Alicia Boole Stott published in 1910 her research [01] on the expansion of 2-, 3- and 4-dimensional polytopes, whose limits are equally moved away from the center (or contracted inwards) until, for each case, a new uniform polytope is outlined. Ball and Coxeter considered Stott’s method to obtain the achiral Archimedeans “far more elegant” [02: 137] than Kepler’s. This webpage shows Grasshopper definitions that I developed [03, 04] to study the expansion e1, that stands for the uniform expansion of the edges (e2 refers to the uniform expansion of the faces, that is dealt with here).
- expansion of the edges of regular polygons
- expansion of the edges of regular polygons of density 2
- expansion of the edges of the convex regular polyhedra
expansion of the edges of regular polygons
In dimension 2, the e1 expansion stands for the edges of a regular polygon {n} being equally moved away from the centre in the direction perpendicular to each side, so that the edges expand into a {2n} of equal edge length or, in Stott’s words: “any regular polygon changes into a regular polygon having the same length of edge and twice as many sides” [01: 5].
the following definition shows how the number of sides of a regular polygon of sides between 5 and 15 expand. at first, we see a pentagon expanding into a decagon; in the end, a heptagon that expands into a regular tetradecagon is shown.
expansion of the edges of regular polygons of density 2
With an adaptation of the previous definition, it was possible to conclude that the edges’ expansion also works for regular pentagons of density 2 (star-gons or polygrams), although with a slightly different result: any {n/2} expands into a regular polygon of density 1, 2n sides and edge length equal to the side of {n/d}. The definition begins by showing a pentagram, that expands into a convex decagon and, in the end, a heptagram {7/2} that expands into a regular tetradecagon.
expansion of the edges of the convex regular polyhedra
the following definitions show the results of the expansion e1 in dimension 3 with convex regular polyhedra. their 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.
the first definition shows the expansion of the edges of a cube, whose faces transform into regular octagons. Through translation of these to a suitable position, the semiregular truncated version of the cube is obtained and the vertices of the cube transform into regular triangular faces. The distance from the edges of the cube to its closest parallel edges of the truncated cube equals the edge length.
The next definition shows the expansion of the edges of a regular tetrahedron in the direction of the corresponding 2-fold symmetry axes. This expansion results in the semiregular truncated tetrahedron iff the distance between the closest parallel edges equals the tetrahedron’s bimedian which is equivalent to the diameter of the mid-sphere of the tetrahedron.
The following video shows 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.
With this Grasshopper definition, it was possible to conclude that the distance between the edges of the dodecahedron and the closest parallel edges of its truncated version (the dashed line segment shown above) equals the edge length multiplied by the golden ratio. The following is an interactive 3D model of the pentagonal dodecahedron and its expanded version, the truncated dodecahedron.
More about this research may be found in [03]. The following table summarizes the uniform expansion of the edges of convex regular polyhedra (with unit edge length):
| base polyhedron (edge 1) | expanded polyhedron (edge 1) | distance between closest parallel edges |
|---|---|---|
| tetrahedron 33 | truncated tetrahedron 3.62 | diameter of the mid-sphere |
| cube 43 | truncated cube 3.82 | 1 |
| octahedron 34 | truncated octahedron 4.62 | 1 |
| dodecahedron 53 | truncated dodecahedron 3.102 | (1+√5) / 2 |
| icosahedron 35 | truncated icosahedron 5.62 | diameter of the mid-sphere |
- 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.
- Ball, W., & Coxeter, H. (1987). Mathematical recreations and essays. New York: Dover Publications.
- 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.
- 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.
geometry addiction 