Similar dissection of sets View Full Text


Ontology type: schema:ScholarlyArticle      Open Access: True


Article Info

DATE

2010-05-06

AUTHORS

Shigeki Akiyama, Jun Luo, Ryotaro Okazaki, Wolfgang Steiner, Jörg Thuswaldner

ABSTRACT

In 1994, Martin Gardner stated a set of questions concerning the dissection of a square or an equilateral triangle in three similar parts. Meanwhile, Gardner’s questions have been generalized and some of them are already solved. In the present paper, we solve more of his questions and treat them in a much more general context. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${D\subset \mathbb{R}^d}$$\end{document} be a given set and let f1, . . . , fk be injective continuous mappings. Does there exist a set X such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${D = X \cup f_1(X) \cup \cdots \cup f_k(X)}$$\end{document} is satisfied with a non-overlapping union? We will prove that such a set X exists for certain choices of D and {f1, . . . , fk}. The solutions X will often turn out to be attractors of iterated function systems with condensation in the sense of Barnsley. Coming back to Gardner’s setting, we use our theory to prove that an equilateral triangle can be dissected in three similar copies whose areas have ratio 1 : 1 : a for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${a \ge (3+\sqrt{5})/2}$$\end{document}. More... »

PAGES

233-247

References to SciGraph publications

  • 2002-09. Trisecting a parallelogram in APPLIED MATHEMATICS-A JOURNAL OF CHINESE UNIVERSITIES
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/s10711-010-9502-y

    DOI

    http://dx.doi.org/10.1007/s10711-010-9502-y

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1015354336


    Indexing Status Check whether this publication has been indexed by Scopus and Web Of Science using the SN Indexing Status Tool
    Incoming Citations Browse incoming citations for this publication using opencitations.net

    JSON-LD is the canonical representation for SciGraph data.

    TIP: You can open this SciGraph record using an external JSON-LD service: JSON-LD Playground Google SDTT

