Abstract Construction Projects and the Imagination

Abstract Construction Projects and the Imagination Hands-on projects for understanding abstract mathematical concepts through the use of polyhedral models and planar designs The 3-dimensional projects use polyhedral models constructed with the Zome System as well as graphics using the computer algebra system, Mathematica, to gain insight into ideas in group theory and graph theory. The 2-dimensional projects investigate various topics using planar models with the Zome System and graphics drawn with the application program, Groups and Graphs. A Stellated Icosahedron and Interior Dodecahedron Dual with Two Tetrahedral Heads and Five Pentagonal Feet Made with the Zome System Raymond F. Tennant, Ph.D. Department of Natural and Quantitative Sciences Zayed University PO Box 4783 Abu Dhabi, United Arab Emirates Raymond.Tennant@zu.ac.ae

2 Introduction A longstanding method for understanding proofs in mathematics involves the creation of two or three-dimensional models which illustrate particular mathematical ideas. The hands-on projects in this paper will discuss models and proof techniques, which can be used in interdisciplinary classes, abstract algebra courses, and thesis projects. The projects involve sketching and building models to describe ideas in group theory, geometry, topology, and hyperspace. The projects are designed as a concrete manner to enhance understanding of abstract mathematical concepts. This paper is divided into three major projects. Each project consists of several exercises, which are designed to direct the student through a series of concepts in a particular area of mathematics. 3D Polyhedral Models Polyhedral & Planar Models Hyperspace Models Project 1 Group Theory and Polyhedral Models Project 2 Historical Proofs of Euler s Formula Project 3 Hypercubes and Graph Theory These projects have been drawn from several courses including the following. Symmetry - a general education mathematics course. Mathematical Symmetry: Connections Between Mathematics and Art - an upper-level mathematics course designed for the university honors program. Visual Mathematics and the Imagination - an upper-level mathematics course designed for the university honors program. Abstract Algebra I - undergraduate group theory course. Modern Geometry II - undergraduate/graduate course in geometry. Graph Theory - undergraduate/graduate course in graph theory. Modern Algebra I - graduate group theory course. Several of the topics have been used for designing thesis projects and independent studies. Several ideas from this collection were employed in hands-on minicourses given at Mathfest 2000, the summer meeting of the Mathematical Association of America at UCLA in Los Angeles, California, and at Mathfest 1999 in Providence, Rhode Island. Raymond Tennant, Ph.D. Department of Natural and Quantitative Sciences Zayed University PO Box 4783 Abu Dhabi, United Arab Emirates Raymond.Tennant@zu.ac.ae

3 Project 1 Group Theory & Polyhedral Models Project Construct Platonic and Archimedean Solids and identify their symmetries. Goal To understand and reinforce concepts and methods of proof in group theory. Definition A symmetry of a polyhedral model is a rotation or reflection, which transforms the model so that it appears unchanged. Definition The rotational symmetries along with the identity transformation form the group of rotations of a polyhedral model. Definition the model. The rotational and reflectional symmetries form the full symmetry group of I. Symmetry Elements and Symmetry Groups of a Model Exercise 1 Construct a model for one of the Platonic or Archimedean solids or another polyhedron. Exercise 2 Describe all of the symmetries of the model as rotations or reflections. Exercise 3 Describe the order of each rotational symmetry of your model. This is the smallest number of times a symmetry motion must be repeated to bring the model to its original position. Exercise 4 Count the number of elements in the group of rotations of the model. Tetrahedron Cube Octahedron Dodecahedron Icosahedron II. Group Actions and Cayley s Theorem Exercise 5 Focus on a set of components for your model, for example, vertices, faces, or some other feature. Describe how the group of rotations acts on this set of components. Cayley s Theorem Every group is isomorphic to a group of permutations.

4 Exercise 6 Look at the group of rotations acting on the components of your model to determine that Cayley s Theorem holds and the rotation group can be expressed in terms of permutations. Describe the elements of this permutation group. Is this permutation group unique? Exercise 7 One proof of Cayley s Theorem for an arbitrary group G involves the creation of a permutation group called the left regular representation of G. How does your investigation of a group of rotations acting on your model compare to this left regular representation? III. Subgroups and the Orbit Stabilizer Theorem Definition Let G be the group of rotations acting on the set I of components of your model. For each i I, the orbit of i under G is defined by orb G ( i) = { ϕ( i): ϕ G}. Definition Let G be the group of rotations acting on the set I of components of your model. For each i I, the stabilizer of i in G is defined by stab G ( i)= { ϕ G: ϕ( i)= i}. Orbit-Stabilizer Theorem Let G be the group of rotations acting on the set I of components of your model. For any i I, G = orb G ( i) stab G ( i). Exercise 8 Describe some of the subgroups of the group of rotations of your model. Exercise 9 Using your subgroups of rotations, describe which combinations of the group axioms would be sufficient for showing a subset of a group is indeed a subgroup. Exercise 10 Use the Orbit-Stabilizer theorem to determine the number of elements in the group of rotations of your model. Also, determine the number of elements in the full symmetry group. IV. Kernels, Shells, Amalgams, and Lagrange s Theorem Definition When one polyhedron is inscribed inside another, the inscribed polyhedron is called the kernel, the circumscribed polyhedron is called the shell, and the kernel and shell taken together is called the amalgam. Exercise 11 Construct, if possible, an inscribed polyhedron inside of your polyhedral model so that the kernel, shell, and amalgam are visible. Exercise 12 Determine the groups of rotations for the kernel, shell, and amalgam, respectively.

