Symmetry Under Transformation

                                   Cubical Maze Module




                                                A Subtle Form of Tunnel Vision

"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)

This is the story of the Sommer Cube (S3), a cubical maze module — leveraging binary and analogical reasoning in a hybrid toy.  Analytic and holistic thinking; a competition of rational (programming desired logic functions) and nonrational.

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. 

Manifested as Full-spectrum COGNITIVE TOOL.  A contribution to Child Development and Gestalt Theory, and a little more. 

An interplay of objective and subjective reality, concept learning where manipulator sets up placeholders for more complicated sets and abstractions, and fills in the details over time. 

For anyone, across the age spectrum.  

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.


Because there are so many issues involved, across so many disciplines, and so many constituencies with competing agendas, we need to get back to learning / development basics:  reality-grounded inquiry (not another planet Vulcan with its narrow, fragmented approach), full-spectrum human / machine study, greater creativity and coordinated long-term involvement of Brain and Psychology Sciences — innovative research    on spatial cognition and transfer of skills and logic acquisition, knowledge transfer, to document claims for the already jammed instructional time in K-12 curricula. 


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.

A too restricted view of human nature … even though only briefly ascendant, 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 Judgment, 1965

Let’s raise the bar.


IMG 0232 - exposure copy copy

A good foundation for what follows is Norbert Wiener’s The Human Use of Human Beings, and Bohm and Peat's Science, Order, and Creativity)  


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.


Swon’t “cooperate”.

While Rubik’s Cube (Machine-think:  about problem solving competence)without that paradox, exercises logic (according to MIT / Rubik’s Cube"Solving the cube becomes almost trivial once a certain core set of algorithms, called macros, are learned.” [mind as pocket calculator])  (note "Brain Games are Bogus.)S(about human learning proficiency) demands logic and intuition (One child exclaimed:  “It’s an inside out Rubik’s Cube!”)

Instead of viewing space as fixed, a passive arena (simple games, puzzles and mazes), the S3 network architecture uses space and geometry as active participants in the problem universe (Animal and Machine) -- balanced, opposing, and apparently random forces.

Enter S3.

A dynamic structure of ideas and their logical expression.

A tangible, lucid demonstration of basic behaviors (consciousness) of the brain through mechanical concepts (switches).  (It’s all about “permutations” of dead-ends and thruways:  particular order matters, like a combination lock.)


               Rolling Ball “Tilt-Switch                                                  Asynchronous Analog-Binary Processor (0/1)   

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.

A paradigm shift, a hybrid, in Marvin Minsky’s terms, simultaneouslly Connectionist and Symbolic, an orchestration of ideas:  a system of simple puzzle, labyrinth / maze and rolling ball (apparently unrelated, but in equilibrium, exploiting the symmetry of geometry and algebra, to higher and higher levels of abstraction)  where intuition, naive commonsense reasoning, is a prerequisite -- logic is necessary but not sufficient.



In other words, S3 is about the tension of switchiness (compound cognitive, perceptual, mechanical flip-flop and schedule of reinforcement):  switching of directed attention, relative motion and form, and navigational strategies.  (Gestalt Engine (analog)  x  Tilt Switch (binary)  =  "Directed Attention" (flip-flop)  =  "Cognitive Flexibility” — think Gestalt Architecture.)


Be it Computer Learning or toy, the story relates to long standing formal ideas / implementations concerning Child Development and Artificial Intelligence, particularly cognitive Constructivism (process discovery / task-based learning) which with rare exceptions (e.g., Friedrich Froebel, 1837, with his “sphere and cube linked by an elucidating cylinder”) never reach the child’s hand and mind. 

                               Peel the conceptual onion to follow Sin science.

                                                  What it is.  What it does. 


The perspective:  Brain as evolutionary union (the whole) of analogical and binary processes, biology and experience (the parts).  The Gestalt engine (de Keyser’s “we are furious pattern-matchers”), resolving dissonance by escalating perception:  bird’s-eye view — like Zen.


