[seqfan] Re: Higher dimensional analogues to plane division by ellipses?

[Hans Havermann]

Yes, thank you. Spinadel's 1999 'The Family of Metallic Means' < http://www.mi.sanu.ac.rs/vismath/spinadel/ > with its bibliography of 30 references, interestingly has a 1997 precursor (The Family of Metallic Means in Design) with *additional* references, including:

[28] Godfrey Gumbs and M. K. Ali, Dynamical Maps, Cantor Spectra, and Localization for Fibonacci and Related Quasiperiodic Lattices, Phys. Rev. Lett., vol. 60, 1988.

[29] Godfrey Gumbs and M. K. Ali, Quasiperiodic dynamics for a generalized third-order Fibonacci series, Physical Review B, vol. 38, Nr. 10, october 1988.

[30] Godfrey Gumbs and M. K. Ali , Electronic Properties of the Tight-Binding Fibonacci Hamiltonian, J. Phys. A: Math. Gen. vol. 22, 1989.

[31] Kolár M. and M. K. Ali, Generalized Fibonacci superlattices, dynamical trace maps, and magnetic excitations, Phys. Rev. B, vol. 39, Nr. 1, 1989.

[32] Kolár M. and M. K. Ali, Attractors in quantum Ising models, Phys. Rev. B, vol. 40, Nr. 16, 1989.

Reference 31 < http://prb.aps.org/abstract/PRB/v39/i1/p426_1 > mentions "spin-wave spectra for Fibonacci superlattices with copper and nickel mean are compared with those for the golden-mean case".


On Jun 2, 2013, at 12:54 PM, allouche at math.jussieu.fr wrote:

> It is that clear that the expression "metallic mean"
> was coined in 1999?
> 
> I seem to remember having heard the expression
> "metallic mean" before (i.e., some years after
> the discovery of quasicrystals).
> 
> In particular I traced back a 1996 paper:
> J. A. G. Roberts, Escaping orbits in trace maps,
> Physica A: Statistical Mechanics and its Applications,
> Volume 228, Issues 1?4, 15 June 1996, Pages 295--325
> where the term "metallic mean" is used (see Page 298
> just before (11)). In that paper "metallic-mean" sequences
> are generated by the substitution rule (morphism of the
> free monoid) a --> b, b --> b^{\ell} a, whose transition
> matrix admits (1 + \sqrt{1+4\ell})/2 as dominant eigenvalue.
> 
> Other "metals" were used before, e.g., bronze in several
> papers including
> G. Gumbs and M. K. Ali, Scaling and eigenstates for a class of
> one-dimensional quasiperiodic lattices, J. Phys. A: Math. Gen.
> 1988, 21 L517--L521.
> 
> I am not even sure that Roberts' paper is the first one where
> the expression is used