of abstraction and relational complexity.


It’s about self-regulation. 

Enlightenment, concocting and evaluating new hypotheses.  The power of Play (here operationally defined as evolutionary regulatory arena).

For anyone, across the age spectrum.  

Danniel C. Dennett (”Making Tools for Thinkingsummed it up:   "This is a fascinating object.” (April 28, 2014, m.s.) 


It’s about synthesis of ideas. 

The full engagement of the physical and mental universe, Hands and Mind; a dynamic structure of nested ideas, their context, and their logical expression:  variations on a theme of square and circle    — continuous flip-flop between 2D and 3D, a counterpoint of logic and intuition with myriad and changing dead-ends and thruways. 


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.

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

Accordingly, each Sreorientation 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.


                                                                             Rolling Ball “Tilt-Switch

                                                                      (Resets according to orientation.)

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 networks. 

A story of "Distributed Adaptive Logic Block Network” , in 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.)

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?)  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.”   



In other words, "dynamic modeling”, constantly updating and making adjustments to one’s own thinking:  learning from myriad and nonlinearly increasing “permutations” of dead-ends, and decreasing thruways, where the evolving errors of interdependent feedback loop information serve as teacher, revising behavior and blueprints for thinking.  Spatial thinking.

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.  (Picture Srotation, slowly, in your mind.)

It’s about cognitive payoff (Scientific American) under uncertainty.  And having fun.

A multidisciplinary benchmark, concrete to abstract, both diagnostic and learning: a new standard.  Analytic and Holistic thinking:  a competition of Rational (programming desired logic functions) and Nonrational; a fundamental change of basic concepts and experimental practices. 

Extracting measurable (psychometric, neuroscience) analogies, structures and rules within an evolving system. 

                                          Scientists in Action - Euler is with us 

                                    (reformulate the problem in abstract terms). 

                                                   We Are All Programmers


Untitled 4

                Let’s Not Underestimate the Mind of the Child

It’s about “objective” science.   

Conventional cognitive study maintains that it is not possible for this five-month-old baby girl to exercise the necessary level of abstraction to work with S3 — but let’s say she did just that; It’s subtle.  (Let’s speculate / conjecture:  her "world" is paradoxical, her cognitive evolution is dynamic, nonlinear; not “stages”, but continuous messy adaptation to paradox, Edelman's "unlabeled place full of novelty"  evolutionary survival skills of Bayesian inference, subjective probabiity as epistemic tool; here choosing and formulating problems,  sets:  holes "same” / ball path exits “different"? — watch her eyes.)  Hands and Mind experience embraced adaptation to create explanatory knowledge:  knowledge as tool.  That mathematician did too — in fact, he makes a living by doing just that.  Each engaged in Play — his / her own evolutionary regulatory arena.


It’s about Mind Waste:  systematic Early Education test bias / failure because early abstract / spatial thinking can not be discerned / measured by the conventional cognitive development model and associated research methods, and is therefore ignored in the education ladder (think learning trajectories). 

Thus, STEM / STEAM gender bias against "girls and minorities”.  

It’s about the acquisition / decline of abstract concepts, in Alzheimers and Autism, not to mention spatial reasoning, and education, independent of age-spectrum, the subtleties of which would be better understood using S  as diagnostic tool.  

It’s about embracing a common research benchmark employing Hands and Mind  (S3 has 24 possible orientations — “switches" reset accordinglyof manipulator-formulated "tests , as Laura Berk describes it below, “calibrating their own complexity of Play”; a nested and evolving problem universe, simplest to most complex abstract reasoning.  

51O9meJRyhL. SX335 BO1,204,203,200  copy

Donald G. Janelle (Space In Mind:  Concepts for Spatial Learning and Education)  suggested, "The ability to inexpensively produce and to broadly disseminate S^3 could, I suspect, greatly facilitate use in STEM learning environments.”  (December 1, 2017, m.s.)        

Thus, Sthesis:  

The mind of the child is unbounded — pertinent inquiry must be unbounded also.  (See below.)  Facilitate them!  Rethink abstract / spatial thinking, and therefore, Play.  Infuse S3 system of logic blocks into educational environments and research.