"For Hegel a Gestalt is a “formation of consciousness” understood as the dissonant unity of a way of thought …. practical activity and subjective thought all aspects of a single whole or figure, that is Gestalt, but always moving, always with internal contradictions."       

                                                                          Anthony Blunden, forward to “ Hegel’s Logic”                                              

Thus, an intiative to see S3 cognitive flexibility get directly into the hands and minds of the children and the researchers; then the researchers can get to work to make the necessary “sea change”.

Hegel for Tots.

"The irony is that the real educational toys are not the flashy gadgets and gismos with big promises, but the staples that have built creative thinkers for decades.” ....

                                                                                 Roberta Golinkoff:  Science Daily, 2000“ 

Let’s kick start our trip.


None of the following excursions is meant to be the final word, but instead the opening for new ideas and discussion.  The picture which should emerge is one of balance:  Man / Woman hunger for abstract reasoning, and its regulation.

As a systems scientist I have crept into turf — the nature of Play (be it Euler, Bach, or baby) — not my own because I couldn’t figure out how to explain S3 issues without some kind of formal framework for the “natural act of abstraction” of human life, which takes us to Artificial Intelligence:  competition between analogical and binary reasoning.  

Science as Art by other means.*


                                                      dialectic -- open your mind

What is S3

A dialectic.

From the perspective of information processing, a system of competing parts, demanding simultaneous analogical and binary reasoning, in ascending levels of abstraction, which kick starts a continuous pattern of cognitive development (not Piaget’s development in stages). 

Combinatory play.

In musical terms, exploration in depth of the contrapuntal possibilities inherent in a cubical maze module:   order, structure and wholeness.  

A 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.

Think like J.S. Bach:  see the “music”.  (Scubedom and fuguedom)


In other words, Sis simply about variations on a theme of      square and circle -- flip-flop, a counterpoint of logic and intuition.  Symmetry, and feedback, under transformation (comfortably understood withinin the framework of music and mathematics).

A do-it-yourself quadruple fugue.   

A generative process.  An “unfolding”.  A switching.  


"Further discussion suggests that in a way, the creative possibilities of the mind are generally dormant.  This state of ‘sleep’ is probably the result of a very widespread and deep confusion between the creative and the mechanical.  As a result of this confusion it is very hard even to perceive that one is not creative.

                                                                                       David Bohm, ”On Creativity, 1968

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.



Thus, S3 is about levels of abstraction, pattern, the rule which governs a system or phenomenon, exactly like numeric, musical, or visual relationships; patterns of thought exceeding the parameters normally experienced in logical operations:  Apex reasoning (think Bayesian inference -- subjective probability as epistemic tool).  Where the strategies and tactics are evolutionary.

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.)

A continuous switching of patterns:  square and circle.


Patterns, a dynamic visuospatial (analogical mapping and processing) cycle of concrete and abstract reasoning:  nested coordinate systems (think Rene Descartes' analytic geometry); Albert Einstein’s “Combinatory Play”;  Mathematics  -- the science of patterns:  nothing but patterns  … Descartes’ "apud me omnia fiunt mathematica.” 

What does Sdo?

It kick starts intuition, the nonrational. 

Creativity -- playfulness, imagination and rigor -- in the face of real and abstract variables:  coordinating mental and manual environment, modifying self-behavior on the basis of experience, through the 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.  (“How complex environments push brain evolution”

It’s about survival of the creativist.

"The paradox of science education is that its goal is to impart new schemata to replace the student's extant ideas, which differ from the scientific theories being taught.

                                                                      Susan Carey, Cognitive Science and Science Education”, 1986


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

Simply put, S(a motile) is a work of art (SPACE AS ARTof bare utility, like mobile and stabile (remember Calder’s play circus), a machine which offers abstraction of form and function, as well as cognitive payoff, for the manipulator.


Four tunnels (trajectories) are as closely packed as possible within a cube (they attempt to “kiss” each other and the faces of the cube) without overlapping and / or kinking and are configured to allow a ball to enter and exit in any of twenty-four perpendicular orientations, with the constraints of gravity and / or specific exits.

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.

Manifested as Full-spectrum COGNITIVE TOOL (manual / mental rotation studies)

An 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.  

