Space As Dialectic 

                                           S3 Sculpture 

                           A Conspiracy Of Pattern -- Hands and Mind

_MG_6368 copy crop

The Sommer Cube (S3) is about variations on a theme of square and circle -- flip-flop, a counterpoint of logic and intuition.  Symmetry, and feedback, under transformation.

It's about generative process.  An “unfolding”.


It’s about Aesthetics.  As a living thing, philosophy in action, classic toy and cognitive tool, following the vision of Leonardo da Vinci to see Arts and Science as one:  

Science as the Arts by other means.

"I shall understand the term "art" in its broadest connotations--perhaps broader than the American Society for Aesthetics would normally recognize: I shall understand art to include all artifice, all human invention.

                                                          Daniel C. Dennett, "Memes and the Exploitation of Imagination

                     Pennsylvania Academy of the Fine Arts 

                 SDay -- April 9, 2002 -- Inventor's Class (six to twelve year olds)

Thirty children and I were seated on the floor, each with a name tag, all more or less at eye level.  I began by introducing myself and explaining that “We are all inventors here.”  They were quiet as mice.  Each child had been required to bring his / her invention and explain it to the group.  Each child had to stand when presenting the device and its reasoning (MIT has never had a more serious presentation of mechanical devices and their underlying processes).

When they were finished, I stood and explained my own little idea (S3), using their own explanations as a foundation to get a little more technical, which they quietly absorbed. 

Then they broke into small groups, and began manipulating S3s, focusing on the idea that principles can evolve into working process -- just like their own inventions (for one example:  a rock which fell on a stick, which moved a leaf, which released a feather) — but in the face of paradox.  And they had fun.

Their faces tell the story … 

Sis a motile, “to touch … and not merely see” -- there is no "right" orientation:  a work of art of bare utility, like mobile and stabile (remember Calder’s play circus), a machine (uses energy to perform intended action) which offers abstraction of form and function, as well as cognitive payoff, for the manipulator.

Exploration in depth of the contrapuntal possibilities inherent in a cubical maze module.

A continuous interweaving, user created, network of tunnel paths (inner voices -- relating to bass line -- cantus firmus), imitative / developed contrapuntally into an orientation-independent and network-interdependent system of harmony.

"This is a fascinating object.” (April 28, 2014, m.s.) 

                                                                      Daniel C. Dennett, The Intentional Stance, Sixth Printing 


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. 


                                                                            Rolling Ball “Tilt-Switch

                                                     allocentric / egocentric flip-flop spatial processing                                                      

In musical (math / pattern) termsSis about polyphony:  simultaneously combining modules, independent in melody (rhythm and contour of ball trajectory) yet interdependent harmonically (periodic variations upon a theme). 

In other words, S3 is about "Artistic Process as Qualitative Problem Solving”, in particular, spatial problem solving.

Sequivocal space   demands introspection, self reference, and employs a tension of switchiness (compound switching -- cognitive, perceptual, flip-flop), demanding simultaneous exercise of alternative navigation strategies:  control under continuously varying context.

Art thrives on that tension.


And Zen koan (cognitive flip-flop) mechanics of enlightenment (contradiction). 

In particular, Sis about variations on a theme of square and circle -- flip-flop, a counterpoint of logic and intuition.

Simultaneous mental (rational and nonrational) and manual rotation, giving form to binary and analogical information, a coupling between physical objects and binary information where bits are directly manipulable and perceptible.


In this vein, the Michael S. Sommer photos, and selected works of Cubist and Renaissance art below all comment on / demonstrate spatial processing, variations on a theme of potential energy: stored energy of position, juxtaposition of perception and cognition.  Active navigation is necessary.


“Art is made to disturb.  Science reassures.  There is only one valuable thing in art:  the thing you cannot explain …”  "tactile … manual space …. This answered to the hankering I have always had to touch things and not merely see them…. "  

                                                                                                        Georges Braque, Illustrated Notebooks


S3 is similar in function and aesthetic to Albert Barnes' fascination with the simple hinge (think Archimedes’ lever, or tilt switch) as art, part of a machine which bears friction between a rotating part and its housing, which rotate relative to each other about a fixed axis; a union of Hands and Mind, and more.


                 MSS door                                                                    Barnes Foundation Collection

Like Cubism, S3 is about uncertainty as tool:  shaping space and motion with simple geometric forms, from multiple, simultaneous, changing perspectives; taking them apart and reassembling them (adaptability:  like algebra, solving geometry problems by analytic means (Cartesian Plane):  illumination of diverse dimensional issues by symmetry and contradiction).

