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


                                   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.

Sommer Cube (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 challenging assumptions.

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

S3 is about uncertainty as tool.

"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

Toward understanding cognitive development by capturing patterns of changing analogical abilities.  A construction block, a cognitive tool for the education-minded consumer.

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 is about simultaneous mental 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. 

"Analogy pervades all our thinking, our everyday speech and our trivial conclusions as well as artistic ways of expression and the highest scientific achievements."

                                                                                                                   G. Polya, How to Solve It (1945)

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

S3 is 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 is about 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).  

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); a topology of paradox.  

While Rubik’s Cube (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.”)S3 (about human learning proficiency) demands logic and intuition. (note "Brain Games are Bogus.) 

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 probabiity as epistemic tool).  Where all the rules change.

(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, “reinvent" the rules.)

Thus, S3 is predicated on switchiness (compound cognitive, perceptual, mechanical flip-flop and schedule of reinforcement):  switching of relative motion and form;

Juxtaposition of square and circle across three dimensions.



                                                                                                               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.


              The Smart Block.     The Upgrade:  Hands and Mind. 

Binary and Analogical Reasoning.

Asynchronous Binary Processor (0/1);

Analogical reasoning between superficially distinct models, which share common underlying structure. 

Structure-mapping, domain-traversing relational engine for Hands and Mind.


The question is:  Machine-think or Human-think?  Remember, we can switch how we think -- re-program ourselves. 

And it’s fun.  (CHALLENGES / GAMES)


                                                        "On the road again …” (PROFILE)

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

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