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 Mat. Sb., 2019, Volume 210, Number 2, Pages 87–114 (Mi msb9014)

Dissections of trapezoids into trapezoids homothetical to trapezoids in a given family

I. S. Zverev

Faculty of Mathematics, National Research University Higher School of Economics, Moscow, Russia

Abstract: We prove several theorems related to the dissection of trapezoids into trapezoids that are homothetical to given ones.
We prove that, using homotheties of a trapezoid with rational ratio of bases, we can tile any trapezoid with rational ratio of bases and the same angles, and no other trapezoid can be tiled.
We also consider trapezoids whose ratio of bases is a quadratic irrationality. For certain pairs of trapezoids we prove that their homotheties can tile any trapezoid with the same angles and the ratio of bases belonging to the same quadratic field. For some other class of trapezoids with a quadratic-irrational ratio of bases, we present a necessary condition on trapezoids that can be tiled with given ones. This condition is remarkable because it contains a transcendental function. This is the first occurrence of a transcendental function in problems of tiling polygons with similar polygons.
Bibliography: 8 titles.

Keywords: dissection of polygons, tiling polygons, homothety, trapezoid.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation ÌÊ-6137.2016.1 This work was carried out within the framework of the Programme of the President of the Russian Federation for State Support of Young Russian Scientists (grant no. MK-6137.2016.1).

DOI: https://doi.org/10.4213/sm9014

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English version:
Sbornik: Mathematics, 2019, 210:2, 245–271

Bibliographic databases:

UDC: 514.112.4
MSC: Primary 52C20; Secondary 05B45

Citation: I. S. Zverev, “Dissections of trapezoids into trapezoids homothetical to trapezoids in a given family”, Mat. Sb., 210:2 (2019), 87–114; Sb. Math., 210:2 (2019), 245–271

Citation in format AMSBIB
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