Hear Albert Barnes (and John Dewey):

"— scientific method as intelligence in operation, art, education — 

are all bound together in A SINGLE ORGANIC WHOLE

To put the matter in other terms, all genuine experience is 

intelligent experience, experience guided by insight

derived from science, 

illuminated by art, and 

made a common possession in education."

                                                                                  Albert C. Barnes (regarding John Dewey's philosophy)

In other words, S3 is about "intelligence in operation”, the need for more learning proficiency than problem solving competence:  situation awareness, creativity, and the careful construction of logical arguments, and their limitations.  Scientifically verifiable Spatial Thinking tools and learning opportunities are needed, for teachers as well as students, diagnostic as well as development. 

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, and self-reflection.   (S3 learning / development research is a first step toward rigorous, scientific demonstration that STEAM is as effecitve as STEM.)

In Hegel’s terms, a dialectic, a flip-flop, of pattern and process, a mixing of different levels of abstraction, self-reference.  According to his Science of Logicthinking which focuses on itself. 

"I liken my study of logic to the study of grammar. You only really see the rewards when you later come to observe language in use and you grasp what it is that makes the language of poetry so evocative”.



In Einstein’s terms, a mystery.   dialectic engine of paradox, demanding logic and intuition.

“The most beautiful thing we can experience is the mysterious.  It is the source of all true art and all sciences.”

                                                                                                     Albert Einstein, Living Philosophies (1931)


Thus, Sis mysterious:  manipulator and S(both the agent and the object) are engaged in a joint  process of change.  

Art as the promise of change, and self-reflection.  

Art as cognitive tool.


"A growing body of data suggests that connections between the art and science of learning are rigorously supportable by quantitative studies. Contemporary research is beginning to explore explicit neuroscientific hypotheses concerning the beneficial effects of activities such as musical performance, drawing, visual aesthetics and dance observation."  

                                                Tyler, C., Levitin, D., & Likova, L., "Art, Creativity, and Learning" / NSF (2008)



                    Potential Energy -- Form and Its Negation


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

Hear Cézanne and  Galileo:

One must first of all study geometric forms: the cone, the cube, the cylinder, the sphere.”

                                                                                  Paul Cézanne, Letter to Emile Bernard



                                                                                                                        Analog-Binary Processor (0/1)

 "[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


S3, both work of art and machine, is about abstration of pattern, the rule which governs a system or phenomenon, exactly like numeric, musical, or visual relationships -- patterns of thought, and its simultaneous mental and manual rotation.

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.


“Symmetry and order do not make a composition.  It is the apparent accident to regularity which the artist actually controls, by which he makes or mars a work.”

                                                                                                                   Alexander Calder: Interview


                                          S3 As Fugue

To repeat, S3 is about pattern, the rule which governs a situation or a phenomenon, exactly like visual, musical, or numeric patterns.

“... a work of instruction and enlightenment, a kit to be assembled ... that looks backward and forward at the same time ...architectural, rotatable, invertible.”   

                                                                                                                    Paul Epstein on J.S. Bach


To repeat, a continuous interweaving, user created, network of tunnel paths (inner voices -- relating to bass line -- cantus firmus), imitative / developed contrapuntally into an orientation-independent and network-interdependent system of harmony.

S3 as Albert Einstein’s "musicality in the sphere of thought.” 

Remember, Sis about the tension of switchiness (compound switching -- cognitive, perceptual, mechanical flip-flop):  counterpoint.

In other words, Sis about polyphony:  simultaneously combining modules, independent in melody (rhythm and contour of ball trajectory) yet interdependent harmonically (periodic variations upon a theme).

“See" the variations 


(George Balanchine's, "Music must be seen, and dance must be heard.”).

As any child knows.



One of a series: same magic (Rilke) place, with different figures “passing through” the space.



Hands and Mind together create rhythmic stimuli to alter and invert desired hypothetical subject lines:  passage / trajectory of ball through space — simultaneously.

Listen.  Multiple simultaneous paths.


                                                                                         J.S. Bach, Die Kunst Der Fugue

The Geometry of Music,9171,1582330,00.html   

"The scientist is always working to discover the order and organization of the universe, and is thus playing a game against the arch enemy, disorganization."

                                                Norbert Wiener, The Human Use Of Human Beings, 1950


A self-generating, self-sustaining, escalating causal loop, where objects / ideas are taken to pieces, analyzed, and reassembled (a cycle of cognitive development where the manipulator / observer is both agent and object of change).

