Titlebook: Geometry of Continued Fractions; Oleg N. Karpenkov Textbook 2022Latest edition Springer-Verlag GmbH Germany, part of Springer Nature 2022

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,Methoden der Str?mungssichtbarmachung,In the beginning of this book we discussed the geometric interpretation of regular continued fractions in terms of LLS sequences of sails. Is there a natural extension of this interpretation to the case of continued fractions with arbitrary elements? The aim of this chapter is to answer this question.

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https://doi.org/10.1007/978-3-662-43199-3In Chap. . we proved a necessary and sufficient criterion for a triple of integer angles to be the angles of some integer triangle. In this chapter we prove the analogous statement for the integer angles of convex polygons. Further, we discuss an application of these two statements to the theory of complex projective toric surfaces.

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Semigroup of Reduced MatricesThere are several ways to construct reduced matrices, however as a rule they are closely related with each other. The reason for that might be the structure of the group. We should mention that the approach here is rather different to the classical approach for closed fields via Jordan blocks.

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Lagrange’s TheoremThe aim of this chapter is to study questions related to the periodicity of geometric and regular continued fractions. The main object here is to prove Lagrange’s theorem stating that every quadratic irrationality has a periodic continued fraction, conversely that every periodic continued fraction is a quadratic irrationality.