Interactive Expansion of Achiral Polyhedra

Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 809)


The representation of geometric concepts of three-dimensional space is a well-identified predicament that can undermine the understanding of geometry in particular, and mathematics, in general. Branco Grünbaum mentioned that no method of presentation is satisfactory for much more than the simplest situations and expressed the hope “that computer-based modes of presentation will alleviate this difficulty in the near future” [1]. Aiming to address specific concepts of polyhedral geometry, we’ve been exploring a 3D modelling software and its graphical algorithm editor as digital tools to illustrate certain concepts through accurate graphical descriptions that are dynamic and interactive and imply the knowledge of several geometric operations. Among the several possibilities that could illustrate the software potential, we have chosen the concepts of expansion and contraction of polytopes conceived by the Irish mathematician Alicia Boole Stott in 1910 [2], in our opinion, one of the most visually interesting for a dynamic description. For the sake of concision, we will restrict our presentation to two- and three-dimensional polytopes and illustrate the possibilities of dynamically interact with virtual models to visualize, in real-time, the expansion and contraction of regular polygons and uniform convex polyhedra. The purpose of this research is thus to graphically clarify Stott’s methods through a dynamic approach made possible with contemporary digital tools and demonstrate how this kind of analysis may simplify further researches on the subject and enhance the didactics of these concepts in particular and, more generally, of polyhedral geometry.


Expansion Contraction Achiral polyhedra Algorithmic modelling 



This assignment is co-financed by the European Regional Development Fund (ERDF) through the COMPETE 2020—Operational Programme Competitiveness and Internationalization (POCI) and national funds by the FCT under the POCI-01-0145-FEDER-007744 project.


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  1. 1.CEAU—Centre of Studies of Architecture and UrbanismPortoPortugal
  2. 2.FAUP—Faculdade de Arquitectura Da, Universidade Do PortoPortoPortugal
  3. 3.UTAD—Universidade de Trás Os Montes E Alto DouroVila RealPortugal

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