the edges of polyhedra

Alicia Boole Stott published in 1910 a research [1] on the expansion and contraction of 2-, 3- and 4-dimensional polytopes in which their limits are equally moved away from the center (or contracted inwards) until, for each case, a new uniform polytope is outlined.

In this webpage, some of the results of the expansion of the edges with convex regular polyhedra are illustrated (the first example shown is the pentagonal dodecahedron). The edges expand to outline faces with twice as many sides, and every vertex transforms into a face with the configuration of the corresponding vertex-figure. Consequently, the base-polyhedron expands into its semiregular truncated version.

Expansion of the edges of the pentagonal dodecahedron

The following 3D model and video show that the expansion of the edges of the pentagonal dodecahedron in the direction of the corresponding 2-fold symmetry axis transforms it into its semiregular truncated version. Its faces and vertices transform into decagons and triangles respectively.

The possibility of modeling this with Grasshopper allowed me to conclude [2] and [3] that the distance between the edges of the dodecahedron and the closest parallel edges of its truncated version equals the edge length multiplied by the golden ratio.

expansion of the dodecahedron’s edges by veraviana on Sketchfab

The distance between the edges of the dodecahedron and the closest parallel edges of its truncated version (the dashed line segment in the video) equals the edge length multiplied by the golden ratio.

Expansion of the edges of the pentagonal dodecahedron
  • [1] Stott, A. (1910) Geometrical deduction of semiregular from regular polytopes and space fillings. Verhan-delingen der Koninklijke Akademie van Wetenschappen te Amsterdam, 11(1) 893-2894.
  • [2] Viana, V. et al. (2018) Interactive Expansion of Achiral Polyhedra. In Luigi Cocchiarella (Ed.). ICGG 2018 – Proceedings of the 18th International Conference on Geometry and Graphics. ICGG 2018. Advances in Intelligent Systems and Computing, 809. Springer. 1116-1128.
  • [3] Viana, V (2020). Aplicações didácticas sobre poliedros para o ensino da geometria / Didactic Applications on Polyhedra for the teaching of Geometry (http://hdl.handle.net/10348/10337) [Tese de Doutoramento, Universidade de Trás-os-Montes e Alto Douro]. Repositório Científico da UTAD.