Cubical Maze Module

                          Symmetry Under Transformation

The Sommer Cube (S3) is a cubical maze module, an exoskeleton, with form and space consistent with Gestalt Illusion and Architecture, predicated on the manipulation of objective and subjective reality. 

Topology -- from planar to spatial environment.  

(Note Descartes’ insight that "smooth surface can be approximated with a polyhedron”.)

Thus, it is analogical in nature.  However, the exoskeleton also circumscribes a system of linkable tunnels which depend on the Law of Gravity.  Thus, it is also binary in nature.

                                                                                Tangram / S3

In other words, S3 is 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.

Thus, S3 is an architecture of nested dichotomies:  contradictions of reason and perception, frames of reference, of logic, pattern and process, motion and stasis; nested, confusing distinctive-feature information.  


A hybrid:  combined puzzle and maze and rolling ball device, where intuition is a prerequisite -- logic is necessary but not sufficient — a classic toy.

A motile (“to touch … and not merely see”:   a work of art of bare utility, like mobile and stabile, a machine (uses energy to perform intended action) which offers abstraction of form and function, as well as cognitive payoff, for the manipulator;

A dynamic router:  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.

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.

    Root (what it is) -- squareness.

    Process (what it does) -- planar rotation of parts — circleness. 



Rotational Motion vs.  Stasis (circleness    vs. squareness).

Tangram morphs from 2D to 3D.

Juxtapose square and circle across three dimensions:

stasis / (binary) motion,

symmetry / asymmetry,


“insideness” / “outsideness”, 

with collective agency of whole and parts.

In other words, S3 is about the tension of switchiness (compound cognitive, perceptual, mechanical flip-flop and schedule of reinforcement — reversal learning":  analogical reasoning, instrumental learning):  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.)


“The basic thesis of gestalt theory might be formulated thus:  there are contexts in which what is happening in the whole cannot be deduced from the characteristics of the separate pieces, but conversely; what happens to a part of the whole is, in clear-cut cases, determined by the laws of the inner structure of its whole.”

                                                                                   Max Wertheimer, "Gestalt Theory”, 1924

                                         The Dialectic  

                    Potential Energy -- Form and Its Negation

Juxtaposition of square and circle across three dimensions.


Contradiction of pattern;


Contradiction of face / quadrants;  


Contradiction of binary motion


Divergent symmetry of face pairs; 


Symmetry under transformation.

While the S3 exterior has six faces (three sets of opposing faces), each "quartered" face with one or two holes (mismatched hole quadrants between modules are dead-ends, and proliferate erratically), the S3 interior has four intertwined chiral tunnels (each tunnel with independent logic gate orientations) with eight entrance / exits.

To optimize multiple simultaneous paths (make most paths begin / cross within a single S3) the manipulator must conceptually drill down” through nested coordinate systems.

To discover order in what is seemingly random.

In other words, navigation of nested levels of abstraction (sets: binary and analog), mental discrimination / integration of different problem models, starkly contrasting conceptual frameworks (each with its own set of rules):

CUBE exoskeleton (six faces)  >>  

HOLES quadrant (eight holes)  >>  

TUNNELS / HOLES (four couplingsgravity gate array; (four binary switches)  >> 

TUNNEL / HOLES (one binary 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.  (Think Field Programmable Gate Array.)


                          ORIGINAL GEDANKENEXPERIMENT

                           Morph Tangram Into 3D Object 

Topology -- from planar to spatial environment.  

(Note Descartes’ insight that "smooth surface can be approximated with a polyhedron”.)



A network of forces and interrelationships. 

(problem) Universe.


Rotational Motion vs. Stasis (circleness     vs. squareness).


    Root (what it is) -- square.

    Process (what it does) -- planar rotation of parts — circleness. 

Rotation (turn).

Reflection (flip).

Translation (slide).

(Note that the parallelogram does not "carry the figure onto itself” suggesting morphing into 3D problem process.) 


                                                                                                                        Two Monks Paradox


Tangram morphs from 2D to 3D.

