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

allouche at math.jussieu.fr allouche at math.jussieu.fr
Sun Jun 2 18:54:40 CEST 2013

Hi,

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

best
jp

Hans Havermann <gladhobo at teksavvy.com> a écrit :

> For those of you (like me) who haven't encountered the expression
> 'Metallic Means' before, this appears to be a 1999 designation
> created by Buenos Aires mathematician Vera W. de Spinadel to extend
> the 'Golden Mean' concept <
> http://www.mi.sanu.ac.rs/vismath/spinadel/ >. The 'silver ratio'
> [1+sqrt(2)] seems to be a better known example. Ukrainian computer
> scientist Alexey Stakhov, author of 'The Mathematics of Harmony'
> (2009), mentions it in section 4.2 <
> http://peacefromharmony.org/docs/7-27_Stakhov_Math_of_Harmony_EN.pdf
>  >. I assume Koshy refers to Thomas Koshy who authored 'Fibonacci
> and  Lucas Numbers with Applications' (2001).
>
>
> On Jun 1, 2013, at 5:35 AM, Jess Tauber <yahganlang at gmail.com> wrote:
>
>>> The (2,1)-sided version of the Pascal Triangle appears to be special in at
>>> least this- diagonal values are identical to numerical coefficients of
>>> power terms that sum to powers of Metallic Means (and the dimensional
>>> labels of the diagonals are the same as the powers the terms are raised
>>> to)- I worked this out about a year and a half ago and still haven't yet
>>> found out if this is an old result, though some work by Koshy appears to
>>> suggest it is known).
>
>
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