Fundamental Structures and coordinate systems
A fundamental bi-radial structure has been identified which makes a very useful coordinate system for analyzing field structures with two origins and in modelling a variety of physical phenomenon. It has two sets of equi-spaced radial lines which are numbered in opposite directions and separated by a finite distance "D".
From this a series of intersection nodes are revealed and assigned the appropriate coordinate pairing based on the ray numbers from each pole.
The coordinates are designated as (x,y).
The coordinates are designated as (x,y).
Here the resulting intersection nodes are isolated and the length of the D segment is given as unit measure.
Here the nodes are shown with their corresponding coordinates. The diagram is on a larger scale to see the coordinates
Connection algorithms
Attraction lines of force
Two approaches to interconnect the nodes:
1) qualitative
2) quantitative.
Qualatative: What is the shortest path between "A" and "B" passing through the unoccupied nodes? Working from the "D" segment outwards the next shortest path through the unoccupied nodes is A, (1,1), B and in the lower hemisphere A, (35,35), B. The next shortest paths passing through the unoccupied nodes are A, (1,2), (2,1), B and A, (34,35), (35,34), B. By repeating this connection algorithm the "attraction" lines are formed.
Quantiative: What are the equations representing the attraction field lines within the bi-radial coordinate system? It turns out that the SUM of any coordinate pair along the "attraction lines" equals a constant.
X+Y=K
Physical interpretation
While these field lines are a purely geometric construct upon casual observation the pattern they exhibit are structurally analogous to attraction magnetic "lines of force". The inverse square analogy will be described elsewhere.
Repulsion lines of force
Qualitative: What paths originating from "A" and "B" passing through the unoccupied nodes approaches the vertical "V axis as close as possible?
Quantitative: What are the equations representing the repulsion lines within the bi-radial coordinate system? It turns out that the difference of the coordinates along any given repulsion line is a constant.
X-Y=K
Quantitative: What are the equations representing the repulsion lines within the bi-radial coordinate system? It turns out that the difference of the coordinates along any given repulsion line is a constant.
X-Y=K
“ On a small enough scale space is just a huge collection of discrete points and actually I think it’s really a
giant network with a changing pattern of connections between points where all that’s specified is how each
point, each node is connected to others.”
From Stephen Wolfram’s 2003 H Paul Rockwood Memorial Lecture on “A New Kind of Science.”
giant network with a changing pattern of connections between points where all that’s specified is how each
point, each node is connected to others.”
From Stephen Wolfram’s 2003 H Paul Rockwood Memorial Lecture on “A New Kind of Science.”
This is an example of underlying structure giving rise to a quantized interference pattern and various "connection algorithms" which yield common field structures. Other hidden harmonic structures are revealed as well. Could these relate to the "nodes" and connection algorithms Wolfram is referring to?