Klein read Gauss's fragments very closely and seems to have been influenced by them, so I think it's not an exaggeration to say that Gauss's drawing was one of his sources of inspiration. Historical significance of Gauss's results:īeing 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). However, I still believe his notes on the mertical relation exhibit knowledge of the Poincare disk model (simply because I cannot explain their correctness in other ways). Therefore, I believe my attempts to answer that question arrived into a deadlock, and I don't understand how stillwell came with his "hypergeometric function conjecture" about the origin of this tessellation. There is another drawing in this notebook depicting a chain of pentagrams, which I believe relates to his thoughts concerning the Pentagramma mirificum, but nothing that gives additional clue about the origin of the (4 4 4) tessellation or the possible motivations behind it. Gauss Varia 2, as can be seen from this link.Īfter gaining access to this notebook, I saw only one more drawing in it related to this tessellation - but again, with no supporting context. Gauss's drawing (together with his related notes on the metrical relations) from p.104 of volume 8 of his works appears in an obscure notebook called Cod. After I asked people at the department for handwritten manuscripts and rare prints of the Gottingen university to search for the location of Gauss's drawing, they managed to find it after many efforts. I don't now on what sources Stillwell bases his conjecture about the origin of Gauss's tessellation. Others were found by Riemann in lectures of 1858-1859 (discovered only in 1897 and published in Riemann ), and the general theory was worked out independently by Schwarz. The Gauss's figure is one such tessellation. In the case of the hypergeometric equations, one is led to groups of automorphisms of certain triangle tessellations of the unit disc. By inverting the quotient of solutions one obtains a function automorphic with respect to a certian group of linear transformations. In addition to the information in fragment, the following formula should also be mentioned $$\frac$. Gauss has repeatedly dealt with composition of other substitutions of the group defined from these generators. The following sentences are a citation of Fricke about the substitutions Gauss used: 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. We exemplify the high potential of this approach by constructing and diagonalizing finite-dimensional Bloch wave Hamiltonians.My purpose is to verify an historical hypothesis I have on Gauss's tesselation of the unit disk as described in John Stilwell "Mathematics and its history". The significance of this achievement needs to be compared to the all-important role played by conventional Euclidean crystallography in the study of solids. This dramatically simplifies the computation of energy spectra of tight-binding Hamiltonians on hyperbolic lattices, from exact diagonalization on the graph to solving a finite set of equations in terms of irreducible representations. Using the mathematical framework of higher-genus Riemann surfaces and Fuchsian groups, we derive a list of example hyperbolic lattices and their hyperbolic Bravais lattices, including five infinite families and several graphs relevant for experiments in circuit quantum electrodynamics and topolectrical circuits. We show that many hyperbolic lattices feature a hidden crystal structure characterized by unit cells, hyperbolic Bravais lattices, and associated symmetry groups. Motivated by recent insights into hyperbolic band theory, we initiate a crystallography of hyperbolic lattices. Their underlying geometry is non-Euclidean, and the absence of Bloch's theorem precludes the straightforward application of the often indispensable energy band theory to study model Hamiltonians on hyperbolic lattices. Hyperbolic lattices are a revolutionary platform for tabletop simulations of holography and quantum physics in curved space and facilitate efficient quantum error correcting codes.
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