Hands And Mind

                              Informal Reasoning:  "Use it or lose it”.

"The aim of heuristic is to study the methods and rules of discovery and invention .... Heuristic, as an adjective, means 'serving to discover'. ... its purpose is to discover the solution of the present problem. ... What is good education?  Systematically giving opportunity to the student to discover things by himself".                                                                                                                                                  

                                                                                                    G. PolyaHow to Solve It (1945)

The Sommer Cube (S3) is an eploration in depth of the contrapuntal (think patterns) possibilities inherent in a cubical maze module.

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.

With Hands and Mind.

                             SEngine of Mathematical Ability

"Grasping the struture of a problem;


Developing chains of reasoning;

Using symbols and language accurately and effectively;

Thinking flexibly - backwards and forwards and switching strategies;

Leaving out steps and thinking in abbreviated mathematical forms;


       generalised relationships,

       problem types, 

       ways of approaching problems, and 

       patterns of reasoning;

Perseverance in problem solving."

                                                                                       Krutetskii V.A. “The Psychology of Mathematical Abilities 

                                                                    in School Children”, University of Chicago Press, 1976 

“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”

"[The universe] cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word." 

                                                                                                        Galileo Galilei, Opere Il Saggiatore, 1623



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?)

(Visualize the necessary systemwide adjustments from the ball’s frame of reference:  changes to the state in a frame of reference moving with the ball.)




S3 is about variations on a theme of square and circle -- flip-flop, a counterpoint of logic and intuition.

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 probabiity as epistemic tool).  Where the strategies and tactics are evolutionary.

(S3 are "pattern blocks", which create porous polyhedron "chains" formed by distinct collections of figurative elements:  circles and squares.)

Consequently, S3 demands recognition of sequence / pattern of repeating events formed in accordance with a definite rule(s) -- but different individuals can perceive the same pattern differently, reach different generalizations.  It’s mysterious. 

Sdemonstrates not just the beauty of the maths product, an optimized distributed processor (S3programming networkbut the process of maths (non-symbolic numerical intuition, naive commonsense reasoning) and the excitement of discovery.  The power of patterns, the abstractions which compose them, make them possible.

With measurable utility.


           (checkered paths allow exit)

For exampe: four paths begin / cross within a single S3.


Utility = 1.00 for the top front S3 (one path S= .25)


Srelationships demand the viewer manipulator reinvent the process of discovery (Plato’s / Archimedes' Polyhedra / Conway’s Game of Life / Kaufman’s Knot, for example -- they already had their excitement of discovery.)  The question is, What is its significance?  Why is it beautiful?

“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 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 


             Discovering and Representing A System Of Rules

         Binary and Analogical Reasoning (switch from fixed to changing values). 

                                     formal (play by the rules) and intuition

The S3 is a programmable logic component (In Field Programmable Gate Array terms); think of bus architecture, plug-ins.  A combinational (technically “permutational", like a combination lock) 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:

A dynamic router, a Analog-Binary Processor:  a system of programmable binary (0/1) switches, a “flying junction” (an over-and-under junction of tunnels which weave past each other) housed within a cubical exoskeleton with multiple axes of rotation and alternative, manipulator-controlled ball trajectories, offering exit in all orientations.


                                                                                                  Rolling Ball “Tilt-Switch

                   allocentric / egocentric flip-flop spatial processing (route following, piloting, ded reckoning) 

Thus, S3 is about continuous synchronous and asynchronous switching:  nested, dynamic, switching of cognitive and perceptual and mathematical dichotomies, where the manipulator must constantly re-examine reigning assumptions, transcend and control nested and evolving recursion and contradiction, by inventing new propositions (Boolean algebra -- a way of seeing new structures -- fundamental to the design of binary computer circuits and programming language).  Note that individual S3 logic elements do not necessarily have a discrete true / false state at any given time; simple Boolean logic is inadequate for this, thus extensions are required. 

(Note also the interesting Sproblem spaces, Boolean Construction:  Atomic Warfare, surgical elimination of command / control node-links, at Challenges / Games.) 

The developing S3 network requires constant revision of operational sets and simultaneous consideration of many possibilities from many perspectives: the order of steps is not fixed, there is no algorithm.  

It demands the manipulator understand and assimilate evolving information about patterns and functional relationships and analyze change in both concrete and hypothetical contexts: a meeting of abstract ideas with the spirit of mathematical rigor.

                                     The Dialectic     

That is, the S3 process requires a cycle of intuition and reasoning, naive commonsense reasoning, to: 

Expand fluency with efficient procedures;

Analyze change in both concrete and abstract contexts;

Connect concrete things and events with their abstract representations;

Develop mathematical arguments about geometric relationships;

Reason about / better understand common-sense and formal logic; and 

Structure proof.

The S3 validation is elegant: success, as characterized by manipulator criteria, is a structure, conceptual or physical, which allows the exit of the ball, or give the mathematics solution.  

And it’s fun.

