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                                   Symmetry Under Transformation

                                   Cubical Maze Module


                                 CONSTRUCTION BLOCKS (Binary Processor) 

                                            Full-spectrum COGNITIVE TOOL 


This is the story of Spatial Thinking (space, representation, process:  multifaceted, interconnecting competencies), manifested by a cubical maze module, the Sommer Cube (S3), the rational in conflict with the intuitive, and why it is important.                                                  

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).  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.

"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 Smart Block.

                                                         Binary and Analogical Reasoning

                                                         (switch from fixed to changing values).

                                                         The Upgrade.     

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

Spatial thinking.

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.

Sis about relative motion (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:  partnership of feedback and symmetry.  Control under continuously varying context.

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?)

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.

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.  



Adventure By Topology   

(technical hints)

Or go directly to OVERVIEW.  

S3 is about exploration of situation awareness, feedback and symmetry, transformation of 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.) 


                                                                         Problem Universe

                              "Control and Communication in the Animal and the Machine

S3 is transformative, a kick-start, a self-generating, self-sustaining, escalating causal loop:  an exploration of deep causal structure; an evolutionary cycle of cognitive development where the manipulator is both agent and object of change. 

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.  Questions ripe for research. 

Spatial Thinking, simply put:  decision space > information > processing.


                                                                       Rolling Ball “Tilt-Switch

Thus, S3 is about human control of complex systems:  dynamic decision making.  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.

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).  



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

A counterpoint of logic and intuition.








         four bit (0/1) -- nibble                    Asynchronous 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.

(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 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)

With Hands and Mind.)  

And it’s fun.  (CHALLENGES / GAMES)

Toy Industry.

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.  STEM WITHOUT BOUNDARIES (STEAM)

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

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

SBusiness.  (PITCH DECK)




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