MULTIPLE SIMULTANEOUS PATHS 

                              (Make most paths begin / cross within a single S3)


The Sommer Cube (S3is about eploration in depth of the contrapuntal possibilities inherent in a cubical maze module.

A system of cubical maze modules (graph theory) where changing questions of function, role and position determine meaning.  (Physical and Virtual Reality)

The challenge — COLLABORATION / COMPETITION / SOLO:  navigation in a topology of paradox, where the strategies and tactics are evolutionary:  optimization of multiple simultaneous paths (make most paths begin / cross within a single S3), in a constant stream of evolving problems, within a topology of paradox.  (“How complex environments push brain evolution” 


What is the minimum number of S3s necessary to make n n PATH S3s (S3s which utilize n conduits as passageways)? 

How many n PATH S3s are possible using n S3s?  (Specify S3 OPTIONS)


A)  Answer the question;  build it.  (If not, advantage to CHALLENGER).

B)  Challenge the CHALLENGER to build it.  (If not, advantage to RESPONDER)

A story of leveraging binary and analogical reasoning in a hybrid toy / cognitive tool.  Analytic and Holistic thinking, a coupling between physical objects and binary information where bits are directly manipulable and perceptible.  

(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 rules are.

In other words, the Sphysical state embodies the binary state of the system, as well as the analogical. (think Tangible User Interface — TUI / Virtual Reality)


In cognitive terms, Cognitive flexibility (across the age spectrum) -- switch in thinking between two different concepts, and multiple concepts simultaneously

(Set Switch:  between task rules; Response Switch:  different resource mapping; Simple Switch:  between a circle and square).  (note photos of individuals manipulating S3

(Specific examples of sets of cognitive flexibility below.)

In Cybernetic terms, a 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 (not to mention the unique powers of particular Smodule clusters -- like chess / go; effective teamwork).  


                                                                                                    Rolling Ball “Tilt-Switch

A dynamic router, a system of programmable binary 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.  Each node has equal power to switch and route.

For anyone, across the age spectrum.

                                           S3:   Low-tech and Virtual Reality




                                          (checkered paths allow exit)  


MULTIPLE PATH                            JUNCTION PATH

[paths cross within a single S3:      [two or more paths converge (Airball)]

0,  1,  2,  3,  4 (good luck!)

Single, Double, Tripple, Quad

SUtility:  .25, .50, .75, 1.00]

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

QUAD:  Utility = 1.00 (for the top front S3)


               (checkered paths allow exit)

AIR BALL                                       TUNNEL BALL

[through air; switches paths]          [through tunnels]

TRANSPARENT                             OPAQUE

[clear S3]                                         [“Black Box” (good luck!)]

OFFSET                                         FACE-TO-FACE

[irregular alignment]                        [square alignment]


[multiple paths within fewest S3s]


[“anything goes”]

                                          S3 OPTIONS



1)  Law of Gravity

2)  Law of Lowest Exit

(Given that a ball must exit at the lowest Slevel and obey gravity.)


Multiple simultaneous paths, but no paths can share an input or output. 

(with the exception of AIR BALL).


[Player has the option of hacking into 

or blocking opponent paths]


[Player has the further option of designating the elimination of specified S3s (routers)

in a fixed 3 x 3 x 3 array]


For Example 

Using the fewest S3s

Make most paths begin / cross within a single S3.

Make two, or three, or four

Simultaneous paths which

Which begin from the SAME S3.

(remember that the ball must exit at the lowest Slevel and obey gravity.)

Make the most paths, or

Longest paths, or

Most multiple paths, etc.

(at least one path traverses all S3s)

Timed / not timed.


Perfect Balance

Create a work of art 

with or without

multiple simultaneous paths.



Position S3s in any way;

Connect tunnels in any way;

Make any combination of paths.

Chicken Little



Crisscross Dunk

AIR BALLS from two or more 

different paths

share single tunnel (switch paths). 


Call the Spot

Using a given number of S3s, 

make the ball exit at a given spot(s).

Atomic Warfare (surgical elimination of command / control node-links)

Designate one (or more) Sas taken out by nuclear strike;


Designated S3 remains in place without apertures; throughput is not possible. 

Reroute multiple simultaneous paths accordingly.

Boolean Construction

Formulate and conceptualize each problem in Boolean (or LOGO) terms.

Demonstrate it.

Two-Color Game

Two players, Red and Blue, take turns connecting S3s,

Linking tunnels to gain strategic / tactical advantage, 

While blocking, or hacking into, the opponent’s paths.


                  (checkered paths allow exit)                                 (virtual reality paths confirmed)



                                S3 VIRTUAL REALITY 

                                       (rotation /  calibration sensors / glowing paths) 

                              A) With or B) Without “THE LAWS"


Escape From The Labyrinth 

Think like the ball.  (Visualize the necessary systemwide adjustments from the ball’s moving frame of reference.)

Levels of difficulty (accelerating rates of change and complexity).

Labyrinths:  Increasingly larger universe of S3s. 

S3 reorientations:  Increasing number  

(random / non-random -- time limited / time free).

Euler’s Revenge

Starting from any aperture, pass through and exit every S3 (or prove that it’s not possible):

A) once and only once;

Variations ...

B) twice and only twice;

C) three times and only three times; 

D) four times and only four times (fasten your seatbelt)

No tunnel may be used more than once.  

The starting and ending points need not be the same.

Roller Coaster Loops 

Loop paths cross within a single S3

0,  1,  2,  3,  4 (good luck!)

Dueling Roller Coasters

Loops vs. Loops

Crisscross Dunk Circuits (Knots)

AIR BALLS from two or more 

path circuits by shared tunnel(s).



Double Back 

In a given cubical block of S3s:

Make a path, from top to bottom, loop back to top, and exit.

Make similar multiple simultaneous paths.

© Michael S. Sommer, Ph.D, 2018