> In process of generating the group table for O (= S4), I found
> that 6 cosets of K4 (the Klein-4 group) within S4, seemed to map perfectly
> fundamental particle labels. This is because there are 5 classes of
> elements in S4 (the number of elements in the classes being 1, 3, 8, 6, and 6).
> The first 3 classes correspond to the T (=A4) subgroup of 12 even
> permutations and seem to echo the bosonic particles 1 photon, 3 weakons, 8 gluons.
> Moreover the odd permutations consisting of the two remaining classes
> map onto 3 families of fermions, with two quarks and two leptons in each
> family).

> This was all finite group modeling, yet I was very intrigued by
> Muses's
> work on hyperspace, especially his paper, "The Amazing 24th Dimension",
> published in the *Journal for the Study of Consciousness* (which he
> edited for Arthur Young), Vol. 5, No. 1 (1972), pp. 82-24. There Muses
> describes the packing of spheres in 24 dimensional space, via the Leech Lattice.
> There are 196,560 spheres packed around a central sphere in this maximally
>dense packing.

> Since S4 is a 24 element group, I thought there might be a
> connection between this group and the magical 24-d space. So my next move was to
> construct the 24-d group algebra of S4. This is done by regarding the
> 24 elements of S4 as vectors in a vector space. Moreover, if we make this
> a complex vector space, then via various theorems (such as Wedderburn's)
> we can see that
> C[S4] = M1 + M2 + M3 + M3 + M1
>
> where C[S4] is the complex group algebra of S4, and Mn is the total
> matrix algebra of n x n complex matrices. Then the unitary elements within
> C[S4] is of necessity the 24-d unitary Lie group:
>
> U = U(1) x U(2) x U(3) x U(3) x U(1)
>
> which we can rewrite as T4 x U(1) x SU(2) x SU(3) x SU(3), and
> thus we recover the gauge groups of the standard theory of particle physics
> (plus and extra SU(3) which may function as a hyperweak gauge group acting
> on the three family labels -- thus breaking CP symmetry -- but this is
> speculative.
>
> We are still a long way from the Leech lattice in 24-d vector
> space that Charles Muses described as important for both physics and
> consciousness.
>
> There is, of course, one force missing in the above unification --
> gravity. In order to make contact with gravity gauge groups (assuming
> such an approach makes sense), I went to the double group version of S4,
> which could be written as DS4, but I prefer to write it as OD, the octahedral
> double group (of the crystallographers). Just as O is a finite
> subgroup of SO(3), OD is a finite subgroup of SU(2). In other words, just as SU(2)
> double covers SO(3), and we write SO(3) = SU(2)/{1,-1}, so we can write

> O = OD/{+1,-1}.
>
> Then if we construct the group algebra of the 48 element OD group, we
> find that:
>
> C[OD] = C[O] + M2 + M2 + M4
>
> But this can be written out and reordered as:
>
> M1 + M2 + M3 + M4 + M3 + M2 + M1 + M2
>
> The reason for writing the direct sums of these matrix algebras in this
> order is that there is a 1-1 correspondence between these matrix
> dimensions and the numbers on the nodes of the extended Coxeter graph for the E7
> Lie algebra (and Coxeter-Weyl group).
>
> 1--2--3--4--3--2--1
> |
> 2
>
> That there is indeed a deep and beautiful connection between these two
> very different mathematical structures, a finite group algebra and a Lie
> algebra is the great discovery of John McKay in 1979. The connection is that
> the 8-d center of C[OD] is equivalent to the 8-d Cartan subalgebra of the
> (infinite dimensional) extended E7 Lie algebra.
>
> I call these numbers on the Coxeter graph balance numbers because
> the positive mirror vectors in the E7 "reflection space" (which Coxeter
> calls the kalleidoscope), when lengthened by these balance number factors
> will all be in balance -i.e., as vectors they sum to zero. These number then
> also define the setting up a a lattice in the 7-d Cartan subalgebra space
> of the 133 dimensional E7 Lie group. Moreover this lattice generates a binary
> error-correcting code -- the Hamming-7 code.
>
> So how do we get to the 24-d Leech lattice? Well, the E7 lattice
> is a sub-lattice of the E8 lattice in the Cartan subalgebra of the 248-d E8
> Lie algebra. And this E8 lattice generates the Hamming-8 code.
>
> This is important for the E8 x E8 heterotic superstring theory. The
> lattice structure in this Lie algebra provides for the "anomaly
> cancellations" for the heterotic string theory. And it was this
> even-self dual lattice structure that provides this "anomaly cancellation". What
> physicists call anomaly cancellation is, in effect, what the
> mathematicians call error-correcting -- every even-self dual lattice corresponds to
> an even self-dual error-correciting code."

Physical reality as software? At least the BIT part?