For example, the Corvid’s ability to use logic to separate, combine or transform ideas is, in the prevailing conceptual framework, comparable to that of a five-year-old human.  Why?  

                                                   Abstract thinker.  Watch.


Thus, Sis about these questions, challenging orthodox assumptions, integrating concepts that have been carefully examined in radically different fields.  A wide-spectrum benchmark.

Abstract reasoning redux. 

Understanding "with the eyes of a child" …  ("making links between seemingly unrelated conceptual realms”).  Thanks, John Conway.

 C4 carson - t copy birthday 2 lens correction

"Part of the underlying motivation for Cognitive Science is a dissatisfaction with the orthodox methods of studying cognition, and an impetus to change the fashion in which we think about the mind and investigate its operations …. Cognitive science, however, needs theories that both cohere and correspond to the facts.

                                                                       P. Johnson-Laird, “Mental Models in Cognitive Science”, 1980

It’s about enlightenment, concocting and evaluating new hypotheses:  science is a question, not an answer.  Any child knows that.

              S3 supporters (who all had fun with their work) speak out at  CONSTRUCTION BLOCKS

S3.  What it is.  What it does. 

The Answers take a little explaining.


It’s all about people. 

Self-regulation, still the magic ingredient — let them buy into it, gently; and enlightenment, concocting and evaluating new hypotheses  not the most profits for the toy companies.

S3 is Science Technology Engineering Mathematics in a low-tech / high-yield, cost-effective format, a system of cubical maze modules (think graph theory, concepts or classes of objects) where changing questions of function, role and position determine meaning; where innovation depends and builds upon its own abstractions.  A slowly escalating war between judgement and perception in an architecture of dynamic nested contexts.  Where an important part of the problem is inside one's own mind.  Hands and Mind.   

"Recognizing Spatial Intelligence", Scientific American (2010)  


"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



“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 

                                                                                         Awakening Children's Minds:  How Parents and Teachers

                                                                                   Can Make a Difference   (May 13, 2008, cc m.s.)



It’s about topology, remember.

Thinking within a topology of paradox.  Reasoning as Play:  directly visualizing S3 (as geometrical objects and the relations between them) becomes secondary to thinking at higher and higher levels of abstraction (think Descartes algebraic representation of Euclidian spatial relations).

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.  


                                                        allocentric / egocentric flip-flop spatial processing

It’s about math. 

“Description:  The faces of the modular Sommer Cube (S3) feed into twisted conduits that comprise a three-dimensional, tunnelling maze. The website provides example arrangements and construction questions such as "Given that a ball must exit at the lowest S3 level and obey gravity, what is the minimum number of cubes necessary to make four quadruple path cubes, or S3s which utilize all four conduits as passageways?" Available with transparent or opaque walls.”

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

Languages:  English

Resource Types:  Games, Manipulatives

Math Topics:  Higher-Dimensional Geometry, Topology”


It's about computer application.

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)

S3 is fascinating because it’s about having fun with representation and processing of relations, a competition of Analytic and Holistic thinking, Rational (programming desired logic functions) and Nonrational (intuition) —  symmetries and relationships and feedback and tasks which demand abstract reasoning from creatures (human, machine and otherwise).  (Hint:  Birds do it too.)  Watch

                                                        test     think S

It’s about Gestalt.


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 (network theory) tunnels  which depend on the Law of Gravity.  Thus, it is also binary in nature.  

Here, logic alone will fail; governing rules, consistent patterns and trends are problematic. 


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 “strange loop” in.  


It’s about Systems Thinking, of course.

system of logic blocks leveraging full-spectrum cognitive / perceptual processing, advanced thinking and intuition.

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

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.


In other words, Zen koans employ the same flip-flop mechanics:  enlightenment by contradiction. 

A switchiness of learning to adjust processes which were previously, but are no longer, successful.  Each S3 reorientation requires the construction of a new field map:  blind alleys, paths, routes and environmental relationships must be continuously revisualized.

