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Discrete & Computational Geometry

, Volume 61, Issue 3, pp 626–652 | Cite as

Corona Limits of Tilings: Periodic Case

  • Shigeki AkiyamaEmail author
  • Jonathan Caalim
  • Katsunobu Imai
  • Hajime Kaneko
Article
  • 106 Downloads

Abstract

We study the limit shape of successive coronas of a tiling, which models the growth of crystals. We define basic terminologies and discuss the existence and uniqueness of corona limits, and then prove that corona limits are completely characterized by directional speeds. As an application, we give another proof that the corona limit of a periodic tiling is a centrally symmetric convex polyhedron [see Zhuravlev (St Petersbg Math J 13(2):201–220, 2002) and Maleev and Shutov (Layer-by-layer growth model for partitions, packings, and graphs, Tranzit-X, Vladimir, 2011)].

Keywords

Tiling Corona Crystal Convex body 

Mathematics Subject Classification

52C22 52A20 20H15 52C23 

Notes

Acknowledgements

We express our cordial gratitude to A.V. Shutov for informing us of the current status of the research on corona limits and for providing us some of the references, which are not easily accessible. We are also largely indebted to the anonymous referee and Fumihiko Nakano who gave us invaluable suggestions and related references. The first author is partially supported by JSPS Grants (17K05159, 17H02849, BBD30028). The third author is partially supported by JSPS Grants (26330016, 17K00015). The fourth author is supported by JSPS Grant (15K17505).

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Copyright information

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Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of TsukubaTsukubaJapan
  2. 2.Institute of Mathematics, College of ScienceUniversity of the Philippines DilimanQuezon City 1101Philippines
  3. 3.Graduate School of EngineeringHiroshima UniversityHigashi-HiroshimaJapan

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