5 Tetrahedron Inscribed in a Dodecahedron with Stellations over the Tetrahedral Faces Hexagonal Prism Inscribed in a Truncated Icosahedron Lagrange s Theorem If G is a finite group and H is a subgroup of G, then H divides G. Further, the number of distinct left (right) cosets of H in G is G H. Exercise 13 For your model with the inscribed polygon, determine if there are subgroup relationships between the groups of rotations for the kernel, shell, and amalgam. When a subgroup relationship exists, verify that Lagrange s Theorem holds. Exercise 14 By removing symmetry from your original model, determine all the possible symmetry subgroups of the model. You might consider possible ways that pyramids, prisms, and other polyhedra might be inscribed in your model. V. Sylow Subgroups, Normal Subgroups and Simple Groups Definition Let G be a finite group and let p be a prime, which divides G. If p k divides G and p k+1 does not divide G then any subgroup of G of order p k is called a Sylow p- subgroup of G. Sylow s Third Theorem Let n p denote the number of Sylow p-subgroups. Then n p 1 mod p ( ) and n p divides G. Exercise 15 Using Sylow s Third Theorem, determine n p the number of Sylow p-subgroups, for each prime p that divides the order of the group of rotations of the model.

6 Definition A group G is simple if its only normal subgroups are the identity subgroup and the group itself. Exercise 16 Construct a polyhedron whose group of rotations is isomorphic to A 4, the alternating group on 4 letters. By inspection, verify that A 4 is not a simple group. Exercise 17 Construct a polyhedron whose group of rotations is isomorphic to A 5, the alternating group on 5 letters. By inspection, is it possible to verify that A 5 is a simple group? Tetrahedron Inscribed in a Dodecahedron Cube Inscribed in a Dodecahedron Exercise 18 Choose a subgroup H of the rotational group G of your polyhedral model. Determine all of the left cosets of H. If H is a normal subgroup, determine the factor group G H. VI. Truncation, Stellation and Classifying Groups of Rotations Definition The process of shaving off the vertices of a polyhedron in a symmetric way is called truncation. Definition The process of extending the edges of a polyhedron until they intersect to form a new polyhedron is called edge-stellation. The process of extending the faces of a polyhedron until they intersect to form a new polyhedron is called face-stellation. Exercise 19 From your original model, construct a new polyhedron by truncating, or stellating. How does the rotational group of the new polyhedron compare to the rotational group of the original model? Is there a subgroup relationship in general?

7 Exercise 20 Determine all of the groups that are possible as rotational groups of polyhedra. How can you prove that you have found the entire list? Pentagonal Pyramid Cyclic Group Z 5 General n-sided base Z n Octagonal Prism Dihedral Group D 8 General n-sided base D n Pentagonal Antiprism Dihedral Group D 5 General n-sided base D n Tetrahedron Alternating Group A 4 Cube Symmetric Group S 4 Dodecahedron Alternating Group A 5 VII. Buckyballs and Planar Graphs Exercise 21 Using a model, determine the group of rotations of the truncated icosahedron, known to chemists as Fullerene C 60 (shown at right), and known to most as a soccer ball. Truncated Icosahedron (Fullerene C 60 )

8 Exercise 22 Describe, possible generators for the group of rotations of the truncated icosahedron as permutations in S 60. Use the embedding in the plane below. 58 57 53 52 42 41 43 34 33 22 21 32 31 50 23 14 13 6 12 11 30 1 49 51 24 15 7 5 2 10 20 29 40 35 36 25 16 4 3 8 9 18 17 19 28 39 56 61 54 26 27 48 59 45 37 38 46 47 55 60 Planar Graph for the Truncated Icosahedron Drawn with Groups and Graphs Exercise 23 Determine another method for describing the group of rotations of the truncated icosahedron as a subgroup of a permutation group. Five Tetrahedra Drawn with Mathematica Exercise 24 Show that a tetrahedron may be constructed inside a dodecahedron so that the vertices of the tetrahedron are vertices of the dodecahedron. Does this mean that the tetrahedral group, A 4, is a subgroup of the icosahedral group, A 5? Exercise 25 Show that a perfect cube may be constructed inside a dodecahedron so that the vertices of the cube are vertices of the dodecahedron. Does this mean that the octahedral group, S 4, is a subgroup of the icosahedral group, A 5?

