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8 Conway s Magic Theorem

9 Session Description This session will look at the mathematics behind Conway's Magic Theorem. This theorem can be used to prove that there are 17 plane symmetry (wallpaper) groups as well as a host of other things. In this session you will encounter orbit manifolds (orbifolds) and learn how Conway's notation works. This session is designed to provide ideas that can be used to enrich the understanding of the most able A level mathematicians.

10 The 17 plane symmetry groups

11 How do you show that there are exactly 17 plane symmetry groups? Conway s magic theorem Each symmetry has a score. The overall score for a group is found using the minimum quantity of symmetries required to create that group starting with rotations. For a group to be a plane symmetry group, the total score for those symmetries has to equal 2 Q: Is this anything to do with V E + F = 2?

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13 A quick example: p31m What is the minimum quantity of symmetries required to create that group (starting with rotations). Start with the rotation that is not on a mirror Now look at a rotation that is on a mirror There s no point in using any of the other centres of rotation as it would leave gaps Now look at one of the mirrors Nothing else is needed.

14 A quick example: p31m

15 We now have a combinatorics question How many ways are there of making a total of 2 if there is a) b) c) d) e) No rotation symmetry The highest order of rotation symmetry is 2 The highest order of rotation symmetry is 3 The highest order of rotation symmetry is 4 The highest order of rotation symmetry is 6 We know these are the only options due to the crystallographic restriction

16 Highest rotation symmetry order Available scores None 2 translation only 1 reflection 1 glide reflection

17 No rotation symmetry Possibilities are 1) 2) 3) 4) Only translation symmetry Two mirror lines Two glide reflection lines One mirror line and one glide reflection line 4 different ways 2 1+1=2 1+1=2 1+1=2

18 Order 2 rotation symmetry

19 Order 3 rotation symmetry

20 Order 4 rotation symmetry

21 Order 6 rotation symmetry Giving the total number of possibilities as = 17

22 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) Only translation symmetry Two mirror lines Two glide reflection lines One mirror line and one glide reflection line 4 different o2 centres not 2 different o2 not and 1 mirror 2 different o2 not and 1 glide 1 o2 not and 2 o2 on and 1 mirror 4 different o2 on and 1 mirror 3 different o3 centres not 1 o3 not, 1 o3 on and 1 mirror 3 different o3 all on and 1 mirror 2 different o4 centres not and 1 o2 not 1 o4 centres not, 1 o2 on and 1 mirror 2 o4 centres on, 1 o2 on and 1 mirror 1 o6 not, 1 o3 not and 1 o2 not 1 o6 on, 1 o3 on, 1 o2 on and 1 mirror p1 pm pg cm p2 p2mg p2gg c2mm p2mm p3 p31m p3m1 p4 p4mg p4mm p6 p6mm

23 But why a score of 2? and why does each symmetry have the score it does? A more fundamental question What do we mean by symmetry?

24 Geometrical congruence That puts an object back where it started Consider a chair Symmetry group Identity Reflection

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26 Federov Schoenflies

27 2 Dimensional Lattices Parallelogrammatic Square Rectangular Rhombic Hexagonal

28 Murray MacBeath William Thurston

29 The four fundamental features 4 features are sufficient to describe all the symmetries of any pattern Wonders Gyrations Kaleidoscopes Miracles Blue: preserves the true orientation of a fundamental region Red: reflect a fundamental region in some way

30 Kaleidoscopes A mirror or kaleidoscopic symmetry Denoted by * * ** (pm)

31 *2 *3

32 *2222

33 Gyrations Looks like *333 but 3 fold rotation and one mirror. 3*3 One kind of 3 fold gyration point and one kind of 3 fold kaleidoscopic point

34 2*22 4*2

35 333

36 Miracles A glide reflection Denoted by Looks like ** but there s another mapping *

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38 Wonders No reflections, gyrations or miracles! Denoted by O

39 Everything has a cost Gyrations and Wonders Symbol Cost Kaleidoscopes and Miracles Symbol O * or N N Cost

40 * *632 Total 2

41 1. Mark any kaleidoscopes in red. If there are mirror lines, consider only one of the regions that they cut the plane into. Put a red * near any one kaleidoscope. Find one corner of each type and write the numbers of mirrors through each of these corners in red. 2. Look for gyration points. Mark one gyration point of each type present in blue with its order. Gyrations never lie on mirror lines. 3. Look for miracle points can you get from a point to a copy of that point without crossing a mirror line. Mark one of the paths with a broken red line and put a cross nearby. 4. If you have found none of the above, mark it with a wonder symbol.

42 The 17 Symmetry Groups

43 O = 2 (obviously) 5 types

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45 ** = = 2 (obviously) 5 types

46 A mixture of gyrations and kaleidoscopes Since 2 = 2 2, 3 = 2 3, 4 = 2 4 and 6 = 2 6, we can replace *nn with n*

47 *632 yields no new results *333 yields 3*3 *442 yields 4*2 *2222 yields 2*22 and 22* 4 types

48 Including miracles We can replace * with where there is a single kaleidoscope. So 22* yields 22 ** yields * and 3 types Total = = 17 types

49 Topological argument

50 The orbifold The orbifold is a hemisphere

51 The topology of the orbifold determines the symmetry group The members of the group are geometrical congruences (i.e. distance preserving) Distance is not invariant in topological terms Geometrical groups are determined by their orbifolds Perelman used this in proving the Poincare Conjecture

52 The Poincaré Conjecture Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

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54 2 *22 * 2

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56 4 * 4 *44

57 4 * 3 *432 2

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59 4 fold cone point 3 fold cone point 2 fold cone point 432

60 So why 2?

61 Any 2 dimensional surface can be obtained from a sphere by Punching holes introducing a boundary * Adding handles O Adding crosscaps (this replaces disks with Moebius bands)

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63 The effect of punching a hole before after Punching a hole reduces the Euler characteristic by 1

64 The effect of an N-fold cone point Using a map where the cone point is a vertex

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66 The effect of an N-fold corner point

67 Where are we now? We have shown how gyrations and kaleidoscopes affect the Euler characteristic. So we can perform surgery on a sphere (adding cone points and mirrors) to create all but 3 of the wallpaper patterns. What about wonders and miracles? What do we have to do to a sphere to create them?

68 Wonders (O)

69 A sphere to a torus

70 The Euler Characteristic of a Torus A wonder O reduces the characteristic by 2

71 Miracles

72 Crosscaps

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74 The effect of a crosscap - a (slight) cop-out To make a crosscap requires a hole. A hole reduces the Euler characteristic by 1 Each crosscap reduces the Euler characteristic by 1

75 A summary of the proof of the Magic Theorem Any pattern can be folded-up into an orbifold by taking all points of the same kind (orbits) to a single point. The 17 wallpaper symmetry patterns map one-to-one with the orbifolds that have a Euler characteristic of 0 (as do the 7 frieze patterns). Every orbifold can be obtained by surgery on a sphere i.e. adding cone points, holes (mirror lines), corner reflectors, handles and cross-caps.

76 A summary of the proof of the Magic Theorem Each feature reduces the Euler characteristic of the sphere by a certain amount (the cost) The Euler characteristic of a sphere is 2 The orbifolds have to have an Euler characteristic of 0 so the combination of fundamental features has to reduce the Euler characteristic of the sphere by 2.

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