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Research on arrangements of geometric figures in space

Research Project

Project/Area Number 17540127
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionUniversity of the Ryukyus

Principal Investigator

MAEHARA Hiroshi  University of the Ryukyus, Faculty of Education, Professor (60044921)

Project Period (FY) 2005 – 2007
Project Status Completed (Fiscal Year 2007)
Budget Amount *help
¥3,270,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥270,000)
Fiscal Year 2007: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2006: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2005: ¥1,200,000 (Direct Cost: ¥1,200,000)
Keywordsco-unit-distance-graph / lattice points on a sphere / unit-sphere-system / chromatic number of balls / knotted cycle of balls / ball number / 単立球面システム / double six lines / 線球変換 / double six spheres / unit-sphere-sytem / escribed sphere / almost halving hyperplane / knotted necklace
Research Abstract

(1) To characterize unit distance graphs on the plane, or to determine the sup of their chromatic numbers is very difficult problem, and no big progress has been made so far. In the present study, we enumerate the unit distance graphs in the planes whose complements are also unit distance graphs in the plane. The total number of such graphs is 69, among them, 55 graphs are connected, and 7 graphs are self-complementary. (Ajoint work with S. V. Gervacio and Y. F. Lim of De La Salle University, Manila)
(2) Concerning a sphere that passes through a prescribed number of lattice points, we have the following result. For every integer n>d>m>l, there is a (hyper) sphere in the d-dimensional Euclidean space that passes through exactly n lattice points, and these n lattice points span an m dimensional polytope. This is a generalization of a result on a circle in the plane obtained by myself and M. Matsumoto in 1998.
(3) Aset of d+2 unit spheres in d-space is called a d-dimensional unit-sphere-system if every d+1 spheres have non-empty intersection, but the intersection of all d+2 spheres is empty. There is no 1-dimensional unit-sphere-system, and there are many 2-dimensional unit-sphere-systems. In the present study, we could prove that for every d>3, there is a d-dimensional unit-sphere-system, and there is no 3-dimensional unit-sphere-system. This settles unit-sphere-systemproblem that has been unsettled since 1989.
(4) Concerning families of solid balls in 3-space, we could slightly improve the bounds on their chromatic numbers, and the bounds on the number of balls necessary to make a knotted cycle.

Report

(4 results)
  • 2007 Annual Research Report   Final Research Report Summary
  • 2006 Annual Research Report
  • 2005 Annual Research Report
  • Research Products

    (55 results)

All 2008 2007 2006 2005

All Journal Article (26 results) (of which Peer Reviewed: 8 results) Presentation (29 results)

  • [Journal Article] Planar unit-distance graphs having planar unit-distance complement2008

    • Author(s)
      S. V. Gervacio
    • Journal Title

      Discrete Mathematics 308

      Pages: 1973-1984

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Pivotal inversions of a finite point-set2007

    • Author(s)
      H. Maehara
    • Journal Title

      Yokohama Mathematical Journal 53

      Pages: 119-126

    • NAID

      120001740837

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] On configurations of solid balls in 3-space2007

    • Author(s)
      H. Maehara
    • Journal Title

      Graphs and Combinatorics 23

      Pages: 307-320

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] The problem of thirteen spheres-A proof for undergraduates2007

    • Author(s)
      H. Maehara
    • Journal Title

      European Journal of Combinatorics 28

      Pages: 1770-1778

    • NAID

      120001371783

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Inversions and Mobius invariants2007

    • Author(s)
      H. Maehara
    • Journal Title

      Ryukyu Mathematical Journal 20

      Pages: 9-23

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Journal Article] The problem of thirteen spheres - Aproof for undergraduates2007

    • Author(s)
      H. Maehara
    • Journal Title

      European Journal of Combinatorics 28

      Pages: 1770-1778

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] On configurations of solid balls in 3-space -Chromatic numbers and knotted cycles2007

    • Author(s)
      H. Maehara
    • Journal Title

      Graphs and Combinatorics 23

      Pages: 307-320

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Pivotal inversions of a finite point-set2007

    • Author(s)
      H.Maehara
    • Journal Title

      Yokohama Mathematical Journal 53

      Pages: 119-126

    • NAID

      120001740837

    • Related Report
      2006 Annual Research Report
  • [Journal Article] The problem of thirteen spheres - A proof for undergraduate2007

    • Author(s)
      H.Maehara
    • Journal Title

      European Journal of Combinatorics (印刷中)

    • NAID

      120001371783

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Partial order on a family of k-subsets of a linearly ordered set2006

    • Author(s)
      S. V. Gervacio
    • Journal Title

      Discrete Mathematics 306

      Pages: 413-419

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Pearl numbers versus stick numbers for knots2006

    • Author(s)
      H. Maehara
    • Journal Title

      Rev. Roumaine Math. Pures Appl. 51-1

      Pages: 49-56

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] On a special arrangement of spheres2006

    • Author(s)
      H. Maehara
    • Journal Title

      Ryukyu Mathematical Journal. 19

      Pages: 15-24

    • NAID

      120001371678

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Partial order on a family of k-subsets of a linearly ordered set2006