    [
      {
        "@context": "https://springernature.github.io/scigraph/jsonld/sgcontext.json", 
        "about": [
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/01", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Mathematical Sciences", 
            "type": "DefinedTerm"
          }, 
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/0101", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Pure Mathematics", 
            "type": "DefinedTerm"
          }
        ], 
        "author": [
          {
            "affiliation": {
              "alternateName": "Department of Mathematics, Faculty of Science, Niigata University, Ikarashi 2-8050, 950-2181, Niigata, Japan", 
              "id": "http://www.grid.ac/institutes/grid.260975.f", 
              "name": [
                "Department of Mathematics, Faculty of Science, Niigata University, Ikarashi 2-8050, 950-2181, Niigata, Japan"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Akiyama", 
            "givenName": "Shigeki", 
            "id": "sg:person.011153327405.03", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.011153327405.03"
            ], 
            "type": "Person"
          }, 
          {
            "affiliation": {
              "alternateName": "School of Mathematics and Computational Science, Sun Yat-Sen University, 510275, Guangzhou, China", 
              "id": "http://www.grid.ac/institutes/grid.12981.33", 
              "name": [
                "School of Mathematics and Computational Science, Sun Yat-Sen University, 510275, Guangzhou, China"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Luo", 
            "givenName": "Jun", 
            "id": "sg:person.0674040142.18", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.0674040142.18"
            ], 
            "type": "Person"
          }, 
          {
            "affiliation": {
              "alternateName": "Department of Knowledge Engineering and Computer Sciences, Doshisha University, 1-3 Tataramiyakodani, Kyotanabe-shi, 610-0394, Kyoto-fu, Japan", 
              "id": "http://www.grid.ac/institutes/grid.255178.c", 
              "name": [
                "Department of Knowledge Engineering and Computer Sciences, Doshisha University, 1-3 Tataramiyakodani, Kyotanabe-shi, 610-0394, Kyoto-fu, Japan"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Okazaki", 
            "givenName": "Ryotaro", 
            "id": "sg:person.011036447511.49", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.011036447511.49"
            ], 
            "type": "Person"
          }, 
          {
            "affiliation": {
              "alternateName": "LIAFA, CNRS UMR 7089, Universit\u00e9 Paris Diderot - Paris 7, Case 7014, 75205, Paris Cedex 13, France", 
              "id": "http://www.grid.ac/institutes/grid.462842.e", 
              "name": [
                "LIAFA, CNRS UMR 7089, Universit\u00e9 Paris Diderot - Paris 7, Case 7014, 75205, Paris Cedex 13, France"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Steiner", 
            "givenName": "Wolfgang", 
            "id": "sg:person.012140615123.03", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.012140615123.03"
            ], 
            "type": "Person"
          }, 
          {
            "affiliation": {
              "alternateName": "Department of Mathematics and Statistics, University of Leoben, Franz-Josef-Strasse 18, 8700, Leoben, Austria", 
              "id": "http://www.grid.ac/institutes/grid.181790.6", 
              "name": [
                "Department of Mathematics and Statistics, University of Leoben, Franz-Josef-Strasse 18, 8700, Leoben, Austria"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Thuswaldner", 
            "givenName": "J\u00f6rg", 
            "id": "sg:person.011406664737.45", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.011406664737.45"
            ], 
            "type": "Person"
          }
        ], 
        "citation": [
          {
            "id": "sg:pub.10.1007/s11766-002-0009-7", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1026186108", 
              "https://doi.org/10.1007/s11766-002-0009-7"
            ], 
            "type": "CreativeWork"
          }
        ], 
        "datePublished": "2010-05-06", 
        "datePublishedReg": "2010-05-06", 
        "description": "In 1994, Martin Gardner stated a set of questions concerning the dissection of a square or an equilateral triangle in three similar parts. Meanwhile, Gardner\u2019s questions have been generalized and some of them are already solved. In the present paper, we solve more of his questions and treat them in a much more general context. Let \\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${D\\subset \\mathbb{R}^d}$$\\end{document} be a given set and let f1, . . . , fk be injective continuous mappings. Does there exist a set X such that \\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${D = X \\cup f_1(X) \\cup \\cdots \\cup f_k(X)}$$\\end{document} is satisfied with a non-overlapping union? We will prove that such a set X exists for certain choices of D and {f1, . . . , fk}. The solutions X will often turn out to be attractors of iterated function systems with condensation in the sense of Barnsley. Coming back to Gardner\u2019s setting, we use our theory to prove that an equilateral triangle can be dissected in three similar copies whose areas have ratio 1 : 1 : a for \\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${a \\ge (3+\\sqrt{5})/2}$$\\end{document}.", 
        "genre": "article", 
        "id": "sg:pub.10.1007/s10711-010-9502-y", 
        "inLanguage": "en", 
        "isAccessibleForFree": true, 
        "isPartOf": [
          {
            "id": "sg:journal.1135919", 
            "issn": [
              "0046-5755", 
              "1572-9168"
            ], 
            "name": "Geometriae Dedicata", 
            "publisher": "Springer Nature", 
            "type": "Periodical"
          }, 
          {
            "issueNumber": "1", 
            "type": "PublicationIssue"
          }, 
          {
            "type": "PublicationVolume", 
            "volumeNumber": "150"
          }
        ], 
        "keywords": [
          "equilateral triangle", 
          "continuous mapping", 
          "certain choices", 
          "solution X", 
          "Martin Gardner", 
          "set X", 
          "function systems", 
          "present paper", 
          "general context", 
          "set", 
          "attractors", 
          "squares", 
          "triangle", 
          "theory", 
          "sense", 
          "Barnsley", 
          "ratio 1", 
          "Gardner", 
          "mapping", 
          "FK", 
          "system", 
          "questions", 
          "similar parts", 
          "exists", 
          "choice", 
          "condensation", 
          "setting", 
          "similar copies", 
          "set of questions", 
          "part", 
          "context", 
          "area", 
          "dissection", 
          "F1", 
          "Union", 
          "copies", 
          "similar dissection", 
          "paper"
        ], 
        "name": "Similar dissection of sets", 
        "pagination": "233-247", 
        "productId": [
          {
            "name": "dimensions_id", 
            "type": "PropertyValue", 
            "value": [
              "pub.1015354336"
            ]
          }, 
          {
            "name": "doi", 
            "type": "PropertyValue", 
            "value": [
              "10.1007/s10711-010-9502-y"
            ]
          }
        ], 
        "sameAs": [
          "https://doi.org/10.1007/s10711-010-9502-y", 
          "https://app.dimensions.ai/details/publication/pub.1015354336"
        ], 
        "sdDataset": "articles", 
        "sdDatePublished": "2022-05-10T10:00", 
        "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
        "sdPublisher": {
          "name": "Springer Nature - SN SciGraph project", 
          "type": "Organization"
        }, 
        "sdSource": "s3://com-springernature-scigraph/baseset/20220509/entities/gbq_results/article/article_509.jsonl", 
        "type": "ScholarlyArticle", 
        "url": "https://doi.org/10.1007/s10711-010-9502-y"
      }
    ]
     

    Download the RDF metadata as:  json-ld nt turtle xml License info

    HOW TO GET THIS DATA PROGRAMMATICALLY:

    JSON-LD is a popular format for linked data which is fully compatible with JSON.

    curl -H 'Accept: application/ld+json' 'https://scigraph.springernature.com/pub.10.1007/s10711-010-9502-y'

    N-Triples is a line-based linked data format ideal for batch operations.

    curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1007/s10711-010-9502-y'

    Turtle is a human-readable linked data format.

    curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/s10711-010-9502-y'

    RDF/XML is a standard XML format for linked data.

    curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/s10711-010-9502-y'


     

    This table displays all metadata directly associated to this object as RDF triples.