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.

"Newton also regarded the geometric approach as more intuitive, certain, and direct, as opposed to algebraic techniques which he once characterized as nauseating. (Mathematical Papers, Volume 4, p. 277)"  

                                                            Robert Fox, The Oxford Handbook of the History of Physics

Hear Newton:  Hands and Mind -- Logic made tangible:  compass and straightedge -- synthetic / axiomatic / pure geometry (as opposed to coordinate / analytic / algebraic geometry). 

Newton for tots.  

Hegel for tots.



“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.)

S3 is about exercising cognitive resources, the essence of programming; it demands, gently at first, the manipulator simultaneously plan and direct multiple lines of thought, "factors of variation" (orientation, clutter, etc.), to mentally and physically rotate, differentiate and process dynamic parts and wholes, separate evolving tasks into manageable, recursive subsequences, all within evolving layers of abstraction.

[Programming] "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) 

A story of "Distributed Adaptive Message 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 monitor / troubleshooting tool testing reachability.)

A story of Spatial Thinking  (space, representation, process:  multifaceted, interconnecting competencies, human and robot), the holistic approach — manifested by a cubical maze module (S3) -- 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 definition of Play, in operational intelligence terms, as Adaptability:  cognitive payoff.

A simple block and ball networking system which leverages full-spectrum cognitive flexibility / perceptual processing   

(Set Switch:  between task rules; Response Switch:  different resource mapping; Simple Switch:  between a circle and square (note photos of individuals manipulating S3).  

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?)

Thus, S3 is about adaptation to uncertainty:  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.  In depth.

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)

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. 



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



                   Children formulate a Systems Problem         (checkered paths allow exit)

Binary and analogical reasoning.  

Let’s Not Underestimate the Mind of the Child.

"The native and unspoiled attitude of childhood, marked by ardent curiosity, fertile imagination, and love of experimental inquiry, is near, very near to the attitude of the scientific mind …"

                                                                                                                   John Dewey, How We Think, 1910


                                                       Scientists in Action


S3 is about the need for more learning proficiency than problem solving competence:  creativity, in operational terms, and the careful construction of logical arguments, and their limitations.  

"Calls for the widespread use of computer-based educational technology often justify themselves by the potential to support some version of constructivism, seen to be a major improvement in education over more behaviorist or information transfer notions of teaching. The strong claim that computer-based educational technologies are inherently constructivist cannot be sustained because current technology largely fails to provide four key components of a constructivist environment: an engaged learner, hands-on interaction with the materials of the task, an authentic problem-solving context, and human interaction during the learning process….”    

     Stephen Gance:  "Are Constructivism and Computer-Based Learning Environments Incompatible?", 2002

Knowledge generation that manipulator can both express and demonstrably use -- avoiding what Whitehead called “inert ideas” -- propositional knowledge that the student could express but not use”.

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.


An exploration from different perspectives, across multiple disciplines.  Where each issue has a different character —  not parallel issues, but orthogonal to each other.

                S3 supporters speak out at  CONSTRUCTION BLOCKS (Analog-Binary Processor)

Human control of complex systems:  dynamic decision making.  Uncertainty as tool.

Thus, S3 is about the genesis of thinking, navigating three-dimensional problem spacecrucial to the development / exercise of visuospatial reasoning (particularly in the formative years) and full-spectrum cognitive engagement and discovery.  (STEM WITHOUT BOUNDARIES:  STEAM)


                                                                            (checkered paths allow exit)

Let’s get back to basics. 

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

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). 

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

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.”


(Slearning / development research is a first step toward rigorous, scientific demonstration that STEAM is as effecitve as STEM.)


         Spatial Thinking and Learning ResearchChildren’s Museum Collaboration.  


"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.)




                                        Geography is dynamic, never static.”  

                                                                                                                                    Zoe A. Thralls,1949

    Marriage of Cognitive Constructivism and Computer Learning

                                                    (CHALLENGES / GAMES)


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

S3 offers a true "parallelism of general cognitive fields”, *

Sis about play and programming a network:  formal (play by the rules) and intuition, and the sense of touch, which overrides the human interaction with complex cognitive and perceptual content in paradoxical ways.  

