Modular Group can be represented by 2*2 matrices;
((a,b),(c,d))
with;
ad-bc=1.
Where a,b,c,d are integers.
Note that Stern-Brocot Tree has the same condition.
The matrix format is obviously polarised, i.e. each element knows where it
sits.
But the symmetry suggests an unpolarised form.
Consider the sequence;
{}
[]
0, 1
{,{}}
[,[]], [[],]
-1, +1
{,{,{}}}
[,[,[]]], [,[[],]], [[,[]],], [[[],],]
-1, +1, -i, +i
Note that this sequence has a 1-step memory (OSM) aspect due to iterating on
the previous step.
OSM can be seen in;
ad-bc=1
since this condition vertically ties a rational node to the next, thus
defining a fully-addressed binary tree.
OSM is in MSet;
x(n+1) = x(n)^2+c
since (n+1)st step remembers (n)th step.
I suggested HFractal and MSet relationship in a previous posting.
http://groups.google.com.au/group/sci.fractals/browse_thread/thread/03cddea07a219797/cbf47ee34af96083?hl=en#cbf47ee34af96083
OSM is in Schrodinger's Equation;
(d/dt)W(x,t) = (-iH/h)W(x,t)
-i represents a delay of pi/2 since;
-i = exp(-i*pi/2)
OSM is the temporal aspect of the iteration.
The sequence above has two aspects; temporal (TA) and spatial (SA).
TA expands the tree and gives it a direction.
SA is realised explicitely after polarisation.
TA defines a HFractal while the SA defines a coordinate system.
TA is like algebra while SA is like geometry.
Newton's third law reiterated;
For every action, there is an equal and opposite reaction, and subsequent
lateral ones for each.
H2 seems to be isomorphic to our universe.
.
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