9 Project 2 Historical Proofs of Euler s Formula Project Construct convex polyhedra and investigate inductive methods for creating a sequences of polyhedra, which retain various graph theoretic properties. Goal To understand graph theoretic techniques for proving theorems including Euler s Formula. I. Euler s Formula for Convex Polyhedra Euler s Formula (for Convex Polyhedra) Let P be a convex polyhedron, and let v, e, and f denote, respectively, the numbers of vertices, edges, and faces of P. Then v e + f = 2. Exercise 1 Euler s first strategy for a proof (c. 1751) of his formula involved starting with a convex polyhedron and removing a vertex along with all of the edges and faces which adjoin it. New triangle faces are added over the hole that has been created. Show that the original polyhedron and the new polyhedron have the same value for v e + f. You may first want to consider the case where all of the faces of the original polyhedron are triangles. Exercise 2 By repeating the process above, one might expect that you will eventually decrease the number of vertices until you are left with four. These four vertices would determine a tetrahedron for which the desired result, v e + f = 2, is achieved. There is a flaw in this argument. Can you find it? Exercise 3 Cauchy s proof (1813) of Euler s formula involves projecting a convex polyhedron onto the plane. Show that the value of v e + f is the same for both the original polyhedron and its projection in the plane. Next, show that it is possible to add edges to the planar graph so that all of the faces are triangles and the value of v e + f does not change. Finally, show that v e + f = 2 for the planar graph with triangle faces. Exercise 4 Create several models of solids for which Euler s Formula does not hold. Euler s Formula (Von Staudt s version) Let P be a convex polyhedron such that i. any two vertices are connected by a path of edges, and ii. any closed curve on the surface separates P into two pieces. Then v e + f = 2. Exercise 5 Karl Von Staudt s version (1847) describes necessary and sufficient conditions for Euler s Formula to hold. To understand a proof of this theorem, construct a polyhedron and color the edges and faces in the following manner.

10 a. Choose any edge of the polyhedron and color it yellow along with the two vertices connected to it. Next choose an uncolored edge with one yellow and one uncolored vertex and color the new edge and vertex yellow. Continue this process until all of the vertices are colored yellow. Determine a relationship between the number of yellow edges and the number of yellow vertices. b. Show that the uncolored region consisting of the uncolored edges and the interiors of all the faces is connected. c. Choose a face and color it red along with the uncolored edges that bound it. Next choose an uncolored face that has exactly one red edge and color it red along with any uncolored edges that bound it. Show that this process will continue until all of the uncolored region of the polyhedron is colored red. Determine a relationship between the number of red edges and the number of red faces. d. Use the relationships for the vertices, edges, and faces of the polyhedron to show that v e + f = 2. e. Describe how Von Staudt s proof shows the dual relationship between vertices and faces. II. Planar Graphs of Convex Polyhedra Any convex polyhedron may be drawn as a planar graph. To visualize this, imagine a convex polyhedron with glass faces. Placing your eye close to one of the faces and peering inside will give you a clear vision of all of the vertices and edges of the polyhedron. This image you see could be projected onto a planar graph. Tetrahedron Cube Octahedron Dodecahedron Icosahedron Exercise 6 Sketch the planar graph of convex polyhedron which is not a Platonic solid. Exercise 7 Describe a nonconvex polyhedron which may be drawn as a planar graph. Exercise 8 Describe a nonconvex polyhedron which may not be drawn as a planar graph. Exercise 9 Can you describe the collection of all polyhedra to include the nonconvex polyhedra which may be drawn as planar graphs. Exercise 10 Determine a planar graph of a polyhedron for each of the possible groups of rotations. How can you prove that you have found the entire list?

11 Pyramid Cyclic Group Prism Dihedral Group Antiprism Dihedral Group Tetrahedron Alternating Group A 4 Cube Symmetric Group S 4 Dodecahedron Alternating Group A 5 III. Duality of Polyhedra and their Planar Graphs The concept of duality of polyhedra may be understood through a sequence of graphs. If P is a connected planar graph and P* then the dual graph of P, call it P*, can be constructed from P in the following manner. First, choose one point inside each face, including the face at infinity, of the planar drawing of P. These points are the vertices of P*. Next, for each edge of P, draw a line connecting the vertices of P* which lie on each side of the edge. These new lines are the edges of P*. That a tetrahedron is dual to itself is seen in the graphs below. P Vertices of P* Edges of P* Remove P Reshape P* The process works for any convex polyhedron as shown in the case of the truncated cube. Truncated Cube P Vertices of P* Edges of P* Remove P Reshape P*