    140 TRIPLES      22 PREDICATES      64 URIs      55 LITERALS      6 BLANK NODES

    Subject Predicate Object
    1 sg:pub.10.1007/s10711-010-9502-y schema:about anzsrc-for:01
    2 anzsrc-for:0101
    3 schema:author Nc4279a1c98014060b660eb0688d0a86b
    4 schema:citation sg:pub.10.1007/s11766-002-0009-7
    5 schema:datePublished 2010-05-06
    6 schema:datePublishedReg 2010-05-06
    7 schema:description In 1994, Martin Gardner stated a set of questions concerning the dissection of a square or an equilateral triangle in three similar parts. Meanwhile, Gardner’s questions have been generalized and some of them are already solved. In the present paper, we solve more of his questions and treat them in a much more general context. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${D\subset \mathbb{R}^d}$$\end{document} be a given set and let f1, . . . , fk be injective continuous mappings. Does there exist a set X such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${D = X \cup f_1(X) \cup \cdots \cup f_k(X)}$$\end{document} is satisfied with a non-overlapping union? We will prove that such a set X exists for certain choices of D and {f1, . . . , fk}. The solutions X will often turn out to be attractors of iterated function systems with condensation in the sense of Barnsley. Coming back to Gardner’s setting, we use our theory to prove that an equilateral triangle can be dissected in three similar copies whose areas have ratio 1 : 1 : a for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${a \ge (3+\sqrt{5})/2}$$\end{document}.
    8 schema:genre article
    9 schema:inLanguage en
    10 schema:isAccessibleForFree true
    11 schema:isPartOf N62cabb7caebd4ffa9e0b4c2a2df56267
    12 Na75aff16979049fb89d0690371376905
    13 sg:journal.1135919
    14 schema:keywords Barnsley
    15 F1
    16 FK
    17 Gardner
    18 Martin Gardner
    19 Union
    20 area
    21 attractors
    22 certain choices
    23 choice
    24 condensation
    25 context
    26 continuous mapping
    27 copies
    28 dissection
    29 equilateral triangle
    30 exists
    31 function systems
    32 general context
    33 mapping
    34 paper
    35 part
    36 present paper
    37 questions
    38 ratio 1
    39 sense
    40 set
    41 set X
    42 set of questions
    43 setting
    44 similar copies
    45 similar dissection
    46 similar parts
    47 solution X
    48 squares
    49 system
    50 theory
    51 triangle
    52 schema:name Similar dissection of sets
    53 schema:pagination 233-247
    54 schema:productId N35ff6af214ef4067bfdffcc4e4f7456a
    55 N52a23d35e62b49baa04cd1c158de2b85
    56 schema:sameAs https://app.dimensions.ai/details/publication/pub.1015354336
    57 https://doi.org/10.1007/s10711-010-9502-y
    58 schema:sdDatePublished 2022-05-10T10:00
    59 schema:sdLicense https://scigraph.springernature.com/explorer/license/
    60 schema:sdPublisher Nc7cb58ef60624fc88267762903098d8a
    61 schema:url https://doi.org/10.1007/s10711-010-9502-y
    62 sgo:license sg:explorer/license/
    63 sgo:sdDataset articles
    64 rdf:type schema:ScholarlyArticle
    65 N33f30e667032450493927b4f407959d4 rdf:first sg:person.0674040142.18
    66 rdf:rest N679330d93a0a4030bd9982fd75eb346a
    67 N35ff6af214ef4067bfdffcc4e4f7456a schema:name doi
    68 schema:value 10.1007/s10711-010-9502-y
    69 rdf:type schema:PropertyValue
    70 N52a23d35e62b49baa04cd1c158de2b85 schema:name dimensions_id
    71 schema:value pub.1015354336
    72 rdf:type schema:PropertyValue
    73 N62cabb7caebd4ffa9e0b4c2a2df56267 schema:volumeNumber 150
    74 rdf:type schema:PublicationVolume
    75 N679330d93a0a4030bd9982fd75eb346a rdf:first sg:person.011036447511.49
    76 rdf:rest Nd731675d6125428786301298c1c012f7
    77 Na75aff16979049fb89d0690371376905 schema:issueNumber 1
    78 rdf:type schema:PublicationIssue
    79 Nc4279a1c98014060b660eb0688d0a86b rdf:first sg:person.011153327405.03
    80 rdf:rest N33f30e667032450493927b4f407959d4
