The construction of approximate grids for quasicrystals using control points

• Project/Area Number: 23540141
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Kochi University
• Principal Investigator: Komatsu Kazushi 高知大学, 教育研究部自然科学系理学部門, 准教授 (00253336)
• Co-Investigator(Kenkyū-buntansha): AKIYAMA Shigeki 筑波大学, 数理物質科学研究科(系), 教授 (60212445) ; GOTO Satoru 東京理科大学, 薬学部, 教授 (50253232) ; EI Hiromi 弘前大学, 理工学研究科, 助教 (60333051)
• Project Period (FY): 2011-04-28 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000); Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2014: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2013: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2012: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2011: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: タイル貼り / 数理モデル / 回転対称性 / 準周期タイリング / grid / 準周期性 / タイリング

Outline of Final Research Achievements

We construct a family of uncountable many non-periodic 3-Archimedean tilings with 6-fold rotational symmetry, that admit three type of vertex configurations by regular triangles and squares. As a limit of the patch sequence, we obtain a tiling or an unbounded configuration. In Danzer tilings, we cover all singular vertex configurations and all wedge-shaped unbounded configurations by up-down generation. We construct non-periodic tilings by gathering wedge-shaped unbounded configurations around a point.
We provide a mathematical model of n-membered hydrocarbon molecules. The configuration space of the model straight-chain hydrocarbon molecules is parametrized by chain lengths. We determine the topological types of fibers of the configuration space of the model by chain lengths when n = 5.

Research Products

• [Journal Article] A note on the construction of non-periodic tilings by attaching wedge-shaped unbounded configurations. (2015)
  • Author(s): Hayashi, H., Komatsu, K. and Yamauchi, M.
  • Journal Title: Kochi Journal of Mathematics Volume: 10 Pages: 53-60
  • Peer Reviewed
• [Journal Article] On non-periodic 3-Archimedean tilings with 6-fold rotational symmetry. (2015)
  • Author(s): Kinoshita, N. and Komatsu, K.
  • Journal Title: Hiroshima Mathematical Journal Volume: 45 Pages: 137-146
  • Peer Reviewed
• [Journal Article] The configuration space of a model for 5-membered straight-chain hydrocarbon molecules parametrized by chain lengths (2015)
  • Author(s): Goto, S., Komatsu, K. and Yagi, J.
  • Journal Title: Nihonkai Mathematical Journal Volume: 26 Pages: 37-45
  • Peer Reviewed
• [Journal Article] An example of a quasiperiodic 3-Archimedean tiling by regular triangles and squares (2014)
  • Author(s): Naoko Kinoshita and Kazushi Komatsu
  • Journal Title: Kochi J. Math. Volume: 9 Pages: 121-125
  • Peer Reviewed
• [Journal Article] On non-periodicity for tilings in the hyperbolic plane (2014)
  • Author(s): Shin'ya Kishimoto and Kazushi Komatsu
  • Journal Title: Kochi J. Math. Volume: 9 Pages: 145-152
  • Peer Reviewed
• [Journal Article] Topology of the Interconversion Pathway Networks of Cycloheptane Conformations and Those of Related n-Membered Rings (2013)
  • Author(s): Satoru Goto, Kazushi Komatsu and Hiroshi Terada
  • Journal Title: Bull. Chem. Soc. Jpn. Volume: 86 Issue: 2 Pages: 230-242
  • DOI: 10.1246/bcsj.20110252
  • NAID: 10031155744
  • Peer Reviewed
• [Journal Article] A remark on the configuration space of a model for ringed hydrocarbon molecules (2012)
  • Author(s): Goto, S., Komatsu, K. and Yagi, J. (Yagi, J.)
  • Journal Title: Kochi Journal of Mathematics Volume: 7 Pages: 89-96
  • Peer Reviewed
• [Journal Article] The configuration space of a model for ringed hydrocarbon molecules (2012)
  • Author(s): Goto, S. and Komatsu, K. (Komatsu, K.)
  • Journal Title: Hiroshima Mathematical Journal Volume: 42 Pages: 115-126
  • Peer Reviewed
• [Journal Article] The subdivision of the window derived from finite subsequences of Fibonacci sequences (2011)
  • Author(s): Hayashi, H. and Komatsu, K. (Hayashi, H.)
  • Journal Title: Nihonkai Mathematical Journal Volume: 22 Pages: 59-66
  • Peer Reviewed
• [Journal Article] Notes on vertex atlas of planar Danzer tiling (2011)
  • Author(s): Hayashi, H. and Komatsu, K. et. al. (Hayashi, H.)
  • Journal Title: Nihonkai Mathematical Journal Volume: 22 Pages: 49-58
  • Peer Reviewed
• [Presentation] Directions of growth in tilings (2011)
  • Author(s): Hayashi, H., Kazumasa Kawatani, K. and Komatsu, K. (Komatsu, K.)
  • Organizer: Rigidity of self-affine tilings and related topics
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] The subdivision of the window derived from finite subsequences of Fibonacci sequences
  • Author(s): Hayashi, H. and Komatsu, K.
  • Organizer: Numeration and Substitution 2012
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] Topology of the configuration space of a model for ringed hydrocarbon molecules
  • Author(s): Satoru Goto, Kazushi Komatsu and Jun Yagi
  • Organizer: Topology of tiling spaces and related topics
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] Strongly non-periodic hyperbolic tilings with only one vertex configuration
  • Author(s): Kazushi Ahara, Shigeki Akiyama, Hiroko Hayashi and Kazushi Komatsu
  • Organizer: Topology of tiling spaces and related topics
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] Non-periodic Archimedean-like tilings with 6-fold rotational symmetry
  • Author(s): Naoko Kinoshita and Kazushi Komatsu
  • Organizer: Topology of tiling spaces and related topics
  • Place of Presentation: 京都大学数理解析研究所