TBe/v/ve ~J0 . ... .,. by Gareth Boreland It is well known to students of basic geometry that the join to form a dodecahedron - we will see later why there sum of the interior angles in a polygon with V vertices is must be 12. (It is also interesting to note that the resulting solid has 20 vertices, each with an angular defect of 36', (V - 2) x 180'. Regular polygons will tessellate the plane if their interior angle divides 360'. Thus the regular polygons giving a total defect of 720' - this is also discussed later.) shown in the table below will tessellate. Let q be the number of polygons meeting at each vertex of the tessellation. Regular polygon Interior angle q equilateral triangle 600 6 square 900 4 regular hexagon 120 0 3 The dodecahedron Other regular polyhedra can be formed by 'closing the gaps' in a similar way with other regular polygons. Let q be the number of the regular polygons meeting at each vertex. Six equilateral triangles meet at a vertex to tessellate the plane; the table shows the polyhedra formed when fewer than six equilateral triangles meet at each vertex and the 'gaps are closed': Regular hexagons will tessellate the plane q Structure formed {p,q} 6 tessellation of plane Bright students will see immediately that since, after 120, 5 icosahedron {3,5} the only remaining factors of 360 are 180 and 360, it is clear 4 octahedron {3,41 that no other regular polygons will tessellate. 3 tetrahedron {3,31 For example, in a regular pentagon each interior angle is 1080. Now 360 = 3 x 108 + 36. So in attempting to tessellate regular pentagons, we find that three will fit together, but leaving a 36' 'gap' or 'angular defect': The icosahedron We can 'close the gap' by lifting two pentagons out of the flat plane. By allowing them to fit together in this way, and The octahedron continuing this '3-D tessellation' we find that 12 pentagons 8 Mathematics in School, November 2007 The MA web site www.m-a.org.uk Cutting the corners off (i.e. truncating) an icosahedron yields the traditional 'football' shape (a truncated icosahedron) as shown: The tetrahedron The symbol {p,q} (called the Schlifli symbol) is used for a polyhedron in which q regular p-sided polygons meet at each The centres of the faces of a cube form the vertices of a vertex. regular octahedron, and vice versa. These two solids thus have the same symmetries. Similarly for squares: A similar duality exists between the icosahedron and the dodecahedron. q Structure formed {p,q} 4 tessellation of plane One of the most remarkable results concerning polyhedra 3 cube {4,31 is surely Euler's Theorem. This states that, for any convex polyhedron with F faces, V vertices and E edges, F + V = E + 2. It can be instructive for pupils to investigate this relationship and try to 'discover' it for themselves. For the more complicated solids, pupils can use models to help them count. Even then it is tricky, merely by counting, to find, for instance, E for an icosahedron; pupils can be encouraged to come up with mathematical relationships that 20 x 3 will help them - e.g. in this instance E-=2 The cube 2 Prisms and pyramids can be investigated easily and, once For completeness we tabulate the result noted already for models or diagrams of specific cases have been used, regular pentagons: students can be encouraged to find E, F and V for prisms and pyramids with n-gonal bases: (Note q cannot be 4 or more, since 4 x 108 > 360; there will be no 'gaps' to close.) E F V n-gonal based prism 3n n + 2 2n q Structure formed {p1q} n-gonal based pyramid 2n n + 1 n + 1 3 dodecahedron {5,3} and finally for regular hexagons: q Structure formed {p,q} 3 tessellation of plane Clearly no other regular convex polyhedra can be constructed in this way - we require q to be at least 3, but for n-gons with n>6, 3 x interior angle > 360'. It is instructive to see if students can be prompted to give similar arguments. So we have found the five regular convex polyhedra, which have been studied by mathematicians since about 350 BC and which are known, of course, as the Platonic solids. Apentagonal prism (E = 15, F = 7 and V = 10) Other authors have spoken of the benefits to learners from working with 3-D models of geometric results to be studied (Olkun and Sinoplu, 2006) and I would agree that perhaps the best way for students to understand the results described above is to see, or even to build, models of them. Indeed, many