    • Author(s)
      S. V. Gervacio
    • Journal Title

      Discrete Mathematics, refereed 306

      Pages: 413-419

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Pearl numbers versus stick numbers for knots2006

    • Author(s)
      H. Maehara
    • Journal Title

      Rev. Roumaine Math. Pures Appl 51 (1)

      Pages: 49-56

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] On a special arrangement of spheres2006

    • Author(s)
      H. Maehara
    • Journal Title

      Ryukyu Mathematical Journal 19

      Pages: 15-24

    • NAID

      120001371678

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Partial order on a family of k-subsets of a linearly ordered set2006

    • Author(s)
      S.V.Gervacio
    • Journal Title

      Discrete Mathematics 306

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Pearl numbers versus stick numbers for knots2006

    • Author(s)
      H.Maehara
    • Journal Title

      Rev. Roumaine Math. Pure Appl. 51

    • Related Report
      2006 Annual Research Report
  • [Journal Article] On a special arrangement of spheres2006

    • Author(s)
      H.Maehara
    • Journal Title

      Ryukyu Mathematical Journal 19

    • NAID

      120001371678

    • Related Report
      2006 Annual Research Report
  • [Journal Article] On the perimeter of a triangle in a Minkowski plane2005

    • Author(s)
      H. Maehara
    • Journal Title

      American Mathemaical Monthly 112

      Pages: 521-522

    • NAID

      120001371786

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Almost halving a family of objects in Euclidean space by a hyperplane2005

    • Author(s)
      H. Maehara
    • Journal Title

      Ryukyu Mathematical Journal 18

      Pages: 1-11

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] On the perimeter of a triangle in a Minkowski plane2005

    • Author(s)
      H. Maehara
    • Journal Title

      American Mathematical Monthly, refereed 112

    • NAID

      120001371786

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Almost halving a family of objects in Euclidean space by a hyperplane.2005

    • Author(s)
      H. Maehara
    • Journal Title

      Ryukyu Mathematical Journal 18

      Pages: 1-11

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] On the perimeter of a triangle in a Minkowski plane2005

    • Author(s)
      H.Maehara
    • Journal Title

      American Mathematical Monthly 112

      Pages: 521-522

    • NAID

      120001371786

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Almost halving a family of objects in Euclidean space by a hyperplane2005

    • Author(s)
      H.Maehara
    • Journal Title

      Ryukyu Mathematical Journal 18

      Pages: 1-11

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Problems on a family of solid balls in 3-space2005

    • Author(s)
      H.Maehara
    • Journal Title

      The Proceedings of Second COE Workshop on Sphere Packings

      Pages: 116-130

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Partial order on a family of k-subsets of a linearly ordered set2005

    • Author(s)
      V.S.Gervacio
    • Journal Title

      Discrete Mathematics (印刷中)

    • Related Report
      2005 Annual Research Report
  • [Presentation] From line-systems to sphere-systems2008

    • Author(s)
      H. Maehara
    • Organizer
      ISM Symposium
    • Place of Presentation
      統計数理研究所
    • Year and Date
      2008-03-11
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] From line-svstems to SDhere-svstems2008

    • Author(s)
      H. Maehara
    • Organizer
      ISM Symposium
    • Place of Presentation
      The Institute of Mathematical Statistics
    • Year and Date
      2008-03-11
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] From line-systems to sphere-systems2008

    • Author(s)
      H. Maehara
    • Organizer
      ISM Symposium
    • Place of Presentation
      統計数理研究所、東京
    • Year and Date
      2008-03-11
    • Related Report
      2007 Annual Research Report
  • [Presentation] 団体戦によるランキング2007

    • Author(s)
      H. Maehara
    • Organizer
      位相幾何学的グラフ理論研究集会
    • Place of Presentation
      横浜ランドマークタワー
    • Year and Date
      2007-11-16
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Ranking by team competition2007

    • Author(s)
      H. Maehara
    • Organizer
      Topological Graph Theory Workshop
    • Place of Presentation
      Yokohama Land Mark Tower
    • Year and Date
      2007-11-16
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Ranldng by team competition2007

    • Author(s)
      H. Maehara
    • Organizer
      International Xu Guangqi Conference
    • Place of Presentation
      PICB, Shanghai, China
    • Year and Date
      2007-10-18
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Ranking by team competition2007

    • Author(s)
      H. Maehara
    • Organizer
      International Xu Guangqi Conference
    • Place of Presentation
      PICB, Shanghai, China
    • Year and Date
      2007-10-18
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Presentation] A new variety of flexatube and reversible polyhedral surfaces2007

    • Author(s)
      H. Maehara
    • Organizer
      Kyoto International Conference on Computational Geometry and Graph Theory
    • Place of Presentation
      京都大学
    • Year and Date
      2007-06-11
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] A new variety of flexatube and reversible polyhedral surfaces2007

    • Author(s)
      H. Maehara
    • Organizer
      Kyoto International Conference on Computational Geometry and Graph Theory
    • Place of Presentation
      Kyoto University
    • Year and Date
      2007-06-11
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] A new variety of flexatube and reversible polyhedral surfaces2007

