On Dissections of Figures into Figures Similar to Given Ones

Student: Sharov Fedor

Educational Programme: Mathematics (Bachelor)

Final Grade: 10

Year of Graduation: 2017

This work consists of three independent parts. In Part I we study the following problem: given a collection of n rectangles, which polygons can be tiled by rectangles similar to one of the given ones? The task is usually complicated and still has no solution. However, particular cases of this general problem are interesting. In Part I we find out which rectangles can be tiled by rectangles similar to one of given n rectangles such that the side ratio of each given one is a quadratic irrationality. In Part II we give a new simple proof of Dehn's theorem by generalizing the notion of area. In Part III we give some other applications of the same method. Actually, the method is a "translation" of the method of additive functions to an elementary language. We considere only the final tillings in the work.

Full text (added June 9, 2017)

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