Thus, S3 as art in action, in process, consistent with Albert Barnes’ idea that "fine art consciously undertaken as such is peculiarly instrumental in quality ... a new training of modes of perception.”  Hands and mind.


                                         Hello.  Hands and Mind (30,000 BC)


The S3 universe consists of an architecture of nested dichotomies:  contradictions of reason and perception, frames of reference, of logic, pattern and process, motion and stasis -- that "game against the arch enemy, disorganization.” 


But, instead of viewing space as fixed, a passive arena, the S3 network architecture uses space and geometry as active participants (think Cubists and Cave Paintings) in that problem universe.  It embodies Braque’s "hankering … to touch things”, that union of hand and mind, the art work of the Cubists, and others as far back as the dark world of the Patagonia, Lascaux, and Altamira Cave paintings (30,000 BC).  


Thus, Sis a dialectic engine of paradox, developing logic and intuition; a tangible, lucid demonstration of basic behaviors (consciousness) of the brain through mechanical, instrumental concepts: a contradiction.  It overrides the human interaction with complex cognitive and perceptual content in paradoxical ways.   Like the the Stone Age cave paintings (variations of pigment intensity to create chiaroscuro, and use of natural cave contours to convey subject three-dimensional effect).

It disturbs.

“Conflict is the gadfly of thought.  It stirs us to observation and memory.  It instigates invention.  It shocks us out of sheep-like passivity, and sets us at noting and contriving…conflict is a sine qua non of reflection and ingenuity.”

                             John Dewey, Human Nature and Conduct: An Introduction to Social Psychology, 1930 



                                                                       Tessellating Space

In other words, the eye presents the mind with that conflict: competing interpretations, paradox; none is satisfactory.  We can be conscious of that tug of war and think about it, and play with it -- the collapse of our expectation of a logical universe.

A “dialectical space”, a topology of paradox.

"Topology is therefore more than a branch of geometry, or geography, or medicine. It deals with art, language, and perception. It is a dialectical space in which ontology and topography meet."

                                                                                  Ben Stol Tzfus, "Robbe-Grillet's Dialectical Topology" (1982) 

"… a kind of Strange Loop, an interaction between levels in which the top level reaches back down towards the bottom level and influences it, while at the same time being itself determined by the bottom level.”   

                                                                                                   Douglas R. HofstadterGödel, Escher, Bach


The paradox facing an infant who hungers to touch to understand.  

In the beginning there was only the nipple, greedy cognition and perception.  Change, and the promise of more change, and self-reflection.

Braque’s “hankering I have always had to touch things and not merely see them…. “  

S3 is all about that magic moment, navigating space, crossing the bridge between Hands and Mind, continuing fascination / puzzlement with the human experience.

Spatial Thinking  (space, representation, process:  multifaceted, interconnecting competencies, human and robot), manifested by a cubical maze module, the Sommer Cube (S3).

                                       "The Hankering  

                                                                    S3 Sculpture      


                                     Rembrandt                        S3 Maquette   ”trompe l’oeil / trompe l’esprit 

Remember, Sis about the tension of switchiness (compound switching -- cognitive, perceptual, mechanical flip-flop):  counterpoint.

The Sconspiracy of pattern, Michael S. Sommer photos, and numerous works of Cubist and Renaissance art shown below all comment on / demonstrate potential energy -- stored energy of position, juxtaposition of perception and cognition:  recursion, and optical illusion; void, solid, motion, stasis; form and function; symbol and object. 

Some engage the viewer in the collapse of the expectation of a logical universe:  the promise of change, and self-reflection.  All interdependent, equal in value, ambiguous and contradictory -- equally figural and important.   Equally false.  Equally self-referential.

To put it another way, it’s about what is not there, “a kind of Strange Loop.”

Like the images below ...



"It turns out that an eerie type of chaos can lurk just behind a facade of order - and yet, deep inside the chaos lurks an even eerier type of order.”  

                          Douglas R. Hofstadter, Metamagical Themas: Questing For The Essence Of Mind And Pattern



                Potential Energy -- Stored Energy of Position

                                    Topology of Paradox

"We believe that great painting --painting that is illustratively full, substantial, and real -- was born with the Renaissance and grew with its flowering …. something about the essential difference between the imagery and intentions of abstract painting and those of representational painting makes it very hard for us to relate abstraction to the past."

