N Boolean Nodes (0, 1) [WHITE, BLACK]

K Connections (INPUTS) per node

Relational Operators

Each node relates the states of its two inputs at time T determine its (output) state at T+1

 

AND, OR, NOR, NAND, XOR, ....

XOR

DETECTS DIFFERENCE

XOR OPERATOR
INPUT 1 at T1	INPUT 2 at T1	OUTPUT at T2
0	0	0
0	1	1
1	0	1
1	1	0

 

Network Infrastructure determines the emergence of a landscape of basins of attraction

Logical Derivations (skipped here) produce the following

Basins

Attractors (single point or cycles)

Tributaries into attractors

A Landscape consisting of Three Basins
Each Basins has an attractor
Two of the basins have Tributaries (Transients) leading into the Attractor

State Vectors
(We rotate them from above picture to be Column Vectors)

Then, in Basin 1:

Attractor Matrix for Basin 1 (Rotated to begin with lowest Boolean value)

We Map Boolean Simulations to a Mental Landscape

Where Stream of Consciousness flows down tributaries
into
Dynamic Eddies
and
Standing Waves

 

TAO MATRIX = Successive XOR between each State Vector in Attractor

Rotate TAO Matrix to begin with Lowest Boolean Value

Note: The TAO Matrices are rotated so that the first column contains the vector with lowest Boolean Value

Attractor Cycle lengthL
is a
Power of 2

or NOT a Power of 2

Some individual nodes have sub-L = to a power of 2

Other nodes have a sub-L = NON power of 2 (L = 6)

System as a whole has a period L = 6

• TAO analyzes change from moment to moment
• From one state vector to the next across all state vectors in attractor
• TAO Produces a Matrix of Moment to Moment Changes in Attractor

Meta-TAO simply compares the differences between any two Matrices cell by cell

Most Frequent Use:

TAO-0 matrix versus Higher order TAO's

Meta-TAO-3 = [TAO-0 XOR TAO-3]

XOR is Reversible

Thus

Meta-TAO [Meta-TAO-3 XOR TAO-0] = TAO-3

SOMETHING A ====> TRANSFORM T ====> SOMETHING B

Where in some sense AFTER the TRANSFORM

SOMETHING B is indistinguishable from SOMETHING A

Symmetry is defined by some kind of invariance over transformation T

What is the this transformation T that leads to this invariance?

One damp morning..

Imagine a spider web coated uniformly with a column of water

Symmetrical with T = Rotation around web (axis)

Symmetrical with T = Translation Horizontally

BREAK THE SYMMETRY

Surface tension of water molecules 'break' the symmetry

A string of equally space beads of water (3 cm apart)

Has lost full horizontal translation

But still has symmetry for horizontal translation by 3 cm

All 144 degree Rotations and Reflections through medial vertical axis are Transformations of Star that stay within the Group

CURIE PRINCIPLE

(1) the symmetries of causes reappear in their effects
and
(2) asymmetries found in effects will be reflected in their causes.

Boolean Infrastructure causes a Landscape of Basins (tributaries and attractors)

Therefore: Symmetries in Infrastructure of the Network will cause symmetries in the the emergent landscape of the system

Description of N7 XOR Ring

N = 7 Nodes

K = 2 Inputs

First Input (Self-referencing) Node checks its own state at T

Second Input: Node checks its nearest clockwise neighbor's state at T

Relational Operator: XOR

Node is ON at T+1 if its state at T is different than its neighbor's state at T

Node is OFF at T+1 if its state at T is the same as its neighbor's state at T

We Start with a Highly Symmetrical Case were it easy to visualize our epistemological principles

And we will move to less symmetrical and more general cases in Part 4

EXTENDED CURIE PRINCIPLE

States of physical systems
Come in sets
and
These sets of states are related by symmetry.

Stewart & Golubitsky: 'a symmetric cause produces one from a set of symmetrically related effects.'

 

WHAT HAPPENS WHEN WE TRANSFORM AN ATTRACTOR MATRIX BY META-TAO?

Specifically, let T = Meta-TAO-3

[Attractor Matrix] ======> T ======> ?

Nine of the Ten Basins generated by our N7 XOR Ring

Note that Meta-TAO-3 matrix is equal to Attractor 4 Matrix

NOTES

Many transforms will generated symmetry groups in this highly symmetrical example

Note that Meta-TAO-5 matrix is equal to Attractor 9
Note that Meta-TAO-6 matrix is equal to Attractor 10
The TAO's are also transforms that produce symmetry groups

Meta-TAO-3 will turn out to be very useful in less symmetrical cases

We are here starting with High Symmetry
and will go to less symmetrical and more general cases in Part 4

Now: Take XOR of A7 and A8

Meta-TAO of attractor 7 and attractor 8 in landscape

Sierpinski Symmetry Lattice:

A lattice symmetry is one in which patterned tiles are translated horizontally and vertically....

Below attractor 6 is highlighted

LIMITATIONS: All has been demonstrated on a System that is designed to be
extremely symmetrical. Not too surprising to find that symmetry matters in a highly symmetrical context.

What happens with less symmetry ? (Part 4 of this symposium)

What happens when a system is built (pseudo) Randomly with no known built in symmetry? (Part 4 if time allows or in discussion)

BATESON: Evolution = Mental Process (=Morphogenesis)

Epistemology
Can be construed to be
The knowing of form

Bateson ====> Symmetry as
The Pattern which Connects forms of life

Morphogenesis
and
the Emergence of Form

In the great stream of Mental Process

Knowing Begets Knowing

within Symmetry Groups

in a Mental Landscape

AND Knowing one Attractor

(defined as Finding differences in the flow of differences)

Is a WAY to other attractors

in the same symmetry group