    • Author(s)
      H. Maehara
    • Organizer
      Kyoto International Conference on Computational Geometry and Graph Theory
    • Place of Presentation
      京都大学、京都
    • Year and Date
      2007-06-11
    • Related Report
      2007 Annual Research Report
  • [Presentation] On a sphere that passes through n lattice points2007

    • Author(s)
      H. Maehara
    • Organizer
      Combinatorics Geometry and Computer Science
    • Place of Presentation
      CIRM、Marseille, France
    • Year and Date
      2007-05-03
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] On a sphere that passes through n lattice points2007

    • Author(s)
      H. Maehara
    • Organizer
      Combinatorics. Geometry and Computer Science
    • Place of Presentation
      CIRM, Marseille, France
    • Year and Date
      2007-05-03
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] On a sphere that passes through n lattice points2007

    • Author(s)
      H. Maehara
    • Organizer
      Combinatorics, Geometry and Computer Science
    • Place of Presentation
      CIRM, Marseille, France
    • Year and Date
      2007-05-03
    • Related Report
      2007 Annual Research Report
  • [Presentation] On a special arrangement of spheres2007

    • Author(s)
      H. Maehara
    • Organizer
      ISM Symposium
    • Place of Presentation
      統計数理研究所
    • Year and Date
      2007-02-27
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] On a special arrangement of spheres2007

    • Author(s)
      H. Maehara
    • Organizer
      ISM Symposium
    • Place of Presentation
      The Institute of Mathematical Statistics
    • Year and Date
      2007-02-27
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] 13球問題の初等的な証明2006

    • Author(s)
      H. Maehara
    • Organizer
      RISM研究集会「群論とその周辺」
    • Place of Presentation
      京大会館
    • Year and Date
      2006-12-21
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] The problem of thirteen spheres -A proof for undergraduates2006

    • Author(s)
      H. Maehara
    • Organizer
      RIMS Workshop
    • Place of Presentation
      Kyoto University
    • Year and Date
      2006-12-21
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] 格子点を通る球面2006

    • Author(s)
      H. Maehara
    • Organizer
      位相幾何学的グラフ理論研究集会
    • Place of Presentation
      横浜ランドマークタワー
    • Year and Date
      2006-11-17
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Reversing a polyhedral surface2006

    • Author(s)
      H. Maehara
    • Organizer
      CMS Summer meeting
    • Place of Presentation
      Calgary, Canada
    • Year and Date
      2006-06-03
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Reversing- a polyhedral surface2006

    • Author(s)
      H. Maehara
    • Organizer
      CMS Summer Meeting
    • Place of Presentation
      Westin Hotel. Calearv. Canada
    • Year and Date
      2006-06-03
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Problems on a family of solid balls in 3-space2005

    • Author(s)
      H. Maehara
    • Organizer
      2^<nd> Pacific Workshop on Discrete Mathematics
    • Place of Presentation
      Honolulu, Hawaii
    • Year and Date
      2005-12-08
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Problems on a family of solid balls in 3-space2005

    • Author(s)
      H. Maehara
    • Organizer
      2^<nd> Pacific Workshop on Discrete Mathematics
    • Place of Presentation
      Tokai University Pacific Center, Honolulu, Hawaii
    • Year and Date
      2005-12-08
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Almost halving a set of objects in Euclidean space by a hyperplane2005

    • Author(s)
      H. Maehara
    • Organizer
      位相幾何学的グラフ理論研究集会
    • Place of Presentation
      横浜ランドマークタワー
    • Year and Date
      2005-11-25
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Almost halving a set of objects in Euclidean space by a hyperplane2005

    • Author(s)
      H. Maehara
    • Organizer
      Topological Graph Theory Workshop
    • Place of Presentation
      Yokohama Land Mark Tower
    • Year and Date
      2005-11-25
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] On a sphere that passes through a given number of lattice points2005

    • Author(s)
      H. Maehara
    • Organizer
      Topological Graph Theory Workshop
    • Place of Presentation
      Yokohama Land Mark Tower
    • Year and Date
      2005-11-25
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Othello-like solitaire on a graph2005

    • Author(s)
      H. Maehara
    • Organizer
      Japan Workshop on Graph Theory and Combinatorics
    • Place of Presentation
      慶応大学
    • Year and Date
      2005-06-22
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Othello-like solitaire on a graph2005

    • Author(s)
      H. Maehara
    • Organizer
      Japan Workshop on Graph Theory and Combinatorics
    • Place of Presentation
      Keio University
    • Year and Date
      2005-06-22
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Problems on a family of solid balls in 3-space2005

    • Author(s)
      H. Maehara
    • Organizer
      The 2^<nd> COE Workshop on Sphere Pacldng
    • Place of Presentation
      九州大学箱崎キャンパス
    • Year and Date
      2005-06-02
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Problems on a family of solid balls in 3-space2005

    • Author(s)
      H. Maehara
    • Organizer
      The 2^<nd> COE Workshop on Sphere Packing
    • Place of Presentation
      Kyushu University
    • Year and Date
      2005-06-02
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary

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Published: 2005-04-01   Modified: 2016-04-21  

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