My purpose is to verify an historical hypothesis iI have on Gauss's tesselation of the unit disk as described in John Stilwell "Mathematics and its history". Looking at the relevant pages in Gauss's Nachlass (volume 8, p.102-105), I read that the commentor (Robert Fricke) on this fragment of Gauss says that Gauss's drawing (the (4 4 4) tesselation) is intended to be a geometrical illustration for composition of substitutions other then the fundamental generators of the modular group. The following sentences are a citation of Fricke about the substitutions Gauss used:
Therefore, it seems that these Mobius substitutions are actually the generators for some tiling of the hyperbolic disk. But this conclusion is a result of a very shallow reading of Fricke's comments and iI lack the proffesional knowledge needed to verify my reading. In addition, there are two drawings in these pages (one on p.103 and the Gauss's tesselation on p.104), and i'm not sure to which drawing Fricke refers. Since Gauss's original notebook (Cereri Palladi Junoni sacrum) has not been published yet, it is virtually impossible for someone with my knowledge to write a conclusive answer.
In other words, Gauss describedthe introduction to the book "Papers on Fuchsian Functions" (in the two notes mentionedwhich is available on internet archive) those Mobius transformations, John Stillwell says several things that correspondseem to pure rotationsme very helpful:
Triangle functions arise in connection with the hypergeometric differential equation of Gauss, as Gauss himself seems to have discovered. This can be guessed from Figure 3, which is an undated fragment found in his Nachlass (see Gauss's werke, volume 8, p. 102-105). The figure is a tessellation of the unit disk by curvilinear triangles with angles $\pi/4$, and it reflects the periodicity of the inverse to a certain quotient of solutions to the hypergeometric equation. The group in this case consists of certain linear fractional transformations which leave the unit disc invariant. Invariance under linear fractional transformations arises quite generally from second order linear differential equations as follows. If $y_1,y_2$ are two linearly independent solutions of $$\frac{d^2y}{dx^2}+p(x)\frac{dy}{dx}+q(x)=0$$ then any other solutions $y'_1,y'_2$ are linear combinations of $y_1,y_2$: $$y'_1 = ay_1+by_2$$ $$y'_2 = cy_1+dy_2$$ hence the quotient of the new solutions is related to the quotient of the old by a linear fractional transformation: $$\frac{y'_1}{y'_2} = \frac{a\eta+b}{c\eta+d}$$ where $\eta = \frac{y_1}{y_2}$. By inverting the quotient of solutions one obtains a function automorphic with respect to a certian group of linear transformations. In the case of the hypergeometric equations, one is led to groups of automorphisms of certain triangle tessellations of the unit disc. The Gauss's figure is one such tessellation. Others were found by Riemann in lectures of 1858-1859 (discovered only in 1897 and published in Riemann [1902]), and the general theory was worked out independently by Schwarz [1872].
I don't now on what sources Stillwell bases his conjecture about the origin of Gauss's tessellation (since I still cannot locate the Riemann sphereoriginal notebook where Gauss drew his tessellation) but it is a direction worth checking.
Being the first drawing of it's kind, the tessellation drawed by Gauss and his related results have planted some the seeds of Felix Klein's "Erlangen program" (with the other influences being Galois's theory of equations and Riemann's geometric ideas). Klein read Gauss's fragments very closely and seems to have been influenced by them, so iI think it's not an exaggeration to say that Gauss's drawing was one of his sources of inspiration.
Therefore iI believe that for a correct historic appreciation of the roots of Erlangen program, it's important to know how Gauss arrived at the metrical relations in his tiling (the results stated by Gauss on the location and radiuses of the centres of the first and secondary circles in his tessellation are already confirmed by my posted answer - although it might not be the original method of Gauss).
