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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
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    $\begingroup$ 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
  • $\begingroup$ "Modular forms" $\endgroup$ – releseabe Sep 19 '20 at 0:49
  • $\begingroup$ Whilst Gauss also discovered non-Euclidean geometry it's unlikely he pursued it to the extent that he saw the connection between this and modular forms. $\endgroup$ – Mozibur Ullah Mar 14 at 11:53
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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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I finally found a reliable source on Gauss's unpublished fragments that actually confirms that this drawing of the hyperbolic tessellation of the unit disk is indeed by Gauss himself: this link - https://kalliope-verbund.info/de/search.html?q=Cereri+Palladi+Junoni+sacrum%2C+Febr.+1805 - gives the notebook in which Gauss drawed this figure. This is a notebook entitled "Cereri Palladi Junoni sacrum", which Gauss started writing at February 1805, and according to the list of contents given there, Gauss's notes on modular functions are on p. 4-8 of it and include the same figures that are referenced in p.103-105 of volume 8 of the digital version of Gauss's werke.

This website actually gives a list of all of Gauss's notebooks, including much material that is not published in any place of the 12 volumes of Gauss's werke (maybe i'd get the answer to many of my questions if only these notebooks would be published...). For example, the famous note in which Gauss started the study of braids is on p.283-284 of "handbuch 7".

As for the non-euclidean implications of Gauss's drawing - my mathoverflow post https://mathoverflow.net/questions/370190/what-is-the-representation-of-the-generators-of-the-triangle-group-for-the-unifo shows that the formulas stated by Gauss on this tesselation can be derived, apart from one step (which involves the use of Cayley-Klein metric for the Poincare disk model), by looking at the figure through euclidean eyes (using the concept of inversion in circle). Therefore, perhaps the use of hyperbolic geometry is not necessary here.

It's quite possible (even likely) that Gauss saw the connection between his figure and hyperbolic geometry, but since this work was done in analytic context (this fragment contains "notes on modular functions"), one cannot know for sure.

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