RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
General information
Latest issue
Forthcoming papers
Archive
Impact factor
Subscription
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Mat. Sb.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


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 МК-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

Full text: PDF file (722 kB)
First page: PDF file
References: PDF file   HTML file

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

Bibliographic databases:

Document Type: Article
UDC: 514.112.4
MSC: Primary 52C20; Secondary 05B45
Received: 25.09.2017 and 25.10.2018

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
\Bibitem{Zve19}
\by I.~S.~Zverev
\paper Dissections of trapezoids into trapezoids homothetical to trapezoids in a~given family
\jour Mat. Sb.
\yr 2019
\vol 210
\issue 2
\pages 87--114
\mathnet{http://mi.mathnet.ru/msb9014}
\crossref{https://doi.org/10.4213/sm9014}
\adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2019SbMat.210..245Z}
\elib{http://elibrary.ru/item.asp?id=37089809}
\transl
\jour Sb. Math.
\yr 2019
\vol 210
\issue 2
\pages 245--271
\crossref{https://doi.org/10.1070/SM9014}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000465078600004}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85067230568}


Linking options:
  • http://mi.mathnet.ru/eng/msb9014
  • https://doi.org/10.4213/sm9014
  • http://mi.mathnet.ru/eng/msb/v210/i2/p87

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles
  • Математический сборник Sbornik: Mathematics (from 1967)
    Number of views:
    This page:96
    References:6
    First page:5

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019