The A5 icosahdral symmetry is the permutathdral group for sporadic groups and E_8. Also the underlying homotopy group describes a nonassociative algebra.
The 24-cell is the minimal sphere packing for four dimensions. The 24-cell (tetrahedachoron) has 24 vertices and 96 edgelinks, where the 24-cell is self dual. So consider the 48-cell given by this cell plus its dual. A reassignment of these according to a golden ratio (dividing each edgelink by phi = (1 sqrt{5})/2 with the assignment of a vertex to each edgelink produces the 120 vertices of the 600-cell --- 96 24 = 120. This is a way in which the construction of the icosian quaternions is arrived at with the simple B_4, D_4 & F_4 group system of the 24-cell. In this way the flat tessellation of the R^4 is mapped to a hyperbolic tesselation of the AdS spacetime.
We might think of the each vertex of the 24-cell as defining a quiver of fields. The associator then gives a nonassociative system for these quaterions
e_i(e_je_k) = -(e_ie_j)e_k = C_{ijk}^le_l.
This factor is particularly important with group transformation between tesselation regions (120-cells) in the spacetime, which produces a nonassociative map between quantum groups of fields (quivers etc), and noncommutative coordinates on the scale of hbar^2. This connects with the gauge-like role of noncommutative geometry and something called zitterbewegung, or the strange motion of a fermion in a region defined by its Compton wavelength. I'd advise reading Hestenes excellent essay on this topic.
The golden mean involved with the 120-tessellation has a Fibonacci sequence to it, which plays a role in the boost of a field from one 120-cell to the next. This boost described the motion of a particle form the boundary of the AdS with E ~ infinity as it arc around back to the boundary with E ---> 0. With the introduction of a BTZ black hole in the space these paths can connect the AdS boundary with the horizon of the black hole.
The running through energy scales then defines a renormalization group, which as yet I honestly don't understand that well. This is connected with Ricci flow equations, such as the Hamilton-Perelman theory used to prove the Poincare conjecture.
As indicated above the extension of the flat 24-cell tessellation to the 120-cell tesselation maps the flat spacetime into the AdS. This is a conformal map. An important issue in general relativity is conformal invariance. A metric is often modified by some scale factor Q so that g_{ab} ---> Q^2g_{ab}. There there is the issue of what is conformally invariant, which in GR is the Weyl curvature. So for the metric line element
ds^2 = g_{ab}dx^adx^b
for a diagonal system we have that the conformal transformed element is
ds^2 = -Q^2(u)(du^2 - dr^2 - r^2dOmega^2).
Now I write the time part as u, because suppose that Q^{-2) = du/dt, then we can write this as
ds^2 = -dt^2 Q^2(dr^2 r^2dOmega^2),
where for this conformal factor Q^2 = exp(sqrt{L/3}t), L = cosmological constant, this gives the deSitter spacetime. So this time dependent conformal transformation can in a special setting define the deSitter cosmology. So this means that in the cosmology the equivalence principle is extended to frames which are conformal. So the comoving frame, which "surfs" on the expansion (Q-dot/Q)^2 = L/3 = H^2(Omega)/c^2, is in effect on a local inertial frame with the expansionary factor. We might call this a cosmological equivalence principle, which generalizes the notion of how we define frames globally.
With the extension to quantum fields in a Maldacena like AdS-CFT duality this golden ratio involved with the local boosts of a field between Voronoi (brillouin zones) cells also defines a renormalization group for running coupling parameters.
I attach a file which gives a picture of how fields are mapped between the AdS boundary and a BTZ black hole. It also indicates how there is an optimal size for the black hole.
Lawrence B. CrowellAttachment #1: dte_ads.gif