                                   Geometry As Paradox

          Asymmetric, Nested Self-Reference and Contradiction

Messy is good.  In Systems Thinking parlance Ashby’s Law of Requisite Variety (counterintuitive) tells us Sis “variety increasing” (manipulator must employ informal as well as formal reasoning)“the variety in the control system must be equal to or larger than the variety of the perturbations in order to achieve control.”  

In other words, the S3 mess is transformative, a kick-start: a self-generating, self-sustaining cycle of spatial reasoning and intuition

In Cybernetic terms, the S3 is a tangible, lucid demonstration of basic behaviors (consciousness) of the brain through mechanical concepts (switches / logic gates); 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.


                 Asynchronous Analog-Binary Processor (0/1)  /  three axis rotation problem space    

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.)

It demands, gently at first, the manipulator simultaneously plan and direct multiple lines of thought, mentally and physically rotate, differentiate and process visual parts and wholes, separate evolving tasks into manageable, self-referential subsequences, all within evolving layers of higher abstraction. 

In logic / mathematical terms, the S3 manipulator must personally (without symbolic / numerical tools) visualize, intuit, reason incommensurate concepts, like Gödel’s Paradox (“this statement is false” requires a broader logical perspective), like the S3, self-referential), like the Pythagorean triangle, 1,1, √2; 1,1,1, √3, and reinvent accordingly (Oops! all numbers are not integers, rational numbers -- let’s look at the bigger picture — think of it as irrational number along a line of possible numbers).


Connect the abstract with reality, deal systematically with changing patterns of:

Structure and transformation (transformation geometry / symmetry; rotation [turns], reflection [flips], translation [slides], etc.);

Change and motion (calculus);

Visual or tactile (geometry);

Operations (algebra); and

Connections (graph theory).

Syematically reinvent the fundamentals of mathematics; create basic strategies, reasoned / intuitive conceptual structures, mathematical abstractions:  (it's worth repeating) 

”The Double-edged Sword of Pedagogy: Instruction limits spontaneous exploration and discovery

        "Two Kinds Of Scaffolding: The Dialectical Process …."


               Tangible Learning -- Not Just Goal-attainment

                                   Systems Perspective

a)   Ratchet: manipulator / learner built. 

Manipulator obliged to focus on process, the interplay of inductive and abductive (intuition) and deductive reasoning, not just outcome: reasoning about reasoning; 

to personally control thinking while learning -- kick-start a series of self-initiated irreversible stages of discovery, calibrating an evolving foundation (patterns, more patterns ...) of inquiry, step by step. 

Thus, voluntary behavior is modified and maintained by its consequences (intermittent reinforcement).

In effect, unlike the tropism of insects, with hard-wired behavioral routines, the S3 demands the manipulator increase its capacity to learn from its errors (without intervention), to confound the manipulator's system of frames of reference (stereotypic themes, paradgmatic scenarios, scripts) and forces the manipultor to redesign them -- to learn. 


In programming terms, Ratcheting is unsupervised learning because the logic hierarchies and changing order of operations (principles have to be unlearned) are not known in advance, demanding:

Conceptual Clustering (inductive / abductive) reasoning), discovering class hierarchy; and 

Concept Formation (deductive reasoning), finding the best place for a new instance within a fixed class hierarchy incrementally.

b)  Scaffold: teacher built. 

Mediator / computer directs process, temporary foundations, levels of difficulty (subject to the fixed assumptions / strictures of authority "learning plan").   




"I think it's bad psychology, when teachers shape our children's mathematics into long, thin, fragile, definition tower-chains, instead of robust cross-connected webs. Those chains break at their weakest links, those towers topple at the slightest shove."

                                                          M. Minsky, "Why People Think Computers Can't", AI (1982) 


This lack of a robust, cross-connected heuristic, a self-initiated and self-regulated progression / regression of re-conceptualization rewards dependency.  

To the contrary, the Sallows the manipulator to Ratchet, calibrate his own design-problem complexity, analyze increasingly complex, dynamic situations and structures:

a)  rewards evolving abstract thought within "robust cross-connected webs" of joint mental and manual rotation, test alternatives in alternative ways;

b)  is less susceptible to mediator / teacher interference.

It’s messy.  It demands the manipulator try a variety of approaches (informal as well as formal reasoning) and learn from mistakes (a self-checking process; the ball does not always exit), to become increasingly adaptable and skilled in abstraction and relationships.

Thus, the fun of S3 and mathematics:  

Do it Yourself.

"Try to acquire the weird practice of savoring your mistakes, delighting in uncovering the strange quirks that led you astray. Then, once you have sucked out all the goodness to be gained from having made them, you can cheerfully set them behind you, and go on to the next big opportunity. But that is not enough: you should actively seek out opportunities to make grand mistakes, just so you can then recover from them.

                                                               Daniel C. DennettIntuition Pumps and Other Tools for Thinking



© Michael S. Sommer, Ph.D, 2017