What does this process and S3 have to do with our new world?  It develops intelligence, the cognitive resources necessary to program a computer (0/1) -- but without the computer; a mechanical user interface (invites interaction and responds to it)

This robust integration of haptics, mental and manual rotation, illusion as operational fact, is missing, from early childhood computer-focused learning (as opposed to child-focused learning) to joint pilot / computer management of aircraft (increasing absence of hand-flying).  

Dynamic interaction of the brain and its environment trumps brain alone.  Be it child, astronaut, surgeon, student:  across the age spectrum.


                                                  Sit on your hands -- then try talking for 15 minutes.

In other words, mental processes reach beyond brain.  Play capitalizes on that.

"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 .… viewing the brain (and hence the mind) as one organ among many, a relatively recent usurper of control, whose functions cannot properly be understood until we see it not as the boss …"

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

S3 as cognitive tool and work of art.  "This is a fascinating object.” (April 28, 2014) 

                                                             Daniel C. Dennett, The Intentional Stance, Sixth Printing  

Thus, S3 is about switching between the two:  manual, and its inseparable partner mental rotation, the ability to rotate mental representations of two dimensional and three dimensional objects, predictor of math / science achievement.

Generating knowledge that manipulator can both express and demonstrably useavoiding what Whitehead’s called “inert ideas” -- propositional knowledge that the student could express but not use”.

Naive Commonsense reasoning, according to Artificial Intelligence precepts, is what makes us smarter, more resourceful, than computers; it's about making assumptions, making assumptions about assumptions, engaging rule-based as well as analogical based reasoning. 

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

To repeat, 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.  

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. 

Survival of the creativist.

Play as ongoing bidirectional exchange of analog and binary signals to the brain along reciprocal avenues.  In Edelman’s neural Darwinism terms, reentrant strengthening of mapping.  

Thus, Sconstruction is about seeing Play in operational intelligence terms (not silliness or behaviorist truism), as adaptability (“neurocognitive plasticity”):  cognitive payoff.  

Play = Ashby’s Cybernetic / homeostatic "good regulator" of Intelligence.

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

Think of play as “defragmentation" / optimization (balancing discrete parts of a dynamic development equlibrium -- intelligence whole -- with Hands and Mind).  A self-regulation, load reduction:  "… NFB (neurofeedback) tunes brain oscillations toward a homeostatic set-point which affords an optimal balance between network flexibility and stability”, "Tuning pathological brain oscillations with neurofeedback”.  

Thus, S(cognitive tool) leverages analogical and binary harmony (coherent states of opposition are preserved while that decoherence is maintained, in an evolutionary fashion).

And intuition — the nonrational — is the cognitive payoff.

S3 is combinatory play.

Ask Einstein.

"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   


Thinking without boundaries.

From developing mind to powerful intellect,

Infant, child or world-class mathematician. 



“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”


On the other hand 

"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 …

                                                                        Laura E. Berk, Child Development, 8th Edition (May 13, 2008, cc mss)

                                          HYBRID TOY 

                                                       Complexity for Tots

Thus, to repeat, 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);

Dedicated to the special genius of each individual, and the opportunity to exercise it.


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

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

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

     Math Topics:  Higher-Dimensional Geometry, Topology”)




THEME PARK / PLAYGROUND (children change design at will)



                 The Smart Block.

                                                      Binary and Analogical Reasoning  

                                                      (switch from fixed to changing values).

                                                      The Upgrade.     


S3 discovery through paradox, uncertainty as tool.  

Freedom to Think.   Mix things up — reassociate them.  Mystical equation.


And it's fun. 


Zen koan (cognitive flip-flop) mechanics of enlightenment (contradiction).

The Very Special Children


Cybernetics equivalent of model railroad network switches.


                                                                        KICK STARTERS 




                                         star light KIC7671081Bhttp://www.ilovetoteachkids.

S3 Play.



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© Michael S. Sommer, Ph.D, 2017