Platonic Solids and Spatial Visualization: Generating Complex Shapes Using Basic Three-Dimensional Operations

Platonic Solids and Spatial Visualization: Generating Complex Shapes Using Basic Three-Dimensional Operations

Beniamino Polimeni
DOI: 10.4018/978-1-5225-0029-2.ch020
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In last decades computer graphics design systems emerged as key players in architectural design, on the one hand leading to a combination of parametric, process-driven architecture and applied geometry, and on the other hand, to a new role of architects as makers, and craft-people, able to bring together architecture as an interplay between art and craftsmanship. This unique dual perspective about architectural practice and education inspires this paper, which tries to analyze how a set of basic volumetric transformations can generate complex spatial outcomes. Using platonic solids as base volumes, we will explore different ideas, applying generalized extrusions, remesh schemes of subdivision and multiplication of the object's faces, and a set of tools to create high-genus meshes. Starting from these new objects, a set of solid wireframe structures will be created as well. The goal of this process is to create a basic guideline to explore the spatial design language: a set of illustrated steps to activate architectural inquiry and to generate innovative design solutions.
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For the last 20 years, the significant impact of tridimensional digital modeling techniques on architecture has been well charted. From the use of ordinary drafting software to the more experimental use of generative design tools and parametric modeling, digital technologies challenged contemporary practice, playing a major role in architectural production from the initial form-finding stages to the final construction.

One of the main advantages of using these tools has been the possibility of creating and exploring complex shapes and geometries that have never been realized before.

The advance of computer graphics has fostered the interaction of architects, mathematicians and artists as well, giving birth to a series of interesting experiments in which geometry is used as a tool to model and fabricate an extraordinary variety of artworks and design solutions.

Many contemporary artists such as George Hart, David Brisson (Brisson, 1992), Helaman Ferguson, Bathseba Grossman, have combined art and geometry to create unusual and high-genus pieces of art (Akleman, Ozener & Yuksel, 2006).

Investigations and researches into geometries has been carried out also by architects such as Andrew Kudless, Marjan Colletti, and Michael Hansmeyer who have explored the use of algorithms and computation to generate original architectural forms.

In this work, we will try to describe and analyze one of the methods commonly used by these artists to construct interactively a wide variety of shapes. Starting from Platonic Solids as base volumes, we will define two design guidelines based on a set of topological mesh modeling operators used as tools to explore different creative possibilities. The aim is to put together the technical possibilities and an operative working knowledge of these methods, bringing new energy to the study of architectural forms. Interdisciplinary fields such as architecture and industrial design can benefit from this approach.

Figure 1.

The lamp “Quin” realized by Bathseba Grossman is an interesting combination of art and geometry applied to an industrial object. (Grossman, 2012). Quin. [Online image]. Retrieved May 1 2015 from:

Figure 2.

Platonic Solids Retrieved May 1 2015 from:



It can be said that for thousands of years there has been a strong interest in geometrical plane shapes and solid forms, especially, those possessing regular features of proportion and symmetry (Emmer, 1982). Among the solid forms that have attracted the curiosity of scientists and artists, Regular Polyhedra have served as art motifs from Prehistoric Times right up to the present.

Regular Polyhedra or Platonic Solids, discovered by the Pythagoreans but described by Plato in the Timaeus consist of regular, convex Polyhedra constructed by congruent regular polygonal faces with the same number of faces meeting at each vertex.

The faces of a Platonic Solid are congruent equilateral triangles (tetrahedron, octahedron, and icosahedron), congruent squares (cube), or congruent regular pentagons (dodecahedron).

Table 1 summarizes the number of faces (F), the number of vertices (V), and the number of edges (E) in each Platonic Solid.

Key Terms in this Chapter

Remeshing: A change in number, shape, size or arrangement of the polygons forming a surface mesh.

Archimedean Solid: The 13 Archimedean solids are the convex polyhedra that have a similar arrangement of nonintersecting regular convex polygons of two or more different types arranged in the same way about each vertex with all sides the same length.

Rapid Prototyping: Rapid prototyping is a group of techniques used to quickly fabricate a scale model of a physical part or assembly using three-dimensional computer aided design (CAD) data. Construction of the part or assembly is usually done using 3D printing or “additive layer manufacturing” technology. However, rapid prototypes, particularly of metal parts, can also be produced through a “subtractive manufacturing” process known as chemical etching.

Regular Polyhedron: A regular polyhedron is a polyhedron with congruent faces and identical vertices. There are only five convex regular polyhedra, and they are known collectively as the Platonic solids.

Maquette: A maquette (French word for scale model, sometimes referred to by the Italian names plastico or modello) is a small scale model or rough draft of an unfinished sculpture.

Skeletonic: Resembling a skeleton. Skeletonic solids allow an easy distinction between front and back.

MeSH: A polygon mesh is a collection of vertices, edges and faces that defines the shape of a polyhedral object in 3D computer graphics and solid modeling. The faces usually consist of triangles (triangle mesh), quadrilaterals, or other simple convex polygons, since this simplifies rendering, but may also be composed of more general concave polygons, or polygons with holes.

Extrusion: Extrusion is a process used to create objects of a fixed cross-sectional profile.

Stellation: Stellation is the process of constructing polyhedra by extending the facial planes past the polyhedron edges of a given polyhedron until they intersect.

Wire-Frame Model: A wire-frame model is a visual presentation of a three-dimensional or physical object used in 3D computer graphics. It is created by specifying each edge of the physical object where two mathematically continuous smooth surfaces meet, or by connecting an object's constituent vertices using straight lines or curves.

Subdivision Surfaces: Subdivision surface is a method of representing a smooth surface via the specification of a coarser piecewise linear polygon mesh. The smooth surface can be calculated from the coarse mesh as the limit of a recursive process of subdividing each polygonal face into smaller faces that better approximate the smooth surface.

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