Juxtapose square and circle across three dimensions:

stasis / (binary) motion,

symmetry / asymmetry,


“insideness” / “outsideness”, 

with collective agency of whole and parts.


  Archimedes’ Tombstone (his pride and joy).


  Tangram + Archimedes' Sphere and Cylinder.  


Patterns of discovery.       

Visualize implicit cube circumscribing sphere, thus cylinders. 

Think tunnels (apparently paradoxical, but not to Archimedes).  "Portions of semicircular cylindrical wedges combined to form Archimedean globes .”  



Think of it this way:   S3 and Tangram are not things, but ideas --                                              

Dialectic of optimal structural and functional relationships (remember trajectory — calculus below).



                      Ball Path (trajectory) Packing Problem

                 Sommer Polyhedron:  Porous Space-Filler


                                      Hungry Balls  +  Cube (jello) = Tunnels

Configure four tunnels, of unlike symmetry, to allow a ball to enter and exit in any of twenty-four perpendicular orientations, where at least one low-quadrant exit is always possible.  

S3 fields are "pattern blocks", which create porus polyhedron "chains" formed by distinct collections of figurative elements.

                                      In Calculus Terms

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.  In other words:

Problem.   “Given [four sets of] two points [tunnel openings in a hollow cube] A and B in a vertical [inclined] plane, what is the curve [four curves, i.e., four tunnels, closely packed and intertwined in harmony] traced out by a point [ball] acted on only by gravity, which starts at A and reaches B in the shortest [longest] time?”    

                                            Bernoulli's Brachistochrone problem [with a twist], calculus of variations.

                                                                                     Note Paul J.Nahin, When Least is Best (2004)

Solution.    S3


                                         In Ratio Terms


     Archimedes’ Tombstone

Ratio of the volume of a sphere to the volume of the smallest cylinder that contains it is 2:3.

Ratio of the surface area of the sphere to the surface area of the same cylinder is also 2:3. 

Think Stunnels:  cylinders / faces is 4:6 = 2:3.

math forum copy

“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”


                                                                          Tessellating Space

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 (among eccentrically rotating local" and expanding “absolute" coordinate systems) — multivariate, celestial mechanics.

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

Binary and Analogical Reasoning.

Thus, Sis a paradigm shift, a hybrid, an orchestration of ideas:  combination 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, are prerequisites -- logic is necessary but not sufficient.

Spatial Thinking in three axis rotation problem space (physical and mental) of nested uncertainty (uncertainty of environment, and cognitive flip-flop and perceptual flip-flop -- "unknown unknowns"), of constant and unexpected contradiction, recursive abstractions and relationships; opposing systems, domains of expectation / reference frames, which exhibit completely different properties and changing order of operations, principles have to be unlearned (not to mention the unique powers of particular Sclusters, in different situations -- like chess / go).

Thus, S-- a topology of synergy

Contradiction of action and nested environment  -- dissonance of conflicting cognitions / actions.

A dynamic router (IField Programmable Gate Array terms, a programmable logic component; think of bus architecture, plug-ins.) 

An organic and reciprocal switching cycle of cognitive development.  

In Buckminster Fuller’s  Tensegrity (tension / compression) terms:  Synergy"coordination of thought and physical action, the genesis of geometry, system, and structure.” 

A self-generating, self-sustaining, escalating causal loop, an evolutionary cycle of cognitive development where Manipulator is both change agent and object of change. 

Transformation.  Equilibrium of disequilibrium.


Genesis:  Gregory Bateson (Cybernetician / Ethnographer) Double Bind”  paradox:  contradictory cognitive and perceptual signals / messages at different levels, where acknowledgement of that contradiction is forbidden. 

Norbert Wieners "pathological oscillations”, in the face of Russell's paradox:  an oscillating, intermittent restriction of cognition and perception within a dynamic logic hierarchy.      

Thus, S(flip-flop of manipulator frame of reference:  egocentric  >>  allocentric) is all about 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.



© Michael S. Sommer, Ph.D, 2018