Skip to main content
SpringerLink
Log in

A Criterion for a Pair of Convex Polygons to be Reversible

  • Original Paper
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

A pair of convex polygons \(\alpha \) and \(\beta \) is said to be reversible if \(\alpha \) has a dissection into a finite number of hinged pieces which can be rearranged to form \(\beta \) under some conditions. In this paper, we give a necessary and sufficient condition for a given pair of polygons \(\alpha \), \(\beta \) to be reversible.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  1. Dudeney, H.E.: The Canterbury Puzzles and Other Curious Problems. W. Heinemann, London (1907)

    Google Scholar 

  2. Bolyai, F.: Tentamen juventutem studiosam in elementa matheseos puræ, elementaris ac sublimioris, methodo intuitiva, evidentiaque huic propria, introducendi, cum appendice triplici. Typis Collegii Reformatorum per Josephum, et Simeonem kali de felső Vist. Maros Vásárhelyini (1832–1833)

  3. Gerwien, P.: Zerschneidung jeder beliebigen Anzahl von gleichen geradlinigen Figuren in dieselben Stücke. J. für die reine und angew. Math. 10, 228–234 (1833)

    Article  MATH  Google Scholar 

  4. Hilbert, D.: Mathematische Probleme, Nachrichten von der Königl. Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-physikalische Klasse, pp. 253–297 (1900)

  5. Dehn, M.: Über den Rauminhalt. Math. Ann. 55, 465–478 (1901)

    Article  MathSciNet  Google Scholar 

  6. Frederickson, G.N.: Dissections: Plane and Fancy. Cambridge University Press, Cambridge (1997)

    Book  MATH  Google Scholar 

  7. Frederickson, G.N.: Hinged Dissections: Swinging and Twisting. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  8. Frederickson, G.N.: Piano-Hinged Dissections: Time to Fold. A K Peters, Wellesley (2006)

    Google Scholar 

  9. Boltyanskii, V.G.: Equivalent and Equidecomposable Figures. D. C. Health and Company, Boston (1963). [translated and adapted from the first Russian edition (1956) by Henn, A.K. and Watts, C.E.]

  10. Mirman, R.: Point Groups, Space Groups, Crystals. Molecules. World Scientific Publishing Co, Singapore (1999)

    MATH  Google Scholar 

  11. Akiyama, J., Nakamura, G.: Dudeney dissections of polygons and polyhedrons—a survey—. In: Discrete and Computational Geometry. Lecture Notes in Computer Science, vol. 2098, pp. 1–30 (2001)

  12. Akiyama, J., Nakamura, G.: Congruent Dudeney dissections of triangles and convex quadrilaterals—all hinge points interior to the sides of the polygons. discrete and computational geometry. Algorithms Comb. 25, 43–63 (2003)

    MathSciNet  Google Scholar 

  13. Akiyama, J., Seong, H.: Operators which preserve reversibility. In: Computational Geometry and Graphs. Lecture Notes in Computer Science, vol. 8296, pp. 1–19 (2013)

  14. Akiyama, J., Rappaport, D., Seong, H.: A decision algorithm for reversible pairs of polygons. Discret. Appl. Math. 178, 19–26 (2014)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the referees, Professor David Rappaport and Professor Joseph O’Rourke for giving us various important comments.

Author information

Authors and Affiliations

  1. Research Center for Math Education, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku, Tokyo, 162-8601, Japan

    Jin Akiyama

  2. Department of Mathematics, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8914, Japan

    Hyunwoo Seong

Authors
  1. Jin Akiyama
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hyunwoo Seong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jin Akiyama.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Akiyama, J., Seong, H. A Criterion for a Pair of Convex Polygons to be Reversible. Graphs and Combinatorics 31, 347–360 (2015). https://doi.org/10.1007/s00373-015-1544-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-015-1544-3

Keywords

Mathematics Subject Classification

Search

Navigation