PROBLEM SPACE / TOPOLOGY


012*let's do lunch credit copy


     

                                                                                     

                                       Turbulent Fields

                                  Network Simulation / Modeling

                                                                                   


The Sis about navigating three-dimensional problem space, with Hands and Mind.

S3 is a hybrid:  combined puzzle and maze and rolling ball device, where intuition is a prerequisite -- logic is necessary but not sufficient (note the Quad Problem).

S3 is a combinational logic block which contains four tunnels, effectively an array of unconnected 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.

The S3 is about:

Pattern

 

the rule which governs a system or phenomenon, exactly like numeric, musical, or visual relationships -- and its simultaneous mental and manual rotation.


"Turbulent  Field ...  the dynamic properties arise not simply from the interaction of the component organizations, but also from the field itself. The "ground" is in motion.” 

                                                    Emery and Trist, "The Causal Texture of Organizational Environments”, 1965.


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.

Transformation.  Equilibrium.

Thus, S3 is about awareness of 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.



                                                              


"The primary purpose in problem solving should be better to understand the problem space, to find representations within which the problems are easier to solve …. once the space is adequately understood, solutions to the problems will more easily be found."

                                                                                            Marvin Minsky, "Frames Systems Theory", 1975


                            


                                              

                                           Mathematician constructs problem space on Quad Problem.


Sis a three axis rotation problem space of regulatory switching (synchronizing concurrent communication / interference among rotating S3s), a variety increasing, heuristic routing (ball or vector) system seeking best paths (optimization) in an unfriendly (blocking) network / environment of switchable links.

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 assymetrical, 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 is a prerequisite.


                     

                                                                                  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.  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

                                                 

In a network of S3s, each node (cubical maze module) is effectively a capricious range of paths, a dynamic system of switches, a router; it connects other nodes through arcs (tunnels) subsumed within the node itself, nodes within nodes within nodes, etc. 

An holarchy, an architecture of nested uncertainty, of constant and unexpected contradiction and recursion.

To optimize multiple simultaneous paths 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 -- what will change, what will remain the same? 

In other words, 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) and repeat the process (while coping with contradiction of action and environment).  

CUBICAL exoskeleton (six faces)  >>  

HOLES quadrant (eight holes)  >>  

TUNNELS gate array (four binary switches)  >> 

TUNNEL (binary switch). 


                           

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

                                                                        

(Given that a ball must exit at the lowest Slevel and obey gravity, exit is possible in all orientations.)

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.  

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 (a 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 all a commentary on / a 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 and ever richer complexities

                                        

(similar to cellular automata, emergent phenomena, 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).

                                                

     

IMG 0869 copy (1) crop 2

                                                                                      Tangram / S3


                                                 

                                                                                 Euler Rules




© Michael S. Sommer, Ph.D. 2015