Abstract
In this paper, we propose an explanation and application of the synthetic method to the representation of the Platonic and Archimedean Solids (PS and AS). The intention is to illustrate the potential of this method in a historical and theoretical context. The study of regular and semiregular polyhedra in this sense is an ideal theme to illustrate the heuristic potential of drawing. Therefore, some synthetic constructions of PS and AS are proposed, defining the constructive algorithms of these figures. The working environment used is the mathematical representation method; for some constructions, parametric and physical simulating tools were used. Particular attention is dedicated to two different synthetic methods: the first, is the construction of the snub cube through paper folding and the second, is a more general method that exploits a physical simulator engine.
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Notes
- 1.
In these experimentations, we used Grasshopper (Version 6 SR19) and Kangaroo 2.
- 2.
By digital representation methods, we mean the set of principles and theories underlying three-dimensional representation software. Among these, we distinguish two different digital representation methods: the mathematical representation method and the polygonal (or numerical) representation method. The first one represents entities in a continuous way by means of parametric mathematical equations (such as, for example, NURBS). The second one approximates shapes by means of polyhedral entities (mesh).
- 3.
For example, the characteristic of being circumscribable in a sphere and, at the same time, of circumscribing a sphere, or the property of having equal solid angles.
- 4.
Semiregular polyhedra are convex polyhedra defined by regular polygons of different types.
- 5.
For example, the rhombicuboctahedron can be generated by dividing the edge of a circumscribed hexahedron into three parts according to the ratio 1: √2: 1. The length defined by the irrational value √2 is easily identifiable from a graphic construction that makes use of the diagonal of a square of side equal to 1 unit.
- 6.
These two polyhedra enjoy the chiral property for which the symmetry generates differences.
- 7.
See note 1.
- 8.
Even if the mathematical digital methods are based on analytical equations (see the NURBS), they can still be counted among the synthetic methods, since the control of the forms and their relationships take place through a language of visual nature. The classical studies of Descriptive Geometry were conducted with the help of a ruler and compass, which made it possible to materialize straight lines and circles on paper without necessarily resorting to their equations. In the same way, nowadays, the available tools to scholars have increased, offering much higher accuracy and allowing the representation of relationships between entities and the control of time dimension and movement.
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Acknowledgments
Leonardo Baglioni and Federico Fallavollita developed the theme, results, and conclusions of the research together. The premises of this study can be found in the chapter of Baglioni [9, pp. 299–422] about the polyhedra. Baglioni dealt in particular with the paragraph The Synthetic Approach For The Construction of the Platonic and Archimedean Solids and Fallavollita, with the Introduction and paragraph The General Synthetic Method to Construct a Polyhedron. Riccardo Foschi worked with Fallavollita on the design of the final procedural algorithm.
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Baglioni, L., Fallavollita, F., Foschi, R. (2022). Synthetic Methods for Constructing Polyhedra. In: Viana, V., Mena Matos, H., Pedro Xavier, J. (eds) Polyhedra and Beyond. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-99116-6_1
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