The holographic structure of the Quaternion
1995 (c) Copyright | this meme may be spread
| freely as long as it is
By Onar Aam | spread in its entirety
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http://www.hsr.no/~onar/ onar@hsr.no
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Hypersets (1), the mathematics of facing mirrors
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In constructing his Principia Mathematica Bertrand Russell made a
rule in set theory stating that a set could not contain itself as
an element. Such sets were called non-well-founded. This was done in
order to avoid paradoxes, but many mathematicians felt uncomfortable
with this rule since it does not naturally derive from mathematics.
Much later the mathematician Paul Aczel justified this unease by
showing that the mathematics which arises from non-well founded sets
is consistent. Hypersets, which these objects are called, are just as
consistent as all other mathematical objects, just like mathematics
once had to accept that the imaginary number _i_, the squareroot of
-1, is just as real as the Reals.
Hyperset theory surely has a lot of applications, but in my opinion
the most intriguing application is the precise description of the
holographic structure of a mirrorhouse. Mirrorhouses are built up
from facing mirrors which reflect each others reflections. The
simplest mirrorhouse possible to construct is made of two facing
mirrors, X and Y. X reflects Y and Y reflects X. In terms of
hypersets this may be written as the following:
Let X and Y be hypersets such that
X = {Y} and
Y = {X}
(ignoring the inversion effect of mirroring)
We now have a finite mathematical description of the mirrorhouse
effect. But if we try to unravel the hyperset by inserting one
element into the other we engage in an infinite regress:
Y = {X = {Y = {X = {Y = {X = {Y = {{X = {Y = {...} } } } } } } } }
This corresponds to the illusory infinite tube which interpenetrates
both mirrors.
Suppose now that we constructed a mirrorhouse from *three* mirrors
instead of two. What hyper-structure would this have? Amazingly it
turns out that it has precisely the structure of the quaternion
imaginaries.
Let i, j and k be hypersets representing three facing mirrors. We
then have that
i = {j,k}
j = {k,i}
and
k = {i,j}
With three mirrors ordering now starts playing a vital role because
mirroring inverts left/right-handedness. If we denote this mirror
inversion by "-" we have that
i = {j,k} = -{k,j}
j = {k,i} = -{i,k}
and
k = {i,j} = -{j,i}
But the above is exactly the structure of the quaternion triple
of imaginaries:
i = jk = -kj
j = ki = -ik
k = ij = -ji
The quaternion therefore is the precise model of the holographic
hyper-structure of three facing mirrors where we see mirror
inversion as the quaternionic anti-commutation. Just like there are
only two possible ways of constructing a mirrorhouse made of three
mirrors there are only two ways to construct the quaternion.
It does not demand a vivid imagination to see that a similar but
more complex mirror constellation plausibly corresponds to the
octonion structure. One thing is certain: at the level of the
octonion the number of structures and constructions virtually
explodes. Unlike the 2 possible quaternions there are 480 different
ways to construct the octonion(2). I therefore suspect that
constructing and understanding the octonion mirrorhouse is an
immensely more demanding enterprise, but if we manage to tame this
monster we should not be surprised if it turns out to enlighten us
with subtle understanding of the reality which interpenetrates us.
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General References:
"The Magic Mirrorhouse", Onar Aam
"Illusory Illusions", Onar Aam
"Steps to the threshold of the social", Kent D. Palmer, Ph.D.
(1)"Chaotic Logic" (Plenum Press), Ben Goertzel, Ph.D.
(2)Tony Smith's application of Octonions to the laws of physics:
http://www.gatech.edu/tsmith/3x3OctCnf.html
Kent D. Palmer:
Software Engineering Technologist
autopoietic social systems theorist
e-mail: palmer@netcom.com
Ben Goertzel:
e-mail: ben@psy.uwa.edu.au
Tony Smith
e-mail: gt0109e@prism.gatech.edu