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Generating Polygons with Triangles

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Book cover Discrete and Computational Geometry and Graphs (JCDCGG 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8845))

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Abstract

A set of triangles \(\mathcal F\) is said to generate a polygon \(P\) if a homothetic transform \(\lambda P\) of \(P\) can be dissected into triangles each congruent to a triangle in \(\mathcal F\). The simplicial element number of a polygon \(P\) is defined to be the minimum cardinality of a family \(\mathcal F\) of triangles that can generate \(P\). The simplicial element number of a set of polygons \(P_1,P_2,\dots ,P_k\) is defined to be the minimum cardinality of a family \(\mathcal F\) of triangles that can generate all \(P_1,\dots ,P_k\). In this paper, we consider simplicial element numbers for several set of regular polygons and generating relations among triangles.

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References

  1. Akiyama, J., Kobayashi, M., Nakagawa, H., Nakamura, G., Sato, I.: Atoms for parallelohedra. Bolyai Society Mathematical Studies 24 (2013); Geometry-Intuitive, Discrete, and Convex, pp. 1–21

    Google Scholar 

  2. Akiyama, J., Hitotumatu, S., Sato, I.: Determination of the element number of the regular polytopes. Geom. Dedicata 159, 89–97 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  3. Akiyama, J., Sato, I.: The element number of the convex regular polytopes. Geom. Didicata 151, 269–278 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  4. Akiyama, J., Maehara, H., Nakamura, G., Sato, I.: Element number of the platonic solids. Geom Dedicata 145, 181–193 (2010)

    Article  MATH  Google Scholar 

  5. Hadwiger, H.: Hillsche hypertetraeder. Gazeta Math. (Lisboa) 12, 47–48 (1951)

    MATH  MathSciNet  Google Scholar 

  6. Hill, M.J.M.: Determination of the volume of certain species of tetrahedron without employing the method of limits. Proc. London Math. Soc. Ser. I 27, 39–53 (1895)

    Article  Google Scholar 

  7. Laczkovich, M.: Tilings of triangles. Discrete Math. 140, 79–94 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  8. Laczkovich, M.: Tilings of convex polygons with congruent triangles. Discrete Comput. Geom. 48, 330–374 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  9. Maehara, H.: Some replicating simplices other than Hill-simplices, Beiträge zur Algebra und Geometrie (to appear). doi:10.1007/s13366-013-0152-8

  10. Niven, I.: Numbers: Rational and Irrational, MAA. Random House, New York (1961)

    Google Scholar 

  11. Soifer, A.: How does one cut a triangle?. Springer, New York (1990). (Center for Excellence in Mathematical Education, Colorado Springs)

    Google Scholar 

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Acknowledgment

Many thanks to the referees for valuable comments.

Author information

Authors and Affiliations

  1. Tokai University, Kanagawa, Japan

    T. Kuwata

  2. Ryukyu University, Okinawa, Japan

    H. Maehara

Authors
  1. T. Kuwata
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  2. H. Maehara
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Corresponding author

Correspondence to T. Kuwata .

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Editors and Affiliations

  1. Tokyo University of Science, Tokyo, Japan

    Jin Akiyama

  2. The University of Electro-Communications, Tokyo, Japan

    Hiro Ito

  3. Tokai University, Tokyo, Japan

    Toshinori Sakai

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Kuwata, T., Maehara, H. (2014). Generating Polygons with Triangles. In: Akiyama, J., Ito, H., Sakai, T. (eds) Discrete and Computational Geometry and Graphs. JCDCGG 2013. Lecture Notes in Computer Science(), vol 8845. Springer, Cham. https://doi.org/10.1007/978-3-319-13287-7_10

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  • DOI: https://doi.org/10.1007/978-3-319-13287-7_10

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-13286-0

  • Online ISBN: 978-3-319-13287-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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