Nexus Network Journal

, Volume 20, Issue 3, pp 741–768 | Cite as

From Solid to Plane Tessellations, and Back

  • Vera VianaEmail author
Geometer’s Angle


In solid tessellations or three-dimensional honeycombs, polyhedra fit together to fill space, so that every face of each polyhedron belongs to another polyhedron. Solid and plane tessellations are intrinsically connected, since any section cut through a solid tessellation always produces some kind of plane tessellation. To clarify this relation, we will mention a short list of convex polyhedra that fill space monohedrally and illustrate the convex uniform honeycombs, focusing on those with structural potential to outline spaceframes. With regular plane tessellations as starting point, we hint at the geometrical possibilities in which the Platonic and two Archimedean solids are explorable in topological interlocking, aiming to expand the repertoire of blocks for monohedral topological interlocking assemblies. This has possible applications in architecture, in relation, for example, to space frames.


Solid tessellations Plane tessellations Regular polyhedra Quasiregular polyhedra Topological interlocking 



All images are from the author, except for Fig. 7, retrieved from in October 2017 and in the public domain.


  1. Andreini, A. 1905. Sulle reti di poliedri regolari e semiregolari e sulle corrispondenti reti correlative. Mem. Società Italiana della Scienze, Ser. 3, 14: 75–129. Retrieved from in October 2017.
  2. Ball, W., Coxeter, H. & Ball, W. 1987. Mathematical Recreations and Essays. New York: Dover Publications.Google Scholar
  3. Chieh, C., 1988. Polyhedra and Crystal Structures. In Shaping Space: A Polyhedral Approach, M. Senechal, M. & G. M. Fleck, G. M., eds., 118-132. Boston: Birkhäuser.Google Scholar
  4. Conway, J., Burgiel, H. & Goodman-Strauss, C. 2008. The symmetries of things. Wellesley, MA: A.K. Peters.Google Scholar
  5. Coxeter, H. 1973. Regular Polytopes. New York: Dover Publications.Google Scholar
  6. Critchlow, K. 1969. Order in Space: a design source book. London: Thames and Hudson.Google Scholar
  7. Cromwell, P. 1997. Polyhedra. Cambridge, U.K.: Cambridge University Press.Google Scholar
  8. Deza, M. & Shtogrin, M. 2000. Uniform Partitions of 3-space, their Relatives and Embedding. European Journal of Combinatorics, 21(6), 807-814.
  9. Dyskin, A., Estrin, Y., Kanel-Belov, A. & Pasternak, E. 2003. Topological interlocking of platonic solids: A way to new materials and structures. Philosophical Magazine Letters, 83(3): 197-203.
  10. Dyskin, A., Estrin, Y., Kanel-Belov, A. & Pasternak, E. 2001. A new concept in design of materials and structures: Assemblies of interlocked tetrahedron-shaped elements. Scripta Materialia, 44(12), 2689-2694.
  11. Estrin, Y., Dyskin, A., & Pasternak, E. 2011. Topological interlocking as a material design concept. Materials Science and Engineering: C, 31(6): 1189-1194.
  12. Gheorghiu, A. & Dragomir, V. 1978. Geometry of structural forms. London: Applied Science Pub.Google Scholar
  13. Glickman. M., 1984. The G-Block System of Vertically Interlocking Paving. Proceedings of the 2nd International Conference on Concrete Block Paving, 10-12. Delft.Google Scholar
  14. Grünbaum, B. 2010. The Bilinski Dodecahedron and Assorted Parallelohedra, Zonohedra, Monohedra, Isozonohedra, and Otherhedra. The Mathematical Intelligencer 32(4): 5-15.
  15. Grünbaum, B. 1994. Uniform tilings of 3-space. Geombinatorics 4: 49-56.Google Scholar
  16. Grünbaum, B. & Shepard, G. 1987. Tilings and Patterns. New-York: W. H. Freeman & Company.Google Scholar
  17. Grünbaum, B. & Shephard, G. C. 1980. Tilings with congruent tiles. Bulletin of the American Mathematical Society 3(3): 951-974.
  18. Holden, A. 1971. Shapes, space, and symmetry. New York: Columbia University Press.Google Scholar
  19. Kanel-Belov A., Dyskin, A., Estrin, Y., Pasternak, E. & Ivanov-Pogodaev, I. 2008. Interlocking of Convex Polyhedra: Towards a Geometric Theory of Fragmented Solids (ArXiv08125089 Math).Google Scholar
  20. Kappraff, J. 1990. Connections: The geometric bridge between art and science. New York: McGraw-Hill.Google Scholar
  21. Lalvani, H. 1992. Continuous Transformations of Non-Periodic Tilings and Space-Fillings. Fivefold Symmetry, 97-128.
  22. Malkevitch, J. 1988. Milestones in the History of Polyhedra. In Shaping Space: A Polyhedral Approach, M. Senechal, M. & G. M. Fleck, G. M., eds., 80-92. Boston: Birkhäuser.Google Scholar
  23. Olshevsky, G. 2006. Uniform Panoploid Tetracombs, Manuscript. Retrieved from in April 2016.
  24. Pugh, A. 1976. Polyhedra: A Visual Approach. Berkeley: University of California Press.Google Scholar
  25. Pearce, P. 1978. Structure In Nature Is A Strategy For Design. Cambridge: MIT Press.Google Scholar
  26. Steinhaus, H. 1960. Mathematical Snapshots. 2nd ed. New York: Oxford University Press.Google Scholar
  27. Towle, R. 1996. Polar Zonohedra, The Mathematica Journal, 1996 Retrieved April 26, 2018
  28. Weizmann, M., Amir, O. & Grobman, Y. 2016. Topological interlocking in buildings: A case for the design and construction of floors. Automation in Construction, 72: 18-25.
  29. Wenninger, M. 1971. Polyhedron models. Cambridge: University Press.Google Scholar
  30. Wikipedia contributors. 2018, January 26. Convex uniform honeycomb. In Wikipedia, The Free Encyclopedia. Retrieved April 7, 2018 from
  31. Williams, R. 1979. The Geometrical Foundation of Natural Structure: A Source Book of Design. New York: Dover Publications.Google Scholar

Copyright information

© Kim Williams Books, Turin 2018

Authors and Affiliations

  1. 1.CEAU (Study Center of Architecture and Urbanism) - FAUP (Faculty of Architecture of Porto’s University)PortoPortugal
  2. 2.Sciences and Technology SchoolTrás-os-Montes e Alto Douro’s UniversityVila RealPortugal

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