Skip to main content
SpringerLink
Log in

Dudeney Dissections of Polygons and Polyhedrons – A Survey –

  • Conference paper
  • First Online:
Discrete and Computational Geometry (JCDCG 2000)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2098))

Included in the following conference series:

Abstract

Given two polygons (polyhedrons) α and β with the same area (volume), the problem of finding a partition of β into parts that can be reassembled to form β is a promising area of study in geometry. We define a new type of dissection, Dudeney dissection, for polygons and polyhedrons. The dissection imposes two restrictions, one based on the reversal of the perimeter (surface area) and the interior (cross-section) of the polygon (polyhedron), and the other based on the hingeability of parts. In this paper, we survey main results on Dudeney dissections of polygons and polyhedrons.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Akiyama, J., Nakamura, G.: A lesson on double packable solids. Teaching Mathematics and Its Applications 18No.1, Oxford Univ. Press (1999) 30–33

    Article  Google Scholar 

  2. Akiyama, J., Nakamura, G.: Dudeney dissection of polygons. Discrete and Computational Geometry (J. Akiyama et al. (eds.), Lecture Notes in Computer Science 1763), Springer-Verlag (2000) 14–29

    Google Scholar 

  3. Akiyama, J., Nakamura, G.: Congruent Dudeney dissections of polygons I,-All hinge points are vertices of the polygon-. submitted to Proc. the 9th Quadrennnial International Conference on Graph Theory and Combinatorics, Algorithms and Applications, Discrete Math.

    Google Scholar 

  4. Akiyama, J., Nakamura, G.: Congruent Dudeney dissections of polygons II,-All hinge points are on the sides of the polygon-. Proc. Mathematics and Art Conference, Bond University (2000) 115–145

    Google Scholar 

  5. Akiyama, J., Nakamura, G.: Congruent Dudeney dissections of polygons III,-Hinge points are on both vertices and sides of a polygon-. (In Japanese) Tech. Reports, Res. Inst. Edu. Develop. Tokai Univ. (June, 2000)

    Google Scholar 

  6. Akiyama, J., Nakamura, G.: Dudeney dissections of polyhedrons I,-Space filings solids-. (In Japanese) The Reports of the Res. Inst. of Educ., Tokai Univ. 8 (2000) 1–12

    MathSciNet  Google Scholar 

  7. Akiyama, J., Nakamura, G.: Dudeney dissections of polyhedrons II,-Three layered type-. (In Japanese) The Reports of the Res. Inst. of Educ., Tokai Univ. 8 (2000) 13–26

    Google Scholar 

  8. Akiyama, J., Nakamura, G.: Dudeney dissections of polyhedrons III,-Mirror image type-. (In Japanese) The Reports of the Res. Inst. of Educ., Tokai Univ. 8 (2000) 27–35

    Google Scholar 

  9. Akiyama, J., Nakamura, G.: Dudeney dissections of polyhedrons IV,-Two layered type-. (In Japanese) The Reports of the Res. Inst. of Educ., Tokai Univ. 8 (2000) 36–57

    Google Scholar 

  10. Akiyama, J., Nakamura, G.: Dudeney dissections of polyhedrons V,-Multi layered type-. (In Japanese) The Reports of the Res. Inst. of Educ., Tokai Univ. 8 (2000) 58–62

    Google Scholar 

  11. Akiyama, J., Nakamura, G.: Dudeney dissections of polyhedrons VI,-Clasping type-. (In Japanese) The Reports of the Res. Inst. of Educ., Tokai Univ. 8 (2000) 63–66

    Google Scholar 

  12. Boltyanski, V. G.: Hilbert’s Third Problem. V. H. Winston & Sons. Translated by R. A. Silverman (1978)

    Google Scholar 

  13. Bolyai, F.: Tentamen juventutem. Maros Vasarhelyini: Typis Collegii Reformatorum per Josephum et Simeonem Kali (In Hungarian) (1832)

    Google Scholar 

  14. Busschop, P.: Problémes de géométrie. Nouvelle Correoindance Mathématique 2 (1876) 83–84

    Google Scholar 

  15. Coxeter, H. S. M.: Introductions to Geometry. Wiley & Sons (1965)

    Google Scholar 

  16. Dehn, M.: Über den Rauminhalt. Nachrichten von der Gesellschft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (1900) 345–354. Subsequently in Mathematische Annalen 55 (1902) 465—478

    Google Scholar 

  17. Dudeney, H. E.: The Canterbury Puzzles. Thomas Nelson & Sons (1907)

    Google Scholar 

  18. Frederickson, G. N.: Dissections: Plane & Fancy. Cambridge University Press (1997)

    Google Scholar 

  19. Gerwien, P.: Zershneidung jeder beliebigen Anzahl von gleichen geradlinigen Figuren in dieselben Stücke. Jouranl für die reine und angewandteMathematik (Crelle’s Journal) 10 (1833) 228–234 and Taf.III

    Article  MATH  Google Scholar 

  20. Grünbaum, B., Shephard, G. C.: Tiling with congruent tiles. Bull. AMS 3 (1980) 951–973

    Article  MATH  Google Scholar 

  21. Hilbert, D.: Mathematische Probleme. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (1900). Subsequently in Bulletin of the American Mathematical Society 8 (1901—1902) 437–479

    Article  MathSciNet  Google Scholar 

  22. Klarner, D. A.: The Mathematical Gardner. Wadsworth International (1981)

    Google Scholar 

  23. Lindgren, H.: Geometric Dissections. D. Van Nostarand Company (1964)

    Google Scholar 

  24. Sydler, J.-P.: Conditions nécessaries et suésantes pour l’équivalence des polyédres de l’espace euclidien à trios dimensions. Commentarll Mathematici Helvetica 40 (1965) 43–80

    Article  MATH  MathSciNet  Google Scholar 

  25. Wallace, W.: (Ed.) Elements of Geometry (8th ed.). Bell & Bradfute. First six books of Euclid, with a supplement by John Playfair (1831)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Research Institute of Educational Development, Tokai University, 2-28-4 Tomigaya, Shibuya-ku, Tokyo, 151-0063, Japan

    Jin Akiyama & Gisaku Nakamura

Authors
  1. Jin Akiyama
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gisaku Nakamura
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Research Institute of Educational Development, Tokai University, 2-28-4 Tomigaya, Shibuya-ku, Tokyo, 151-0063, Japan

    Jin Akiyama

  2. Department of Computer and Information Sciences, Ibaraki University, Hitachi, 316-8511, Japan

    Mikio Kano

  3. Department of Mathematics, Tokai University, 3-20-1 Orido Shimizu-shi, Shizuoka, 424-8610, Japan

    Masatsugu Urabe

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Akiyama, J., Nakamura, G. (2001). Dudeney Dissections of Polygons and Polyhedrons – A Survey –. In: Akiyama, J., Kano, M., Urabe, M. (eds) Discrete and Computational Geometry. JCDCG 2000. Lecture Notes in Computer Science, vol 2098. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-47738-1_1

Download citation

  • DOI: https://doi.org/10.1007/3-540-47738-1_1

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42306-5

  • Online ISBN: 978-3-540-47738-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation