Exploring the Vertices of a Triacontahedron
Year: 2013 Authors: Robert Weadon Rollings
Core claim
The vertices of all five Platonic solids can be realized from a rhombic triacontahedron’s circumsphere or inscribed sphere.
Topics
Platonic solids, rhombic triacontahedron, geometric modeling, golden ratio, wood craftsmanship
Domains
solid geometry, polyhedra, spherical geometry, golden ratio, sculpture, woodworking, geometric art, craft design
Methods
physical model making, geometric construction, comparative placement of vertices
Media
babinga wood, brass rods, cocabola wood spheres
Paper text
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Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture
Exploring the Vertices of a Triacontahedron
Robert Weadon Rollings 883 Brimorton Drive, Scarborough, Ontario, M1G 2T8 E-mail: bobsturn@bell.net
Abstract
The rhombic triacontahedron can be used as a framework for locating the vertices of the five platonic solids. The five models presented here, which are made of babinga wood and brass rods illustrate this relationship.
Introduction
My initial interest in the platonic solids began with the work of Cox [1]. My initial explorations of the platonic solids lead to the series of models Polyhedra through the Beauty of Wood [2]. I have continued this exploration by using a rhombic triacontahedron at the core of my models and illustrating its relationship to all five platonic solids.
Rhombic Triacontahedron
The rhombic triacontahedron has 30 rhombic faces where the ratio of the diagonals is the golden ratio. The models shown in Figures 1, 2, 4, and 5 illustrate that the vertices of an icosahedron, dodecahedron, hexahedron and tetrahedron all lie on the circumsphere of the rhombic triacontahedron. In addition, Figure 3 shows that the vertices of the octahedron lie on the inscribed sphere of the rhombic triacontahedron.
Creating the Models
Each of the five rhombic triacontahedrons are made of 1/4 inch thick babinga wood. The edges of the rhombi were cut at 18 degrees on a band saw. After the assembly of the triacontahedron 1/8 inch brass rods were inserted a specific vertices and then fitted with 1/2 inch cocabola wood spheres. Different vertices (or faces in the case of the octahedron) were chosen for each platonic solid. The total size of each model is about 5 inches by 5 inches by 5 inches.
Figure 1: Tetrahedron
Rollings
Figure 2: Hexahedron
Figure 3: Octahedron
Figure 4: Dodecahedron
Figure 5: Icosahedron
References
[1] J. Cox, Beyond Basic Turning, Linden Publishing Co, Fresno, 1993. [2] R. Rollings (2010). Polyhedra expressed through the beauty of wood, Journal of Mathematics and the Arts, Vol. 4, 191 - 200.