    81 Nc7cb58ef60624fc88267762903098d8a schema:name Springer Nature - SN SciGraph project
    82 rdf:type schema:Organization
    83 Ncbf6289ac3424172999cbc4f075e4e79 rdf:first sg:person.011406664737.45
    84 rdf:rest rdf:nil
    85 Nd731675d6125428786301298c1c012f7 rdf:first sg:person.012140615123.03
    86 rdf:rest Ncbf6289ac3424172999cbc4f075e4e79
    87 anzsrc-for:01 schema:inDefinedTermSet anzsrc-for:
    88 schema:name Mathematical Sciences
    89 rdf:type schema:DefinedTerm
    90 anzsrc-for:0101 schema:inDefinedTermSet anzsrc-for:
    91 schema:name Pure Mathematics
    92 rdf:type schema:DefinedTerm
    93 sg:journal.1135919 schema:issn 0046-5755
    94 1572-9168
    95 schema:name Geometriae Dedicata
    96 schema:publisher Springer Nature
    97 rdf:type schema:Periodical
    98 sg:person.011036447511.49 schema:affiliation grid-institutes:grid.255178.c
    99 schema:familyName Okazaki
    100 schema:givenName Ryotaro
    101 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.011036447511.49
    102 rdf:type schema:Person
    103 sg:person.011153327405.03 schema:affiliation grid-institutes:grid.260975.f
    104 schema:familyName Akiyama
    105 schema:givenName Shigeki
    106 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.011153327405.03
    107 rdf:type schema:Person
    108 sg:person.011406664737.45 schema:affiliation grid-institutes:grid.181790.6
    109 schema:familyName Thuswaldner
    110 schema:givenName Jörg
    111 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.011406664737.45
    112 rdf:type schema:Person
    113 sg:person.012140615123.03 schema:affiliation grid-institutes:grid.462842.e
    114 schema:familyName Steiner
    115 schema:givenName Wolfgang
    116 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.012140615123.03
    117 rdf:type schema:Person
    118 sg:person.0674040142.18 schema:affiliation grid-institutes:grid.12981.33
    119 schema:familyName Luo
    120 schema:givenName Jun
    121 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.0674040142.18
    122 rdf:type schema:Person
    123 sg:pub.10.1007/s11766-002-0009-7 schema:sameAs https://app.dimensions.ai/details/publication/pub.1026186108
    124 https://doi.org/10.1007/s11766-002-0009-7
    125 rdf:type schema:CreativeWork
    126 grid-institutes:grid.12981.33 schema:alternateName School of Mathematics and Computational Science, Sun Yat-Sen University, 510275, Guangzhou, China
    127 schema:name School of Mathematics and Computational Science, Sun Yat-Sen University, 510275, Guangzhou, China
    128 rdf:type schema:Organization
    129 grid-institutes:grid.181790.6 schema:alternateName Department of Mathematics and Statistics, University of Leoben, Franz-Josef-Strasse 18, 8700, Leoben, Austria
    130 schema:name Department of Mathematics and Statistics, University of Leoben, Franz-Josef-Strasse 18, 8700, Leoben, Austria
    131 rdf:type schema:Organization
    132 grid-institutes:grid.255178.c schema:alternateName Department of Knowledge Engineering and Computer Sciences, Doshisha University, 1-3 Tataramiyakodani, Kyotanabe-shi, 610-0394, Kyoto-fu, Japan
    133 schema:name Department of Knowledge Engineering and Computer Sciences, Doshisha University, 1-3 Tataramiyakodani, Kyotanabe-shi, 610-0394, Kyoto-fu, Japan
    134 rdf:type schema:Organization
    135 grid-institutes:grid.260975.f schema:alternateName Department of Mathematics, Faculty of Science, Niigata University, Ikarashi 2-8050, 950-2181, Niigata, Japan
    136 schema:name Department of Mathematics, Faculty of Science, Niigata University, Ikarashi 2-8050, 950-2181, Niigata, Japan
    137 rdf:type schema:Organization
    138 grid-institutes:grid.462842.e schema:alternateName LIAFA, CNRS UMR 7089, Université Paris Diderot - Paris 7, Case 7014, 75205, Paris Cedex 13, France
    139 schema:name LIAFA, CNRS UMR 7089, Université Paris Diderot - Paris 7, Case 7014, 75205, Paris Cedex 13, France
    140 rdf:type schema:Organization
     




    Preview window. Press ESC to close (or click here)


    ...