Tiling the Lateral Surface of the Concave Cupolae of the Second Sort

  • Marija ObradovićEmail author


As architectural structures, concave cupolae of the second sort (CC II) are suitable for prefabrication, thanks to the uniformity of their elements—regular polygons. This paper discusses ways to subdivide the lateral polyhedral triangle (LPT) of CC II into new regular polygons, triangles and hexagons, which can then tile the whole deltahedral lateral surface, forming patterns applicable in architectural design. The number of different arrangements of triangles and hexagons depends on the number of unit hexagonal cells (tiles) that can be placed within the triangular grid of the equilateral triangle. Considering solutions that involve fragments of k-uniform Euclidean tilings, we explored the ones that arise when the edges of the LPT are subdivided into 3 ≤ b ≤ 9 (b \(\in {\mathbb{N}}\)) segments. Without attempting to analyse all the possible cases, we focused on the ones with the D3 symmetry, in order to propose the simplest solutions for the assembly, so it will not matter if the face of the CC II is rotated or flipped. This results in 30 different solutions, excluding the ones consisting of triangles alone. The solutions found are applicable to all the representatives of CC II. We chose several examples of these results for an architectural design proposal.


Equilateral triangle Hexagon Tiling Deltahedron Cupola Architecture 



This research has been supported by the MPNTR of Republic of Serbia, grant No. III44006. The Fig. 4 is based on Wikimedia: By Tomruen—Own work, CC BY-SA 4.0, edited by the author.


