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Until recently, i thought that the first to observe the connection between modular forms and the hyperbolic plane (a model of non-euclidean geometry) was Felix Klein. But in the last week i read the chapter on non-euclidean geometry in the book of John Stillwell - "Mathematics and it's history", where it's mentioned that using his theory of modular forms Gauss discovered a tessellation of the Poincare disk model by equilateral triangles of angles: $$\pi/4,\pi/4,\pi/4$$.

So how Gauss could accomplish this without knowing the hyperbolic plane? i understand that this work was done in the context of elliptic functions and modular forms (not hyperbolic geometry), but the fact that he used triangles with angles sum less than pi, suggests that he might suspect that it's connected with hyperbolic geometry. Also, the fact that Klein studied very closely the writings of Gauss might suggest that this accomplishment has historical influence.

So i'll be glad to clarify this "historical corner".

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    $\begingroup$ I seriously doubt that Gauss knew about Poincaré model. $\endgroup$ – Moishe Kohan Jul 30 '17 at 15:51
  • $\begingroup$ I think, Stilwell is over-interpreting what Gauss knew. He refers to Gauss' "Werke", Band 8, p. 104. Once you look there (assuming that you read German), you realize that most of what is written there is by Fricke who explains how Gauss' work can be interpreted using Poincare unit disk model (and the language of discontinuous group actions). It is entirely possible (or even likely) that Gauss knew much more than he wrote, but we will never know... $\endgroup$ – Moishe Kohan Aug 1 '17 at 1:07
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The best source on this is Klein's Lectures on mathematics in XIX century, vol. I. It has a whole chapter on Gauss, with main focus on elliptic integrals and modular forms. Klein can be trusted because he himself worked on this and really read a lot of Gauss's writings. It seems to me from this book that Gauss did not know about the connection with non-Euclidean geometry, and Poincare's model is indeed Poincare's invention.

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  • $\begingroup$ So one can only say that Gauss discovered a tessellation of a circle using certain modular figures. I accept your answer since there is really not much evidence that Gauss systematiclly developed hyperbolic geometry. Can you answer another question of mine; Did Gauss know Jacoby's four squares theorem? the evidence there is much more solid since certainly Gauss knew the connection between certain quadratic forms and the theory of theta functions. The crucial identity which Jacoby used to prove his theorem is stated in Gauss's nachlass. $\endgroup$ – user2554 Aug 1 '17 at 18:53
  • $\begingroup$ @user2554: I cannot tell anything about 4 squares, I do not have Klein with me at this time. But from what I read I do not remember any claims that Gauss knew it. $\endgroup$ – Alexandre Eremenko Aug 1 '17 at 19:44
  • $\begingroup$ that's why i ask... i dont ask questions that have easy answers. I hope you read my post; the point of debate is in my opinion whether knowledge of the crucial part of the proof implies understanding of the connection between sum of squares functions and the coefficients of powers of the theta function. My question is therefore intended to clarify if the derivation of the main identity of the proofs already identifies quantatively this connection. $\endgroup$ – user2554 Aug 1 '17 at 20:06
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Just to shed new light on Gauss's possible thoughts regarding hyperbolic geometry, i had to throw into this post two pieces of information i found. From modern viewpoint, these two pieces are connected to elliptic geometry, not to the geometry that Gauss termed "non-euclidean geometry" (which is actually hyperbolic).

According to p.17 in the article "Loxodromic Spirals in M. C. Escher's Sphere Surface", Gauss found in 1819 (in his unpublished fragment "Die Kugel") that rotations of the Riemann sphere correspond to unitary Mobius transformations - that is, by assigning complex numbers to the sphere using the stereographic projection, than a rotation can be represented as a fractional linear transformation:

$$z \mapsto \frac {{az + b}}{{-\bar{b}z + \bar{a}}}$$

This piece of information is important in two aspects: it shows that Gauss was the first to appreciate the importance and usefulness of the extended complex plane $C\cup (\infty)$ for geometric problems on the sphere, and secondly because it shows that Gauss understood that certain Mobius transformations generate the isometries of the Riemann sphere. Since the Riemann sphere is intimately connected to models of elliptic geometry, and since Gauss also encountered similar transformations in his analytic (modular forms) and number-theoretic (reduction of binary quadratic forms) studies, one can argue there is a faint relation between Gauss's meditations on non-euclidean geometries and his analytic work.

The second fragment of Gauss dates from 1840, and is entitled "Der Kreis". It's less important than the first one, and deals with the so called "Cross-ratio" of four points, which is of significance in models of non-euclidean geometry. I don't understand exactly what the significance of it, but Stackel comments on it, stating that Gauss improved in it the notions of "Double-ratio" and "harmonic conjugates". This fragment might be entirely tangled in Gauss's mind with synthetic geometry considerations and Mobius's barycentric calculus (not to hyperbolic geometry), but from a modern point of view, there is a relation.

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