                                                                                                        Frank Stella, Working Space 

                          Barnes Foundation Study Group 

       Studio Lecture -- November 13, 2012 -- S3 /  Michael S. Sommer photos




                        Visual Illusion / Analytic Abstraction

                                                 Always a tension.

Rearranging Reality                                                            


Giovanni Piranesi -- engineer, architect, etcher (rearranged reality: "capriccio" / multiple view points / experience of real space / spatial movement / planar characteristics of objects)

Georges Braque -- sculptor, painter (geometricization of motion and form / succession of viewpoint -- reduction by cube, sphere, cylinder, cone as building blocks / "tactile … manual space …")

Alexander Calder -- sculptor; stabile / mobile (flowing energy / succession of viewpoint / "drawing in space")

Sandro Botticelli -- painter (rearranged reality / flowing energy, tactile movement and sense)

Rembrandt -- painter (illusion of form and tactile space / opposition of light and shadow: chiaroscuro / geometric clarity)

Paul Cezanne -- painter (flowing energy / succession of viewpoint -- reduction by cube, sphere, cylinder, cone as building blocks / sculptured, combinatory, "tactile … manual space")

M. C. Escher -- graphic artist (mathematical / visual relationships / manipulation of space: viewpoint, recursion and symmetry -- spheres, cubes, rings and spirals


Albert C. Barnes —  Cognition and perception (seeing and learning from art)

William James -- Psychology, (anticipated much of Cognitive Psychology) human subjectivity (Umwelt / self-world) (primacy of space perception)

John Dewey -- Psychology, philosophy, and educational reform (new way of thinking)

Georg Hegel -- Philosophy (contradiction / opposition) (unfolding of the idea in space)

Marvin Minsky -- Artificial Intelligence (one of the fathers of)  (new way of learning) (advanced technologies for exploring cognitive / perceptual space) 


  Rembrandt circles / planes.  Why circles?                      Braque geometry / planes  

application-51 med-2


                                                                          geometry / planes                                                      



                                                                          energy: critical mass 

“Painting isn’t an aesthetic operation; it’s a form of magic designed as mediator between this strange hostile world and us.”

                                                               Picasso, Quoted in Mario Livio, The Golden Ratio (2002)


                      geometry /  time                                                          geometry / motion 


                                                                      poetry in motion


                                                      geometry /  time


              points of reference / parallax                                               geometry /  time



                         Cezanne motion / space                                        Botticelli motion / space 




                                          Piranesi                                                        Escher


                                                             play of cognition / perception








                                                                             without waking  


" … fine art 

consciously undertaken as such is peculiarly instrumental in quality.  

It is a device in experimentation

carried on for the sake of education.  

 it exists for the sake of a specialized use, use being a 

new training of modes of perception.

                                                                              John Dewey, Experience and Nature, 1929

“Art” does not create the forms; it is their selection and organization in such ways as to enhance, prolong and purify the perceptual experience.  It is not by accident that some objects and situations afford marked perceptual satisfactions; they do so because of their structural properties and relations.  An artist may work with a minimum of analytic recognition of these structures or “forms;” he may select them chiefly by a kind of sympathetic vibration.  But they may also be discriminatively ascertained; and an artist may use his deliberate awareness of them to create works of art that are more formal and abstract than those to which the public is accustomed.

 "Tendency to composition in terms of the formal characters marks much contemporary art, in poetry, painting, music, even sculpture and architecture.  At their worst, these products are “scientific” rather than artistic; technical exercises, sterile and of a new kind of pedantry.  At their best, they assist in ushering in new modes of art and by education of the organs of perception in new modes of consummatory objects; they enlarge and enrich the world of human vision.

“Thus, by only a slight forcing of the argument, we reach a conclusion regarding the relations of instrumental and fine art which is precisely the opposite of that intended by seclusive estheticians; namely, that fine art consciously undertaken as such is peculiarly instrumental in quality.  It is a device in experimentation carried on for the sake of education.  It exists for the sake of a specialized use, use being a new training of modes of perception.  The creators of such works of art are entitled, when successful, to the gratitude that we give to inventors of microscopes and microphones; in the end, they open new objects to be observed and enjoyed….

"In creative production, the external and physical world is more than a mere means or external condition of perceptions, ideas and emotions; it is subject-matter and sustainer of conscious activity; and thereby exhibits, so that he who runs may read, the fact that consciousness is not a separate realm of being, but is the manifest quality of existence when nature is most free and most active.”

                                                                                                 John Dewey, Experience and Nature, 1929

© Michael S. Sommer, Ph.D, 2018