Revisiting Mat Board Models for A Physical Proof of Five and Only Five Regular Solids or Polyhedron
Year: 2010 Authors: Robert McDermott
Core claim
Mat board models provide a stable, hands-on spatial proof of the five regular solids and their regular corners.
Topics
physical models, regular solids, polygon nets, tactile learning
Domains
geometry, polyhedra, Platonic solids, regular polygons, educational craft, 3D model making, visualization
Methods
mat board construction, polygon tracing and cutting, net assembly, hands-on demonstration
Media
mat board, paper polygons, physical polyhedron models, figures
Paper text
The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.
Bridges 2010: Mathematics, Music, Art, Architecture, Culture
Revisiting Mat Board Models for A Physical Proof of Five and Only Five Regular Solids or Polyhedron
Robert McDermott (Retired) Center for High Performance Computing University of Utah Salt Lake City, Utah, 84112, USA E-mail: robert.mcdermott@.utah.edu
Abstract
Physical models are invaluable aids for visualizing concepts clearly in space. In this paper mat board built more stable physical models. Polygons, polygon corners, and polyhedra are used to present a physical proof of regular solids. A physical proof shows five and only five regular corners, and subsequently five regular solids.
1. Introduction to Polygons
An understanding of these solids is achieved when students make physical models and handle these models. The models are made using mat board, which is stronger than poster board, which improves the durability of the models [Fig. 1]. Regular polygons are the simplest of two-dimensional shapes having straight edges and equal length sides [Ref. 1, 2, & 3]. An equilateral triangle, a square, and a regular pentagon, [Fig. 1], were used to construct regular corners in this paper. Polygons were traced and cut with a cleaver.
Figure 1: equilateral triangles, squares, pentagons, traced, and cut with a cleaver.
2. Polygon Nets for Polyhedra
Nets of regular polygons produce regular polyhedra [Fig. 2]. Two different nets produce a tetrahedron. Eight equilateral triangles produce an octahedron, and twenty equilateral triangles produce an icosahedron. For the remaining platonic solids squares produce a cube, and pentagons produce a dodecahedron.
Figure 2: 4 triangles each for 2 nets, 8 triangles, 20 triangles, 6 squares, 12 pentagons.
McDermott
4. Regular Polygons Used to Form Corners for Regular Solids
Equilateral triangles form three regular corners in [Fig 3] for a tetrahedron, an octahedron and an icosahedron. Three squares form a corner for a cube and 3 regular pentagons form a corner for a regular dodecahedron
Figure 3: a tetrahedron, a cube, a octahedron, an icosahedron and a dodecahedron
5. Handling a Regular Polyhedron
If we hold a regular polyhedra between our fingers in [Fig 4] we can see each of its vertices, edges, and faces.
Figure 4: fingers touching vertices, edges and faces for five regular solids
6. Conclusion
I have presented a spatial proof using physical models to a wide range of students over many years, and I have been well received by both students and teachers. The teachers have shown their appreciation by wanting to keep the models for their classrooms to serve as constant examples for future study. Once, when I presented the models and the proof to seventh and eight grade students, they wanted to take the proof home to their parents in order to prove a point for themselves.
Emphasizing the tactile experience of handling each of the regular solids provides students with a recognizably different experience than merely viewing three-dimensional images of the regular solids on a piece of paper or on a computer screen
References
[1] McDermott, R., “A Physical Proof for Five and Only Five Regular Solids”, Bridges 2005. [2] Coxeter, H.S.M., Regular Polytopes, Third Edition, Dover Publications, Inc., NY, 1973. [3] Holden, A., Shapes, Space and Symmetry, Columbia University Press, New York, 1971.