  1. Akiyama, Jin, Takayasu Kuwata, Stefan Langerman, Kenji Okawa, Ikuro Sato, and Geoffrey C. Shephard. 2011. Determination of all tessellation polyhedra with regular polygonal faces. In: Computational Geometry, Graphs and Application (CGGA 2010) 7033 of Lecture Notes in Computer Science, eds. Jin Akiyama, Jiang Bo, Mikio Kano, Xuehou Tan, 1-11. Berlin: Springer-Verlag.Google Scholar
  2. Arfken, George B. 1985. Mathematical methods for physicists. San Diego: Academic Press.Google Scholar
  3. Aslaksen, Helmer. 2006. In Search of Demiregular Tilings. Bridges: Mathematical Connections in Art, Music, and Science (Bridges Conference, London, August 2006), eds. R. Sarhangi, and J. Sharp, 533–536. London: Tarquin Books.Google Scholar
  4. Chavey, Darrah P. 1984 a. Periodic tilings and tilings by regular polygons. PhD Thesis, University of Wisconsin, Madison.Google Scholar
  5. Chavey, Darrah P. 1984 b. Periodic tilings and tilings by regular polygons. I: Bounds on the number of orbits of vertices, edges and tiles. Mitteilungen aus dem Mathematischen Seminar Giessen, 164: 37–50.Google Scholar
  6. Chavey, Darrah P. 1989. Tilings by Regular Polygons II: A Catalog of Tilings, Computers & Mathematics with Applications. 17 (1-2): 147–165.Google Scholar
  7. Clinton, Joseph D. 1971. Advanced structural geometry studies. Part 1: Polyhedral subdivision concepts for structural applications. Technical report, NASA CR-1734/35 I: 1–96. Washington, D.C.Google Scholar
  8. Conway, John H. 1992. The Orbifold Notation for Surface Groups. In: Groups, Combinatorics and Geometry, eds. Martin W. Liebeck, Jan Saxl, ch. 36, 438–447. Cambridge [England]; New York: Cambridge University Press.Google Scholar
  9. Conway, John H., Heidi Burgiel and Chaim Goodman-Strauss. 2008. The Symmetries of Things. Wellesley, MA: A K Peters, Ltd.Google Scholar
  10. Coxeter, Harold Scott Macdonald. 1973. Regular polytopes. New York: Dover Publication.Google Scholar
  11. Critchlow, Keith. 1969. Order in space: a design source book. London: Thames and Hudson.Google Scholar
  12. De Bruijn, Nicolaas Govert. 1964. Pólya’s Theory of Counting. In: Applied Combinatorial Mathematics, ed. E. Beckenbach, 144–184. New York: John Wiley and Sons, Inc.Google Scholar
  13. Fedorov, Evgraf Stepanovich. 1891. Cиммeтpiя нa плocкocти (Symmetry in the plane), Зaпиcки Импepaтopcкoгo C.-Пeтepбypгcкoгo минepaлoгичecкoгo oбщecтвa (Proceedings of the Imperial St. Petersburg Mineralogical Society), 2 (28): 345–390 (in Russian).Google Scholar
  14. Fuller, Buckminster Richard. 1975. Synergetics: explorations in the geometry of thinking. New York: Macmillan Publishing Co. Inc.Google Scholar
  15. Galebach, Brian L. 2002. N-uniform tilings. ProbabilitySports. (accessed 29 June 2018).
  16. Gardner, Martin. 1958. Mathematical games. Scientific American. 198.6: 108–114.Google Scholar
  17. Ghyka, Matila Costiescu. 1946. The geometry of art and life. New York: Sheed and Ward.Google Scholar
  18. Goldberg, Michael. 1937. A class of multi-symmetric polyhedra. Tohoku Mathematical Journal, First Series, 43: 104–108.Google Scholar
  19. Grünbaum, Branko, Geoffrey C. Shephard. 1977. Tilings by regular polygons, Mathematics Magazine 50 (5): 227–247.Google Scholar
  20. Hamkins, Joel David. 2016. There are no regular polygons in the hexagonal lattice, except triangles and hexagons (accessed 4 July 2018).
  21. Johnson, Norman W. 1966. Convex Solids with Regular Faces. Canadian Journal of mathematics 18 (1): 169–200.Google Scholar
  22. Kepler, Johannes. 1619. Harmonices mundi. Libri V. Reprint. Bruxelles: Culture et Civilisation. 1968.Google Scholar
  23. Misić, Slobodan, Marija Obradović and Gordana Đukanović. 2015. Composite Concave Cupolae as Geometric and Architectural Forms, Journal for Geometry and Graphics 19 (1): 79-91. Vienna: Heldermann Verlag.Google Scholar
  24. Mišić, Slobodan. 2013. Generisanje kupola sa konkavnim poliedarskim površima (Generation of the cupolae with concave polyhedral surfaces), PhD thesis. University of Belgrade, Faculty of Architecture.Google Scholar
  25. Obradović, Marija and Slobodan Mišić. 2008. Concave Regular Faced Cupolae of Second Sort, In: Proceedings of 13th ICGG (ICGG 2008, Dresden, August 2008), ed. Gunter Weiss, El. Book: 1–10. Dresden: ISGG/Technische Universität Dresden.Google Scholar
  26. Obradović, Marija, Slobodan Mišić and Maja Petrović. 2012. Investigating Composite Polyhedral forms obtained by combining concave cupolae of II sort with Archimedean Solids. In: Proceedings of 3rd International Scientific Conference MoNGeometrija 2012 (MoNGeometrija 2012, Novi Sad, June 2012), ed. Ratko Obradović, 109–123. Novi Sad: Faculty of Technical Sciences.Google Scholar
  27. Obradović, Marija, Slobodan Mišić, Branislav Popkonstantinović, Maja Petrović, Branko Malešević and Ratko Obradović. 2013. Investigation of Concave Cupolae Based Polyhedral Structures and Their Potential Application in Architecture, Technics Technologies Education Management 8 (3): 1198–1214.Google Scholar
  28. Obradović, Marija, Slobodan Mišić, Branislav Popkonstantinović and Maja Petrović. 2011. Possibilities of Deltahedral Concave Cupola Form Application in Architecture. Buletinul Institutului Politehnic din Iaşi sectia Constructii de Masini LVII (LXI) (3): 123–140.Google Scholar
  29. Obradović, Marija, Slobodan Mišić and Branislav Popkonstantinović. 2014. Concave Pyramids of Second Sort - The Occurrence, Types, Variations. In: Proceedings of 4th International Scientific Conference on Geometry and Graphics MoNGeometrija 2014 – Volume 2 (MoNGeometrija 2014, Vlasina, June 2014), ed. Sonja Krasić, 157–168. Niš: Faculty of Civil Engineering and Architecture in Niš and Serbian Society for geometry and graphics (SUGIG).Google Scholar
  30. Obradović, Marija. 2012. A Group Of Polyhedra Arised as Variations Of Concave Bicupolae of Second Sort. In: Proceedings of 3rd International Scientific Conference MoNGeometrija 2012 (MoNGeometrija 2012, Novi Sad, June 2012), ed. Ratko Obradović, 95–132. Novi Sad: Faculty of Technical Sciences.Google Scholar
  31. Obradović, Marija. 2006. Konstruktivno – geometrijska obrada toroidnih deltaedara sa pravilnom poligonalnom osnovom (Constructive-geometrical elaboration on toroidal deltahedra with regular polygonal bases). PhD thesis, University of Belgrade, Faculty of Architecture.Google Scholar
  32. Obradović, Marija, Slobodan Mišić and Branislav Popkonstantinović. 2015. Variations of Concave Pyramids of Second Sort with an Even Number of Base Sides. Journal of Industrial Design and Engineering Graphics (JIDEG) – The SORGING Journal, 10 (1) Special Issue: 45–50.Google Scholar
  33. Pickover, Clifford A. 2009. The math book: from Pythagoras to the 57th dimension, 250 milestones in the history of mathematics. New York: Sterling.Google Scholar
  34. Pólya, George. 1924. Über die Analogie der Kristallsymmetrie in der Ebene. (On the analog of crystal symmetry in the plane), Zeitschrift für Kristallographie, 60: 278–282.Google Scholar
  35. Sloane, Neil J. A. 2003. The on-line encyclopaedia of integer sequences. (accessed 4 July 2018).
  36. Steinhaus, Hugo. 1950. Mathematical snapshots. Mineola, New York: Oxford University Press, rpt. Dover Publications 1999.Google Scholar
  37. Wenninger, Magnus J. 1979. Spherical models. Cambridge [England]; New York: Cambridge University Press.Google Scholar
  38. Wikipedia. 2018. Euclidean Tilings by Convex Regular Polygons. (accessed 29 June 2018).

Copyright information

© Kim Williams Books, Turin 2018

Authors and Affiliations

  1. 1.Faculty of Civil EngineeringUniversity of BelgradeBelgradeSerbia

Personalised recommendations