PROBLEM SPACE / TOPOLOGY



     

                                                                                                

        


                                        Turbulent Fields

                                  Network Simulation / Modeling

                                                                

                                                                            


The Sommer Cube (S3) is navigating three-dimensional problem space with Hands and Mind.

Thus coordinating mental and manual environment, modifying self-behavior on the basis of experience, through the optimization of multiple simultaneous paths, in a constant stream of evolving problems, within a topology of paradox.

A dynamic cycle of abstractionnested recalibration and reflection:  symmetry under changes of coordinate systems.

Spatial Thinking   (space, representation, process:  multifaceted, interconnecting competencies, human and robot).

                           

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?)

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.

Adventure by topology.


                                                



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 (note the Quad Problem).

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

Synergy,


Buckminster Fuller's "coordination of thought and physical action, the genesis of geometry, system, and structure.


                            


                                              

                                           Mathematician constructs problem space on Quad Problem.


Sis a three axis rotation problem space of regulatory switching (synchronizing concurrent communication / interference among rotating switching stations), a variety increasing (informal as well as formal reasoning), heuristic routing (ball or vector) system seeking best paths (optimization) in an unfriendly (blocking), gravity-dependent network of switchable links.

In other words, a distributed processor (S3) programming network.

Think of the ball as “ping” :  a network monitor / troubleshooting tool testing reachability.

                            

                                         


Arguably, in the classic topology of networks problem, while Euler only saw the seven bridges of the city of Konigsberg as a simple


                                                       


choice of optimal routes, the S3 manipulator has to reformulate the dynamic problem of  S“bridges” / nodes and their increasing interplay of dynamic wholes and parts; tactics and strategy have to be continuously reinvented with a cubical maze module offering a development of choices (control flow) to create linearly independent / dependent paths, (using a ball, or symmetry in mathematics).


                                                        

                                           (checkered paths allow exit)  


Euler's problem was straightforward, the bridges never change orientation or connectivity, while S3 "bridges” (distributed processors) regularly change attitude, causing cascading, unexpected change, symmetrical and asymmetrical, partial and / or complete loss of connectivity for the entire network.

Here, conventional / "practical" reasoning will not go very far; the combinatoric issues rapidly become overwhelming.  A robust flexibility of logic and intuition, naive commonsense reasoning, are prerequisites.


                     

                                                                                  Tessellating Space


In Euler's (Topology / Graph Theory -- mathematical study of networks) terms, S3 demands the manipulator distill a geographic system to its underlying graph, its conceptual abstraction: two fundamental components, simple line and point, abstracted into a set of edges and junctions (paths and switches), a geometry problem independent of distance; a continuously updated network connectivity with rules that control which network objects can properly connect to each other.

Each new S3 attitude  simultaneously reprograms four "gravity feed" chiral tunnels differentially, nonlinearly.  Each tunnel, with twenty-four possible attitude combinations (impede (0) / allow (1) ball flow), is effectively a two-state rolling ball "tilt-switch", a binary (0/1) logic gate

To optimize multiple simultaneous paths (make most paths begin / cross within a single S3) the manipulator must simulate the future, conceptually "drill down” through nested coordinate systems.  But what is the manipulator’s reference frame?  Like the Frame Problem:  What information is relevant for each action, what will change, what will remain the same), the problem of delimiting the consequences and non-consequences of actions.

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)  /  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.  (Think Field Programmable Gate Array.)

(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.)


Each of these nested frames of reference establishes conditions / opportunities as the focal point of each frame is moved in any of three perpendicular directions.

With each rotation (precession of the rotational axis of each tunnel), and each additional S3, the manipulator must focus on the binary issues (0/1) of a given tunnel (part), proceed up the logic hierarchy to the S3 (whole), then to the ever expanding universe of S3 (overarching whole / system of systems).  Then go back down the hierarchy of nested coordinate systems, (each with different properties and order of operations, principles have to be unlearned) and repeat the process (while coping with contradiction of action and environment)

Unlike the stationary bridges, S3 routes change as a consequence of existing geometry; the S3 is not within a field; it is the field (each S3 is both whole and part).

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.  

This problem, sets of problems, must be constantly reformulated in abstract terms; 

a system of rules must be discovered and represented.  

Not just once like Euler's. 

S3 play,

A topology of paradox.

                                         

                                      

                                (checkered paths allow exit via Air Balls and linked tunnels)



                            Mind And Machine As Fugue

                         Symmetry Under Transformation

                                                                                


"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) 


To repeat, S3 is about patterns, the rules 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


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

Contrapuntal in two or more voices, built on a self-referential theme repeated differently, closed and recursive, recurring frequently. 

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 [Einstein's] "musicality in the sphere of thought.”



               


 

                                                                                          J.S. Bach, Die Kunst Der Fugue


S3, as [Glen Gould's] “uncompromised invention.”

   

                                                                                                                             J.S. Bach, Well Tempered Clavier


Each voice appearing successively, sharing the same melodic material -- building blocks to be altered or transposed:  variations in twenty-four major and minor keys (play with Pythagoras's Circle of Fifths), and S3 topological play ("Combinatory Play") and Jazz play, all are commenting on / taking flight from their own underlying structure.  

In a joyful way.


The Geometry of Music 

http://www.time.com/time/magazine/article/0,9171,1582330,00.html   

 

S3  —

A form of play, a secret plan or clever trick (not necessarily competitive).  

An evolving topological conundrum predicated on the first S3 module.

A network which evolves in accordance with simple local rules:

The Law of Gravity, and 

The Law of Lowest Exit,

which govern emerging global patterns.   

Thus, the cooperation of neighboring S3s leads to unexpected utilities (the unique powers of particular S3 clusters, in different situations -- like chess / go) and ever richer complexities.

                                        

(emergent phenomena, like John Conway’s Game of Life, etc.)  except that here the field of play (S3) is “aggressive”, while the ball / actor is “passive” (the only option is down).

                                                

     

                                           

                                                                                 Tangram / S3


                                                 

                                                                                 Euler Rules

                                                                          



© Michael S. Sommer, Ph.D, 2017