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                                   Symmetry Under Transformation

                                   Cubical Maze Module


                                                  Classic toy.  (PITCH DECK)  

                           Full-spectrum Cognitive engagement.  Cognitive tool

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).

"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

The Sommer Cube (S3is about formulating, then answering, the question, What is a “creative” (in operational terms) toy?  Vast amounts of money are wasted by hard-working education-minded consumers, and children left at cognitive disadvantage, because every toy claims to be creative.  But that’s just not true. 

S3 is 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 (not "mere facts", but principles); functional relations, particularly goals and feedback (What information is relevant?  What are my assumptions?  Are they justified?)

S3.  What it is.  What it does.  

The Answers take a little explaining:  follow multiple perspectives -- mental and manual rotation:  the sciences, art, and music -- with Links.  



Adventure By Topology  

(technical hints)

Or go directly to OVERVIEW.

S3 is about exploration of feedback and symmetry, influences among parts and whole, square and circle -- 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.

(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.) 

Thus, S3 is about the need for more learning proficiency than problem solving competence.

In technical terms -- Sis predicated on switchiness (compound cognitive, perceptual, mechanical flip-flop and schedule of reinforcement):  switching of relative motion and form, flip-flop of manipulator frame of reference:  egocentric  >>  allocentric; patterns of thought (higher levels of abstraction) in a paradoxical environment, and how to get the most out of them:  a partnership of feedback and symmetry.  

Uncertainty as tool.

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 (symmetry under arbitrary changes of coordinate systems).

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).  


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. 

Thus, S3 is about variations on a theme of square and circle, juxtaposition across three dimensions.

A counterpoint of logic and intuition.






             Rolling Ball Switch



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

Each S3 reorientation simultaneously reprograms the four "gravity feed” tunnels differentially; each acts as a binary (0/1) logic gate (rolling ball "tilt-switch") to impede (0) / allow (1) ball flow.

Thus, S3 is about simultaneous mental (rational and non-rational) and manual rotation, giving form to digital and analogical information, a coupling between physical objects and digital information where bits are directly manipulable and perceptible. 

In other words, the S3 physical state embodies the digital state of the system, as well as the analogical. (think Tangible User Interface -- TUI)

"I cannot state strongly enough my conviction that the preoccupation with Consistency, so valuable for Mathematical Logic, has been incredibly destructive to those working on models of mind .…This obsession has kept us from seeing that thinking begins with defective networks that are slowly (if ever) refined and updated.”  

                                                   Marvin Minsky,  A Framework for Representing Knowledge, 1975


                                Control and Communication in the Animal and the Machine

Thus, S3 is about uncertainty as tool.

Topology of paradox"contradiction that follows correct deduction from consistent premises”:  mixed up abduction (intuition) / deduction / induction.  Reasoning as play.  (Think of it as "Pre-K math -- capture the idea / excitement of “numberness”, in a powerful way, before introducing integer.  

With Hands and Mind.)  

And it’s fun.  (CHALLENGES / GAMES)

             The Smart Block.     

The Upgrade.  

Binary and Analogical Reasoning.

Thus, S3 is about challenging assumptions, integrating concepts that have been carefully examined in radically different fields.

Consumer Knows Best ...

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. 

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

Innovation.  Classic Toy.  Full-spectrum cognitive engagement across market segments.





How to cite this material:

Sommer, M. S., (2016).  Sommer Cube:  Symmetry Under Transformation.  Retrieved from

"North Haven:   A State of Mind




                                                                                 "l'espoir vis en moi


© Michael S. Sommer, Ph.D, 2016