Geodesic
Dissections of trapezoids into trapezoids homothetical to trapezoids in a given family
Sbornik. Mathematics, Tome 210 (2019) no. 2, pp. 245-271 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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. 
@article{SM_2019_210_2_a3,
     author = {I. S. Zverev},
     title = {Dissections of trapezoids into trapezoids homothetical to trapezoids in a~given family},
     journal = {Sbornik. Mathematics},
     pages = {245--271},
     year = {2019},
     volume = {210},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2019_210_2_a3/}
}
TY  - JOUR
AU  - I. S. Zverev
TI  - Dissections of trapezoids into trapezoids homothetical to trapezoids in a given family
JO  - Sbornik. Mathematics
PY  - 2019
SP  - 245
EP  - 271
VL  - 210
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SM_2019_210_2_a3/
LA  - en
ID  - SM_2019_210_2_a3
ER  - 
%0 Journal Article
%A I. S. Zverev
%T Dissections of trapezoids into trapezoids homothetical to trapezoids in a given family
%J Sbornik. Mathematics
%D 2019
%P 245-271
%V 210
%N 2
%U http://geodesic.mathdoc.fr/item/SM_2019_210_2_a3/
%G en
%F SM_2019_210_2_a3
I. S. Zverev. Dissections of trapezoids into trapezoids homothetical to trapezoids in a given family. Sbornik. Mathematics, Tome 210 (2019) no. 2, pp. 245-271. http://geodesic.mathdoc.fr/item/SM_2019_210_2_a3/

[1] M. Dehn, “Über Zerlegung von Rechtecken in Rechtecke”, Math. Ann., 57:3 (1903), 314–332 | DOI | MR | Zbl

[2] R. L. Brooks, C. A. B. Smith, A. H. Stone, W. T. Tutte, “The dissection of rectangles into squares”, Duke Math. J., 7 (1940), 312–340 | DOI | MR | Zbl

[3] C. Freiling, D. Rinne, “Tiling a square with similar rectangles”, Math. Res. Lett., 1:5 (1994), 547–558 | DOI | MR | Zbl

[4] M. Laczkovich, G. Szekeres, “Tiling of the square with similar rectangles”, Discrete Comput. Geom., 13:3-4 (1995), 569–572 | DOI | MR | Zbl

[5] C. Freiling, M. Laczkovich, D. Rinne, “Rectangling a rectangle”, Discrete Comput. Geom., 17:2 (1997), 217–225 | DOI | MR | Zbl

[6] F. Sharov, Dissection of a rectangle into rectangles with given side ratios, arXiv: 1604.00316

[7] B. Szegedy, “Tilings of the square with similar right triangles”, Combinatorica, 21:1 (2001), 139–144 | DOI | MR | Zbl

[8] R. Kenyon, “Tilings and discrete Dirichlet problems”, Israel J. Math., 105:1 (1998), 61–84 | DOI | MR | Zbl