That is, 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. 


It’s about Cybernetics.

A Cybernetic switch, a dynamic router, 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 synthesis of Marvin Minsky’s Connectionist and Symbolicpredicated on Spatial Thinking  (space, representation, process:  multifaceted, interconnecting competencies, human and robot), as manifested by S3 — the rational in conflict with the intuitive. 

Abstract reasoning:  food for the hungry, creativist mind of the child … the scientist.


Thus, 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” ladder, from 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)

With S3, the human mind embraces a Gestalt confusion, which demands an evolutionary cycle of cognitive development.

The foundation of cognitive process and Artificial Intelligence.  

Think of the human mind as Gestalt Engine (mind as machine), one of the most powerful forces in the universe, sorting the evolving, increasingly chaotic Sproblem space into organized patterns, filling in missing information (Pragnanz).  Problem space as evolving construct:  a furious evolutionary  competition of patterns, structure and logic at multiple levels of abstraction and relational complexity, recursive processes of self-observation and self-reference.

Above all, it's about synergy.

S3 is both agent and object of change, and the mind is both agent and object of change.  An eminently powerful, researchable partnership.


What seems to be not understood is that science is a question, not an answer — any child will understand that 

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.

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

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:  “Solving the cube becomes almost trivial once a certain core set of algorithms, called macros, are learned." 



                                                                          Watch the algorithm.

It’s about opening our minds.

Let’s jump beyond algorithm. 

Enter S3,  a "complex system”.              

S3 won’t “cooperate” — causal interaction within dynamic core. 

There is no winning algorithm.  

To proceed successfully the manipulator must test and explain the apparently purposeful thwarting behavior of S(Shannon's Ultimate Machine  think like a child, not too different from a Cybernetician).

It’s about self-regulation.



It’s about Hands and Mind.

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

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.


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

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.  (Think endogeneity:  the solution is married to the error.)

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.”


Thus, the S3 Initiative:  introducing SPlay (within an operational definition of Play as evolutionary regulatory arena) to a wide audience; proposing Spatial Thinking / neuroscience Learning Research / Children’s Museum (below) creative partnership, as well as S3 availability in homes and schools.  Evidence-based practice for teaching and learning.




It’s about need (how to think). 

More learning proficiency than problem solving competence: situation awareness, creativity, and the careful construction of logical arguments, and their limitations.

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).

Cognitive and educational assessment:  in a low-tech, high-yield, cost effective format which requires fluid intelligence, and nurtures it. 

Like S3STEM 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).  

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

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

Thus, learning / development research as first step toward rigorous, scientific demonstration that STEAM is as effective as STEM.  Evidence-based practice for teaching, and learning.  

And finally, it’s about working together.

STEAM / STEM is really just about people of good will seeing the issues from a richer perspective, the larger whole — It should not be (but it usually is) an argument; it should be a coming together within the constraints of whatever budgets are in play.  An opportunity for innovation.

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


S3 is waiting. 

                                         HYBRID TOY

                                       Challenges / Games 


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

PROGRAMMABLE LOGIC BLOCKS (Analog-Binary Processor:  binary gate arrays / switches)




THEME PARK / PLAYGROUND (children change their problem space at will)

It’s about corporate responsibility beyond dollars and cents.

                    Hello, Toy Industry.  Let’s Lead the Way! 

Consumer Knows Best … 

Scientifically verifiable Spatial Thinking toys are needed -- today.

(Think evidence-based practice for teaching and learning.)

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.  Challenge of uncertainty.  Full-spectrum cognitive engagement across market segments.

Let’s raise the bar.

                       The Smart Block.

                                                             Binary and Analogical Reasoning

                                                             (switch from fixed to changing values).

                                                             The Upgrade. 


SBusiness.  (PITCH DECK)



How to cite this material:

Sommer, M. S., (2018).  Sommer Cube:  Symmetry Under Transformation.  Retrieved from

"North Haven:   A State of Mind




                                                                                 "l'espoir vis en moi

© Michael S. Sommer, Ph.D, 2018