What mathematical techniques Gauss used in order to tessellate the unit disk?

Question

This question is a continuation of my previously posted question: Was Gauss aware of the non-euclidean implications of his work on moduler forms?, and is based on the information given in John Stillwell's book "Mathematics and it's history", where it's mentioned that using his theory of modular forms Gauss discovered a tessellation of the unit disk by "equilateral triangles" (but curved) of angles: $\pi/4,\pi/4,\pi/4$.

My previous question was about how Gauss interpreted his own computations with modular forms. In this question i don't wish to know if Gauss was aware of the Poincare disk model of hyperbolic geometry, but it's more about placing his calculations in the right context.

From looking in his writings i gained the impression that he was quite aware of the action of the modular group in the upper-half plane (although he didn't view it as a model of the hyperbolic plane) - his nachlass contains drawings of the fundamental domain of the modular group, and he, for example, proved a remarkable result on the mutual connection of all values of the arithmetic-geometric mean of two complex numbers (see David Cox's paper: "AGM of Gauss").

But i didn't find things that look like computations of tessellations of the unit disk (by curved triangle). In his commentary on p. 99-102 of the 8th volume of Gauss's werke ("Three Fragments About Elliptic Modulfunctions"), Fricke gives some related sources (such as one of Gauss's articles on the orbit of pallas, where he is supposed to use some of his analytic work in implicit way), and mentions that the drawing of tessellation of the unit disk was found among his writings. I guess one of the problems is that Google translate doesn't make very good transations.

So my question is:

• Since i'm not sure that p. 99-102 in volume 8 is really the place where he is supposed to make calculations about tessellations, i'm asking for reference - if anyone who is famililar enough with these themes can help, i'll be grateful. If someone sees a formula that resembles a formula he knows from the theory of tessellations of the unit disk in one of Gauss's writings,i'll be glad if he will mention it here.

Answer 1

It is impossible to answer precisely what Gauss was thinking. All we have is his collected papers (this includes unpublished pieces). A good online source is Fortunately they are all easily available online: https://gdz.sub.uni-goettingen.de/volumes/id/PPN235957348. The same edition includes extended commentaries. The topic you are asking about was commented on by Klein. If you do not read German, there is a book of Klein, Lectures on history of mathematics in 19th century, which has a large chapter on Gauss, and this book is available in English.

As I understand (from Klein's lectures and other sources), Gauss knew about the action of the modular group on a half-plane, (pictures in his manuscripts suggest this), but there is no evidence that he was aware of the interpretation of this action in terms of non-Euclidean geometry).

Answer 2

Google translation to Ludwig Schlesinger's essay on Gauss's contribution to analysis reveals the answer - on p.105-106 of his essay, appears his explanation of Gauss's result - according to which Gauss was aware of the importance of reflection in circles (inversion with respect to circles) for the composition of certain fundamental substitutions. He than explains the drawing of tessellation of the unit disk from his nachlass as a network of circular arc triangles completed by their orthogonal circle.

Although it appears first that the drawing from p.104 of volume VIII of Gauss's nachlass was made by Fricke (what makes this historical thesis on Gauss's understanding of modular forms weaker), Schlesinger mentions that this drawing was actually found among Gauss's posthumous papers, and published with only few modifications by Fricke.

Related directly to the questions sorrounding Gauss's drawing is my Mathoverflow post and partial answer: https://mathoverflow.net/questions/370190/what-is-the-representation-of-the-generators-of-the-triangle-group-for-the-unifo. The main conclusions made in this post are that Gauss's handwritten notes confirm that he recognized the key notion of "inversion with respect to a circle" (which can be understood as a generalization of the concept of reflection), as well as the principle of generation of curved triangles network by successive reflection of curved triangles with respect to their sides, a process which tiles the whole hyperbolic plane with copies of a fundamental triangle.

In addition, the values he gives for the radiuses of curvature of the sides of the first and secondary circles (which give the scale of the fundamental triangle, when it viewed euclidly) show that the border curve for his network is exactly the unit circle, a fact which i'm unable to explain without assuming that Gauss was using the Cayley-Klein metric for the hyperbolic plane (this fact shows that Gauss knew how to calculate the scale of the border circle in terms of the scale of the fundamental triangle).

However, it seems that Gauss was led to these consideration solely by his analytic work and it's number theoretic counterparts (i.e the intimate relation to quadratic forms). Gauss attempted to appreciate and illustrate the different aspects of the modular action by casting it in geometric form. Even though he might eventually view these considerations as connected with his meditations on non-euclidean geometry, this connection almost certainly didn't come to him so early in his career (Schlesinger and Fricke date Gauss's drawing to before 1805).