12 Exercise 11 Use the planar method to find the dual of a convex polyhedron. IV. Planar Proofs of Euler s Formula Exercise 12 Devise a proof of Euler s Formula using a sequence of planar graphs in which the value for v e+ f remains the same. Two sequences of planar graphs are shown below. Icosahedron Remove a vertex Add edges Remove a vertex Add edges Remove a vertex Add edges Remove a vertex Remove a vertex Remove a vertex Remove a vertex Add edges Remove a vertex

13 This sequence may be viewed as the face at infinity being the ocean and the edges being dams. Planar graph, ocean in gray First dam removed Several dams removed Process continues Compartments disconnected Final stage V. Euler s Formula for Polyhedra which are not Convex Definition A sphere with g handles or equivalently, a torus with g holes is denoted as a surface of genus g. Definition be embedded. The genus of a graph is the smallest genus of a surface on which the graph can Euler s Formula Generalized Let G be a connected graph with genus g, and let v, e, and f denote, respectively, the numbers of vertices, edges, and faces in an embedding of G on a surface of genus g. Then v e + f = 2 2g. Pentagonal Torus Octagonal Quasi-Torus

14 Pentagonal Torus Drawn with Mathematica Torus Drawn with Mathematica Exercise 13 Describe why the value for v e + f is the same for both of the tori shown above. Exercise14 Using techniques similar to those used to prove Euler s Formula in the previous exercises, describe a method for proving the general version of Euler s Formula for the case where the genus of the graph is one. In other words, focus on polyhedral models, which are not planar but can be embedded on a torus. Exercise 15 Describe methods for proving the generalized version of Euler s Formula for n = 2,3,4,.

15 Project 3 Hypercubes and Graph Theory Project Construct hypercubes and planar and nonplanar graphs. Goal To use hypercube models to answer questions in graph theory and to understand methods of mathematical induction. Exercise 1 Sketch a 3-dimensional cube on paper as a square in a square. Construct a 4-dimensional hypercube as a cube in a cube. Describe a possible construction of a 5-dimensional hypercube. Cube Hypercube Exercise 2 Fill in the following chart. Dimension Model Vertices Edges Faces Solids 0 Point 1 Line Segment 2 Square 3 Cube 4 4D Hypercube Definition A sphere with g handles or equivalently, a torus with g holes is denoted as a surface of genus g. Definition The genus of a graph is the smallest genus of a surface on which the graph can be embedded.

16 Euler s Formula Let G be a connected planar graph, and let v, e, and f denote, respectively, the numbers of vertices, edges, and faces in a plane drawing of G. Then v e + f = 2. Euler s Formula Generalized Let G be a connected graph with genus g, and let v, e, and f denote, respectively, the numbers of vertices, edges, and faces in an embedding of G on a surface of genus g. Then v e + f = 2 2g. Exercise 3 Determine the genus for each of the cubes and hypercubes for dimensions 0, 1, 2, 3, 4, 5, 6,, n. Exercise 4 Construct a surface on which the 4-dimensional hypercube can be embedded. Exercise 5 Describe a solution for solving the Towers of Hanoi puzzle with four disks by looking at a Hamiltonian path on a 4-dimensional hypercube. Describe the method for solving the puzzle with n disks. According to legend, the original puzzle had 64 disks. With your method, how many moves would be required to solve the puzzle? A Hamiltonian Path on a Three-Dimensional Cube Exercise 6 Describe how your Hamiltonian path solution for the Towers of Hanoi puzzle may translated into a proof by mathematical induction. Imagination Topics Extending Ideas from Flatland to Higher Dimensions Exercise 7 Describe how a sphere passing through Flatland would appear to the Flatlanders. Describe how a hypersphere passing through 3-dimensional space would appear to its inhabitants.

17 Zome Quasi-Sphere Passing Through a Plane Exercise 8 In 3-dimensional space, it is possible to have two Flatland universes perpendicular to one another and which intersect in a line. Describe how Flatlanders in one universe appear to the Flatlanders in the other. In 4-dimensional space, is it possible to have two 3-dimensional universes perpendicular to one another and which intersect in a plane. How might the inhabitants look to one another?

18 References [1] Abbott, Edwin, Flatland: A Romance of Many Dimensions, Dover Publications, Inc., Dover Thrift Editions, 1992 (originally published in 1884). [2] Cromwell, Peter, Polyhedra, Cambridge University Press, 1997. [3] Gallian, Joseph, Contemporary Abstract Algebra, 4 th edition, Houghton Mifflin Co., 1998. [4] Rucker, Rudy, The Fourth Dimension: A Guided Tour of Higher Universes, Houghton Mifflin Co., 1984. [5] Weeks, Jeffrey, The Shape of Space: How to Visualize Surfaces and Three-Dimensional Manifolds by Jeffrey Weeks, Marcel Dekker Inc., 1985.