>
> "Here's where the 24-d Leech lattice comes in: the Leech lattice
> is also an error-correcting code lattice; in fact it is the lattice for the
> even self-dual 24-bit Golay code. This code can be generated by three
> copies of the E8 lattice: E8 + E8 + E8. This is important because in the
> heterotic string theory, the bosons see the full 26 dimensions of the original
> 26-d string theory of 1970, in which the fermions see only the 10
> dimensions of the superstring theory of 1971. Since the strings sweep out a 2-d
> world sheet, it is the transverse vibrations of this world sheet that must
> account for the particle states. The bosonic transverse vibrations are in a
> 24-d space, while the fermionic transverse vibrations are in an 8-d space.
> The 24-d space has the Leech lattice corresponding to the Golay code (or
> E8 + E8 + E8 lattice), while the 8-d fermionic vibratory space corresponds to
> the Hamming-8 code of the a single E8 lattice. The E8 + E8 lattice (16-d)
> interpolates between the Leech lattice and the single E8 lattice in the
> lattice direct sum E8 + E8 + E8, from which one can derive the 24-d
> Leech lattice.
>
> So although, my 24-d group algebra C[O] seemed to contain much of
> the structure of the standard model of particle physics, it did not make
> direct contact with the 24-d Leech lattice. However, there is a kind of
> apotheosis for the E7 Lie algebra (which is closely associated with C[OD] via
> McKay's correspondence (mentioned above)).
>
> The E8 x E8 heterotic string theory is just one of five competing
> string theories. But since 1995, due to the work of Edward Witten (and others)
> there is an overarching theory called M-theory which unifies the 5
> competing string theories as subtheories. Moreover, the 11-d supergravity theory
> (of point particles is contained as a subtheory of 11-d M-theory. As a
> consequence E7 is brought in as the basic symmetry group of the ll-d
> theory.
>
> So I feel that I was barking up the right tree after all. My
> starting point in all this was the work I did at the ISC under the influence (in
> somewhat different direction) of Arthur Young and Charles Muses.
>
> In any case, it is now abundantly clear that Muses's fascination
> with hyperspace in general and the 24-d Leech lattice in particular was a
> move in the right directing for advance in physics.
>
> I would even say that Muses's speculation that consciousness
> should be modelled via hyperdimensional mathematics is on the right track. My
> approach to this is deeply involved in E7 and indeed in the complete set of
> A-D-E Coxeter graphs.
>
> This is written up in great detail in my appendix
> paper"Consciousness: a
> Hyperspace View" in Jeffrey Mishlove's book, *Roots of Consciousness* (2nd Edition), Marlowe, 1993.
>
> A summary (and update) is "A Mathematical Strategy for a Theory of
> Consciousness" pp, 579-588 of *Toward a Science of Consciousness: the
> First Tucson Discussions and Debates*, edited by S.R. Hameroff, A.W.
> Kaszniak, & A.C. Scott (MIT, 1996).
>
> I should mention that Jeffery Mishlove also worked with Arthur
> Young at the ISC, and interacted with Charles Muses there and elsewhere. And
> there is material on both Charles Muses and Arthur Young his book *Roots of
> Consciousness*
>
> You can find out more the McKay correspondence between finite
> subgroups of SU(2) and Lie groups at
>
> http://math.ucr.edu/home/baez/ADE.html
>
> where John McKay has the short paper "A Rapid Introduction to ADE Theory."
>
> There is a reference to *Roots of Consciousness* at the end of McKay's paper.
>
> My mathematician friend Tony Smith had sent McKay of copy of my long
> appendix paper of 1993. So that is apparently why McKay refers to
> *Roots of Consciousness*
>
> BTW: Tony Smith has also become interested in the work of Charles
> Muses, and you can find some of Tony's thinking on this in the long paper "Del
> Pezzo Surfaces and the A-D-E series" at:
>
> http://www.innerx.net/personal/tsmith/NDxNDalg.html
>
> On Superstring theory, see the November issue of *Scientific American*
> for an interview with Brian Greene, who wrote *The Elegant Universe* and is
> hosting a Nova series on string theory. Parts I & II on Oct. 28, Part
> III on
> Nov. 4.
>
> All for now.
>
> Saul-Paul

Correct me if I am wrong Saul-Paul but what is your picture of the connection of this group structure to consciousness. Do you have in mind orthodox quantum measurement theory with the results of measurement being "eigenvalues" along lines of Wigner & Stapp i.e. "collapse." I have argued against that view for consciousness. However, the A-D-E structure also applies to catastrophies and in my picture there is a MACRO Q-BIT landscape in ordinary space. Think of a giant Bohmian real quantum potential Q in ordinary space made from PSI(x) the coherent order parameter that is literally the physical mind field. This PSI may have a fractal structure. Actually what we have is not PSI(x) but a wavelet transform PSI(x,L) at scale L and so we have a family of complex functions PSI at different scales L and there are topology changes of this what sheaf of landscapes with coupling at different scales - gets very complex and that's I think how V.I. Arnold's theory of singularities come in. Indeed the dynamics of the mental landscape, say in the real representation of Q(x,L) ~Q(x,L)^-1 Grad^2Q(x,L) (or maybe the wave propagation generalization) can actually be pictured in computer simulations allowing us to literally read thoughts! There is a great simplification in that the MACRO-QUANTUM mental field of the living mind in my theory is a local complex field in ordinary 3D space and a lot of its features can be pictured in terms of the dynamics of the real MACRO-QUANTUM potential Q. Of course both the amplitude and phase of PSI will also have physical importance. Since MACRO-QUANTUM PSI is literally a fusion of IT and qubit in the condensate one has to ask if there is any analog to a localized ball rolling on it as there is in micro-quantum theory. The answer I think is no. But there is a distributed say electric dipole moment IT distribution D(x,L) in two-way relation to the BIT PSI(x,L). D(x,L) at some scale L will be directly related to nerve impulse firing patterns for example. On smaller scales to MRI and so on.