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Andrei Petrovich Kiselyov
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The book is a unique phenomenon in Russian geometry education. It was first published in 1892; there have been more than 40 revised editions, and dozens of millions of copies (by these parameters it is trailing only Euclids Elements). Our edition is based on 41st edition (the stable edition of Nil Aleksandrovich Glagolev; its been in public domain since 2015). At a few places we reverted changes to the earlier editions; we also made more accurate historical remarks.
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We present some episodes from the history of interactions between geometry and physics over the past century.
David Mumford made groundbreaking contributions in many fields, including the pure mathematics of algebraic geometry and the applied mathematics of machine learning and artificial intelligence. His work in both fields influenced my career at several key moments.
We indicate that Herons formula (which relates the square of the area of a triangle to a quartic function of its edge lengths) can be interpreted as a scissors congruence in 4-dimensional space. In the process of demonstrating this, we examine a number of decompositions of hypercubes, hyper-parallelograms, and other elementary 4-dimensional solids.
Felix Kleins so-called Erlangen Program was published in 1872 as professoral dissertation. It proposed a new solution to the problem how to classify and characterize geometries on the basis of projective geometry and group theory. The given translation was made in 1892 by Dr. M. W. Haskell and transcribed by N. C. Rughoonauth. We replaced bibliographical data in text and footnotes with pointers to a complete bibliography section.
The statement of the Gauss-Bonnet theorem brings up an unexpected form of reflexivity (major concept of philosophy of mathematics), so that geometry contemplates itself in it. It is therefore the revolutionary and multifaceted concept of Gaussian curvature that triggers a new conceptuality above Euclidean geometry. Here, the equality between integral of total curvature and Euler characteristic indicates that a concept of topological nature is equal to a number which expresses a concept of geometric nature. This further demonstrates that mathematics develops through the intervention of different disciplines on top of each other, as observation tools, formal structuring, new unifying points of view.
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