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

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.

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 All we have is his collected papers (this includes unpublished pieces).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).

• please make a carefull reading of my question, because all the details you've mentioned are already in my question, and i also emphasized that i don't care whether or not Gauss was aware of the non-euclidean interpretation. I say so because once a question recieves an answer, other users might gain an impression there is no need for additional efforts (so it lowers the chance i'll get an answer). – user2554 Aug 29 '18 at 20:44
• My question is essentially about Gauss's calculations of tessellations of the unit disk (reference request and an explanation as a bonus) , and not about whether or not he viewed this disk as a model of the hyperbolic plane (i.e Poincare'ss disk model). – user2554 Aug 29 '18 at 20:46
• @user2554 This is what happens when you don’t make a question precise (in the body; titles don’t count). Yours is “asking for reference” but doesn’t say for what (under “my question is:”). – Francois Ziegler Aug 29 '18 at 23:04
• @Francois Ziegler - i apollogize if i didn't make my question precise enough - that's because i was very confused about the the meaning of Gauss's results and the references to them. To make my question more precise i'll ask it again here: where in Gauss's work appear his calculations related to tessellation of the unit disk (not the upper half plane!)? and how to view his techniques in modern terms? – user2554 Aug 30 '18 at 15:22
• As far as i understand, Mobius transformation generate the isometries of the upper half plane and Gauss was aware of this, but i ask about his formulas related to tessellation of the unit disk. – user2554 Aug 30 '18 at 15:24

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 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.

However, 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 understand the geometric aspects of the modular action (the figure of fundamental domains), and didn't view these consideration as connected in anyway with his meditations on non-euclidean geometry. Poincare disk model (and other models of hyperbolic geometry) was Beltrami's innovation, and Gauss didn't know of it.