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Constuction of Some Archimedean Solids from Platonic Solids

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CONSTUCTION OF SOME ARCHIMEDEAN SOLIDS FROM PLATONIC SOLIDS Eva Barcíková Department of Mathematics, Faculty of Natural Sciences, Constantine the Philosopher University in Nitra, Tr. A. Hlinku 1, 949 74 Nitra, Slovakia Correspondig author: [email protected] Abstract In this paper we deal with obtaining of Small rhombicuboctahedron and Great rhombicuboctahedron , which are Archimedean solids, from Platonic solids. We focus on one of several ways of construction which is expansion. We show the particular steps of this process. We think that the process of expansion of Platonic or Archimedean solids can be used as the manipulating activities in a geometry lesson. Key words: Small rhombicuboctahedron, Great rhombicuboctahedron, Archimedean solids 1 Introduction The Archimedean solids took their name from Archimedes, who discussed them in a now- lost work. Later Pappus describes the 13 Archimedean solids in his fifth book of his Mathematical Collection, but it was Kepler who rediscovered the whole set of 13 solids and gave them the names by which they are known today. There are several ways to obtain an Archimedean solid from Platonic solids. Seven of them can be obtained by truncation of a Platonic solid. They are the truncated tetrahedron, truncated cube, truncated octahedron, truncated icosahedron, truncated dodecahedron, cuboctahedron and the icosidodecahedron. The other two Archimedean solids, snub cube and snub dodecahedron, are formed by a process called snubbing. Four additional solids can be obtained by expansion of Platonic solids or by expansion of one of the previous 9 Archimedean solids. Truncation is process of removing all the corners of a figure in a symmetrical fashion. Truncation is in most cases the easiest way to obtain an Archimedean solid from Platonic solid. Snubbing is a process of taking all the faces of the solid, pulling them outward so they no longer touch, then giving them each a small rotation on their centers (all clockwise or all counter-clockwise) until the spaces between can be filled with equilateral triangles. Very similar process to snubbing is expansion. We show the particular steps of this process in the following text. 2 Small rhombicuboctahedron The small rhombicuboctahedron is a 26-faced Archimedean solid with 24 vertices and 48 edges. His faces consist of 8 equilateral triangles and of 18 square faces. It is sometimes also called the truncated icosidodecahedron but this name is inappropriate. By truncation, we would obtain rectangular instead of square faces. Fig. 1 Small rhombicuboctahedron 275 There are several ways to obtain the small rhombicuboctahedron. The first option is truncation of cube. We remove all the corners of cube with edge length a. The length of small rhombicuboctahedron edge will be a. 2 1 . This truncation is not so intuitive as in other case of Archimedean solids. There is also other possibility. The small rhombicuboctahedron can be constructed as the convex hull of the centers of edges of cuboctahedron. Fig. 2 Small rhombicuboctahedron as the convex hull of the centers of edges of cuboctahedron. The third option to obtain this solid from Platonic solids is the expansion of cube. The process of expansion includes three steps: 1) Pull the faces of the polyhedron apart. 2) Replace the edges of each of the original face with squares. 3) Replace vertices where n faces meet with n-sided polygons. In our case the original polyhedron is cube. The following illustrate these three steps in detail: First we pull the faces of the cube apart. Thus, we have six separated squares. Fig. 3 In next step we simply replace the edges of each original face with squares. Finally, we replace vertices with n-sided polygons. Pulling the faces of the cube apart forms empty spaces between the three faces sharing a common vertex. The common vertex splits into three vertices, one for each of the three faces that meet at the vertex. So we can place an equilateral triangle such that the vertices of the equilateral triangle meet the three vertices that were split from the common vertex. The second and third steps of expansion are illustrated in figure 4. 276 Fig. 4 2.1 Great rhombicuboctahedron The process of expansion can be applied although on some Archimedean solid and the result will be other Archimedean solid. One of the obtained solid can be Great rhombicuboctahedron. It is also called the truncated cuboctahedron. The name suggests that the solid was created by truncation of cuboctahedron, which is not correct. Let’s look at the Great rhombicuboctahedron and find the right solid from which it can be created. Fig. 5 In second step of the process of expansion, we replace the edges of each of the original face with squares. 12 of all faces of Great rhombicuboctahedron are squares which linking together 8 regular hexagons or 6 regular octagons. This means that the original solid consists either of 8 regular hexagon faces or 6 regular octagon faces. So we have two options the truncated octahedron or truncated cube. Look what happen if we use the process of expansion on truncated octahedron. First we pull the faces of the truncated octahedron apart. Thus, we have eight separated regular hexagon faces. Next we replace the edges of each original face with squares. Instead of original squares faces forms empty space between eight new edges. So we can place an regular octagon faces there. Created solid is Great rhombicuboctahedron. 277 Fig. 6 Analogically, we can use the process of expansion on truncated cube and the result will be as well Great rhombicuboctahedron. It is illustrated in figure 7. → → Fig. 7 3 Results and discussion There are several ways how to work with Archimedean solids on a mathematics lesson. We show some possibilities of construction of selected solids by the process of expansion. Of course, its depend on teacher’s creativity which way he chooses for their lessons. Students could try these constructions using Polydron, which is geometric construction product. Polydron can be used for inquiry based learning and manipulating activities. Students can discover the Archimedean solids through the game. “From the most basic early shapes of cubes and prisms, and progressing right up to the full set of Archimedean solids, the world of Polydron shapes leaves no stone unturned in its quest to help children to understand and enjoy geometry.“ (Krishnan, 2002). 4 Conclusion Although Archimedean solids have been known for millenniums, they are quite little used in school. We think that the manipulating activities in geometry give us the platform to introduction of new knowledge using the constructivist approach to teaching. There are several possibilities how to construct Archimedean solids from Platonic solids, whose could 278 be used in school by this way. In this article, we show some of them. Purpose to deal with these constructions is the development of logical thinking, spatial imagination and other skills during the discovery of the Archimedean solid’s characteristics. Acknowledgement This work has been supported by UGA VII/38/2012 "Komparácia výstupov vedeckých prác doktorandov, mladých vedeckých a pedagogických pracovníkov formou prezentácie výsledkov" 5 References Cromwell, P. R. 1997. Polyhedra. New York: Cambridge University Press, 1997. ISBN 9- 521-55432-2 Krishnan, K. Polyhedra. [online]. Singapore : National University of Singapore. 2002 Available at: <http://www.math.nus.edu.sg/~urops/Projects/Polyhedra.pdf > Coxeter, H.S.M. 1973. Regular Polytopes, Dover, New York, 1973 Vallo, D. - Záhorská, J. - Ďuriš, V. 2011. Objavujeme sieť štvorstena. In: Acta Mathematica 14: zborník príspevkov z IX. nitrianskej matematickej konferencie, FPV UKF v Nitre v dňoch 22. - 23. septembra 2011 - Nitra : UKF, 2011. - ISBN 978-80-8094-958-7, s. 231- 236. 279 .
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