Study of Isohedral tilings by polyominoes and polyiamonds

2010 Fiscal Year Final Research Report

• Project/Area Number: 20540127
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Kitasato University
• Principal Investigator: FUKUDA Hiroshi Kitasato University, 一般教育部, 准教授 (70238484)
• Project Period (FY): 2008 – 2010
• Keywords: 離散幾何学

Research Abstract

We described computer algorithms that produce the complete set of isohedral tilings by n-omino or n-iamond tiles in which the tiles are fundamental domains and the tilings have 3-, 4-, or 6-fold rotational symmetry. The symmetry groups of such tilings are of types p3, p31m, p4, p4g, and p6. There are no isohedral tilings with symmetry groups p3m1, p4m, or p6m that have polyominoes or polyiamonds as fundamental domains. We display the algorithms' output and give enumeration tables for small values of n.

Research Products

• [Journal Article] Three-Body Choreographies in GivenCurves (2009)
  • Author(s): Hiroshi Ozaki, Hiroshi Fukuda, Toshiaki Fujiwara
  • Journal Title: J.Phys A42 Pages: 395205(16)
  • Peer Reviewed
• [Journal Article] Enumeration of Polyominoes,Polyiamonds and Polyhexes for Isohedral Tilings with Rotational Symmetry (2008)
  • Author(s): Hiroshi Fukuda, Nobuaki Mutoh, Gisaku Nakamura, Doris Schattschneider
  • Journal Title: Lecture Notes in Computer Science (Springer) 4535 Pages: 68-78
  • Peer Reviewed
• [Journal Article] Jin Akiyama, Hiroshi Fukuda, Chie Nara, Toshinori Sakai, Jorge Urrutia (2008)
  • Author(s): Universal Measuring Boxes with Triangular Bases
  • Journal Title: American Mathematical Monthly 115 No.3 Pages: 195-201
  • Peer Reviewed
• [Presentation] Polyominoes and Polyiamonds as Fundamental Domains of Isohedral Tilings for Crystal Class D_2 (2010)
  • Author(s): Hiroshi Fukuda, Chiaki Kanomata, Nobuaki Mutoh, Gisaku Nakamura, Doris Schattschneider
  • Organizer: The China-Japan Joint Conference on Computation Geometry, Graphs and Applications (Session 2A: (11:00-12:00,November 3))
  • Place of Presentation: Dalian, China
  • Year and Date: 20101103-20101106
• [Presentation] Symmetryof Isohedral Tilings of Polyominoes and Polyiamonds as Fundamental Domains (2009)
  • Author(s): Hiroshi Fukuda, Chiaki Kanomata, Nobuaki Mutoh, Gisaku Nakamura, Doris Schattschneider
  • Organizer: The 7th Japan Conference on Computational Geometry and Graphs (Session 5A: Tiling and Packing(16:20-18:00, November 11))
  • Place of Presentation: Kanazawa, Japan
  • Year and Date: 20091111-20091113
• [Presentation] Fixed Center of Mass Configurationsof Three Points in Given Curves (2009)
  • Author(s): Hiroshi Ozaki, Hiroshi Fukuda, Toshiaki Fujiwara
  • Organizer: The 7th Japan Conference on Computational Geometry and Graphs (Session 7B: Physics-Motivated Geometryand Graph Theory (11:00-12:00, November12))
  • Place of Presentation: Kanazawa, Japan
  • Year and Date: 20091111-20091113
• [Presentation] Three-Body problem on GivenCurve (2009)
  • Author(s): H. Ozaki, H. Fukuda, T.Fujiwara
  • Organizer: The internatonal Bogolyubov conference"Problems of theoretical and mathematical physics" To the 100th anniversary of N. N. Bogolyubov's birth (18:20-18:40, Aug24, Monday)
  • Place of Presentation: Moscow-Dubna, Russia
  • Year and Date: 20090821-20090827
• [Book] 離散数学のすすめ(『ゴスパー曲線とその一般化』)(福田宏, 中村義作)(A5版330頁)(第11章) (2010)
  • Author(s): 伊藤大雄・宇野裕之編著
  • Total Pages: 152-167
  • Publisher: 現代数学社
• [Book] 理系への数学(離散数学のすすめ15『ゴスパー曲線とその一般化』)(第41巻第6号通巻498号) (2008)
  • Author(s): 福田宏, 中村義作
  • Total Pages: 55-60
  • Publisher: 現代数学社