geometric facts concerning polyhedra are most easily demonstrated by models; in fact some are difficult to visualize without them. As examples, consider the following results, each of which can grasp the attention and interest of students: SA cube has order 3 rotation symmetry about a body A pentagonal pyramid (E = 10, F = 6 and V = 6) diagonal. Mathematics in School, November 2007 The MA web site www.m-a.org.uk 9 Many proofs of Euler's Theorem exist. Tony Gardiner 180,(nF -2F )=360V- 720 (2003) presents a proof using spherical geometry. See also n David Eppstein's web page Nineteen Proofs of Euler's (multiplying by 360 and rearranging). Theorem (wwwl.ics.uci.edu/- eppstein/junkyard/euler/) which gives proofs based mostly on topology and graph Thus S=180oF (n-2)= (V-2)x360. n theory. There are proofs, however, which are accessible to school students - see, for instance, www.wikipedia.org/wiki/ Tony Gardiner (2003) uses spherical geometry, rather than a Euler characteristic. prior assumption ofE + 2 = F + V, to derive results which can be shown to be related closely to this. He proceeds to run The theorem can be used to prove some of the results which a similar argument in reverse to prove Euler's Theorem. were merely stated before. For example, it was noted that 12 regular pentagons form the faces of a dodecahedron. Euler's (It can also be noted that the result can be generalized for all Theorem can be used to prove that the polyhedron {5,3} polyhedra as S = (V- X) x 3600 where X = V + F - E is the must have 12 faces. Euler characteristic of the polyhedron.) In this case we must have E = 5_ and V = 5F (because two pentagons meet along each polyhedron edge and three References pentagons meet at each polyhedron vertex). Gardiner, T. 2003 'The Euler Descartes Theorem'. In Pritchard, C. (Ed.) The Changing Shape of Geometry, Cambridge University Press, pp. Then, using F + V = E + 2, 333-338. 5F 5F Olkun, S. and Sinoplu, B. 2006 'Internalizing Three-Dimensionality we obtain F+ 3F= --+2 Through Making Toys', Mathematics in School, 35, 5. 3 2 and so F = 12. I conclude with a result that may not be so well known. We have mentioned that in any polygon the sum of the interior Keywords: Polyhedra; Euler; Platonic solids. angles is given by (V - 2) x 180'. An analogous result exists for the sum of all the face angles in a polyhedron. Author Gareth Boreland, Sullivan Upper School, Holywood, Co. Down BT18 9EP. Theorem e-mail: [email protected] In any convex polyhedron the sum of all the face angles, S, is given by S = (V - 2) x 360', where V is the number of vertices. This result can be rewritten as 3600x V- S = 720'. This shows that it is completely equivalent to Descartes' law of angular defect. The angular defect at a vertex is defined to be From Little Acorns 360' minus the sum of the face angles at that vertex. by Lesley Ravenscroft, Diana Cobden, Descartes' law states that the sum of the angular defects at all Colin Abell and Elaine Griffin the vertices of a convex polyhedron is 720'. We saw this earlier in the case of a regular dodecahedron. I hope the way The purpose of From Little in which the result is stated here makes it more accessible to ettle Acorns is to share with teachers students. in Primary, Middle and Secon- dary schools some of our ideas ra 9 Proof 0" about the uses of Spreadsheets to enhance the teaching of Mathe- Let such a polyhedron have F faces, E edges and V vertices. matics. Spreadsheets 9 to 13 We especially wished to support Let Fn be the number of n-gonal faces. winu~irr r~ru~iur the inexperienced Maths / I.T. C~Z~L~ 1 teacher. And those interested in Thus F = Fn andE =- 1nFn The Mathematical developing the use of I.T. to n 2 n Association support pupils' learning. (The 1 is required because two polygons meet at each polyhe2dron edge.) Members Price 07.59 S=-YF x1800(n-2)=180 F (n - 2). Non Members Price 010.99 ISBN: 0 906588 36 7 n n Now, the sum of the face angles on each n-gonal face is MATHEMATICAL ASSOCIATION Order from: (n - 2) x 180'. Thus the sum of all the face angles is given by The Mathematical Association Euler's Theorem is then applied: 259 London Road Leicester LE2 3BE Telephone 0116 2210014 E+2-F+VI E + 2 = F + V Email [email protected] or order Online at www.m-a.org.uk supporting mathematics in education - (nFn)+2=i (F)+V 2 n n n 10 Mathematics in School, November 2007 The MA web site www.m-a.org.uk.