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Quote from Don Zagier (Mathematicians: An Outer View of the Inner World):

" I like explicit, hands-on formulas. To me they have a beauty of their own. They can be deep or not. As an example, imagine you have a series of numbers such that if you add 1 to any number you will get the product of its left and right neighbors. Then this series will repeat itself at every fifth step! For instance, if you start with 3, 4 then the sequence continues: 3, 4, 5/3, 2/3, 1, 3, 4, 5/3, etc. The difference between a mathematician and a nonmathematician is not just being able to discover something like this, but to care about it and to be curious about why it's true, what it means, and what other things in mathematics it might be connected with. In this particular case, the statement itself turns out to be connected with a myriad of deep topics in advanced mathematics: hyperbolic geometry, algebraic K-theory, the Schrodinger equation of quantum mechanics, and certain models of quantum field theory. I find this kind of connection between very elementary and very deep mathematics overwhelmingly beautiful."

The recursion described here is a(n+1)=(1+a(n))/a(n-1) with a(0)=1, a(1)=3. It's fairly straightforward to show that for every positive a(0),a(1) the sequence repeats after 5 steps (and a(0)=a(1)=golden ratio is a fixed point). One can also think of this recursion as the planar map f(x,y)=((1+x)/y,x), and every point in the plane is a period-5 point of this map.

So what is the connection with the deep topics listed in the quote?

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Hyperbolic geometry / algebraic K-theory must mean that $(1+x)/y$ and its iterates appear in the functional equations of one of the dilogarithm functions (Abel's relation or similar). There is a 5-term functional equation with some cyclic symmetry.

Models of QFT may refer to the "pentagon relation" from integrable models. This again reflects the order 5 property.

[added:]

This does appear to be what Zagier was pointing to. Back in the 1990's,

ADE functional dilogarithm identities and integrable models F. Gliozzi, R. Tateo http://xxx.lanl.gov/abs/hep-th/9411203

formulated a functional equation generalizing some identities for the Rogers dilogarithm evaluated at roots of unity. The latter identities were known to correspond to torsion in $K$-theory of fields (see papers of Frenkel and Szenes on dilogarithm identities, also early and mid-90's, on arxiv, including proofs of Gliozzi-Tateo formulae and some comments on $K_3(\mathbb{C})$).

On page 4-5 of the paper it is pointed out that the 5-fold periodicity comes from an inspection of the cyclic symmetry of the Abel 5-term functional equation of the Rogers dilogarithm. If the terms are numbered correctly then $(1+x)/y$ is the generator of this symmetry. The periodicity of this transformation was probably observed about 200 years ago.

Cluster algebras appeared later and provided the technology to prove (among many other things) more general periodicity conjectures from integrable models of QFT, going back to earlier papers of Zamolodchikov. However, the connection of this one period-5 transformation to the dilogarithm is probably easiest to see in the Gliozzi/Tateo paper above. K-theory and hyperbolic geometry have no known direct connection to cluster algebras, though the latter are related to triangulations of polygons and this is sometimes connected to hyperbolic 2- and 3-dimensional geometry.

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Google "cluster algebra", e.g. see this paper Combinatorial interpretations for rank-two cluster algebras of affine type.

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    $\begingroup$ See also the paper in which cluster algebras were first defined: arxiv.org/abs/math/0104241 $\endgroup$ – Qiaochu Yuan Nov 24 '10 at 10:22
  • $\begingroup$ Cluster algebras are an additional "deep topic" for the (1+x)/y transformation, but they do not account for the connections given in Zagier's quotation: hyperbolic geometry, K-theory, and models of QFT. It is still very much an open question what cluster algebras do relate to, fundamentally. For now they are an amazingly fruitful algebraic construction "ex machina" with an ADE classification. $\endgroup$ – T.. Nov 25 '10 at 8:30
  • $\begingroup$ @T..: I recommend that you study cluster algebras a bit more before underestimating their role here. $\endgroup$ – Bill Dubuque Nov 25 '10 at 8:49
  • $\begingroup$ @Bill, regarding your comment "they do" (now deleted), cluster algebras are not known to have any relation to hyperbolic geometry or K-theory. The connections that Zagier indicated are through the dilogarithm, which is known to have close relations to hyperbolic geometry (e.g., to volumes or Lobachevsky's function) and to regulator maps on K_2 of algebraic number fields. Zagier has written papers on this. If you think that cluster algebras also have connections to those two "deep topics", please elaborate. $\endgroup$ – T.. Nov 25 '10 at 8:50
  • $\begingroup$ What is true is that cluster algebras are the current state of the art framework for proving generalizations of the order-5 property of the (1+x)/y iteration -- some of which were conjectured in the physics literature on integrable models and conformal field theory, as mentioned in the other answer. That is, cluster algebras are relevant and natural for understanding the recurrence, but they also are not the parts of mathematics to which Zagier was principally alluding. $\endgroup$ – T.. Nov 25 '10 at 9:03
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About the connection with Schrodinger equations of quantum mechanics and certain models in QFT: you can have look to the papers: arXiv:hep-th/9812211, arXiv:hep-th/9906219

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