From Solid to Plane Tessellations, and Back

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.


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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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