Elements of Polygons and Polyhedra

  • Jin Akiyama
  • Kiyoko Matsunaga


In this chapter, I'd like to show you the building blocks for Platonic solids. We can call them ‘elements’, in the same way that in chemistry, materials are made up of, or synthesized from chemical elements. In physics, particle physicists study the elementary particles (physical elements, so to speak), which make up matter.


Dihedral Angle Element Number Regular Tetrahedron Platonic Solid Regular Polytopes 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    M. Aigner and G. M. Ziegler, Proofs from THE BOOK, Springer (1996)MATHGoogle Scholar
  2. [2]
    J. Akiyama, H. Maehara, G. Nakamura and I. Sato, Element Number of the Platonic Solids, Geom. Dedicata 145(1) (2010), 181-193CrossRefMATHGoogle Scholar
  3. [3]
    J. Akiyama, and I. Sato, The element numbers of the convex regular polytopes, Geom. Dedicata, 151 (2011), 269-278MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    J. Akiyama, S. Hitotumatu and I. Sato, Determination of the element numbers of the convex regular polytopes, Geom. Dedicata 159 (2012), 89-97MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    V. G. Boltyanskii, Equivalent and Equidecomposable Figures, D.C. Heath and Co. Translated and adapted from the first Russian edition (1956) by Alfred K. Henn and Charles E. Watts (1963)Google Scholar
  6. [6]
    V. G. Boltyanskii, Hilbert's Third Problem, V. H. Winston & Sons. Translated by A. Silverman (1978)MATHGoogle Scholar
  7. [7]
    V. Chvátal, A combinational theorem in plane geometry, J. Combinatorial Theory, Ser. B, 18 (1974), 39-41CrossRefMATHGoogle Scholar
  8. [8]
    M. Dehn, Über den Rauminhalt, Math. Annalen 55 (1902), 465-478CrossRefMATHGoogle Scholar
  9. [9]
    S. Fisk, A short proof of Chvátal's watchman theorem, J. Combinatorial Theory, Ser. B24 (1978), 374MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    D. Hilbert, Mathematical Problems, Lecture delivered at the International Congress of Mathematicians at Paris in 1900, Bulletin Amer. Math. Soc. 8 (1902), 437-479MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    J. O'Rourke, Art Gallery Theorems and Algorithms, Oxford University Press (1987)MATHGoogle Scholar
  12. [12]
    E. Schönhardt, Über die Zerlegung von Dreieckspolyedern in Tetraeder, Math. Annalen 98 (1928), 309-312MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    S. Tatekawa, Shinosuke Rakugo in PARCO (in Japanese), January 2011Google Scholar

Copyright information

© Springer Japan 2015

Authors and Affiliations

  • Jin Akiyama
    • 1
  • Kiyoko Matsunaga
    • 2
  1. 1.Tokyo University of ScienceTokyoJapan
  2. 2.YokohamaJapan

Personalised recommendations