CHAPTER 6 TWO-COLORED WALLPAPER PATTERNS 6.0 ...

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Is the half turn (at) K color-preserving or color-reversing? In view of K = G 1 ∗G (G applied upwards (P) followed by G + (P)) and 1 K = G 2 ∗G (G applied downwards (P) followed by G − (R)) we conclude 2 that the half turn at K must be both color-preserving and colorreversing; that is, the situation featured in figure 6.55 (‘mixed’ horizontal axes) is impossible. We conclude that each of the two pg-like ‘factors’ of a pgg-like pattern could be either a pg or a pg′ , but not a p b ′ 1g. This should allow for four possibilities, but since the outcome of this ‘multiplication’ is not affected by the order of ‘factors’, we are down to three types: pgg = pg × pg, pgg′ = pg × pg′ = pg′ × pg, pg′ g ′ = pg′ × pg′ 6.6.3 Another way of looking at it. The discussion in 6.6.2 was very useful in terms of analysing the structure of the pgg pattern, but it is certainly not the easiest way to see that any two of its glide reflections parallel to each other must have the same effect on color. Indeed that follows at once from our Conjugacy Principle (6.4.4): every two adjacent parallel axes are mapped to each other by any half turn center lying half way between them! It might be a good idea for you to see how the Conjugacy Principle works in this special case, though: you should be able to provide your own proof, arguing in the spirit of figure 6.36. In another direction now, let’s revisit the pgg example of 4.8.3 and figure 4.43. We state there, with the Conjugacy Principle in mind (4.11.2), that it appears that there are two kinds of glide reflection axes in both directions: our reservations are now further justified by the impossibility of coloring that pattern in such a way that any two parallel glide reflections would have opposite effect on color! 6.6.4 Further examples. First, three pgg-like ‘triangles’ that you should compare to the p2-like patterns of figure 6.46:
pg′ g ′ pgg′ pgg′ Fig. 6.56 The ‘proximity’ between the two families (pgg and p2) is further outlined through the following curious example of a pg′ g ′ that is a close relative of the p2 example in figure 6.5: Fig. 6.57 pg′ g ′ This is an example that many would classify as a p2: after all, the rotations of both the pg′ g ′ and the p2 are color-preserving only. Well, the advice offered in 4.8.2 remains valid: after you locate all (hopefully!) the rotation centers, check for ‘in-between’ diagonal glide reflection! Instead of applying this ‘squaring’ process to the p2′ in figure 6.39 for a pgg′ , we offer a fancy pmg-generated pgg′ :
Page 1 and 2: © 2006 George Baloglou first draft
Page 3 and 4: Fig. 6.3 In the pattern of figure 6
Page 5 and 6: 6.0.4 Symmetry plans and types. Tri
Page 7 and 8: 6.1.2 Infinitely many color-reversi
Page 9 and 10: otations of both kinds, the corresp
Page 11 and 12: eversing 90 0 rotation -- not to me
Page 13 and 14: Fig. 6.13 In view of these facts, t
Page 15 and 16: Fig. 6.15 pg → pm All reflections
Page 17 and 18: Fig. 6.18 pg′ → pm′ All refle
Page 19 and 20: produced six pm-like two-colored wa
Page 21 and 22: 6.3.4 Translations and hidden glide
Page 23 and 24: 6.4 cm types (cm, cm′ , p c ′ g
Page 25 and 26: Fig. 6.28 pm′ → cm′ Fig. 6.29
Page 27 and 28: Fig. 6.31 p b ′ 1 Fig. 6.32 p b
Page 29 and 30: So, while our first two colorings i
Page 31 and 32: pattern ‘look the same’. It’s
Page 33 and 34: p c ′ g p c ′ m R P R P R P R P
Page 35 and 36: Fig. 6.42 pg′ → p2′ Fig. 6.43
Page 37 and 38: proof will be given in section 7.6.
Page 39 and 40: Fig. 6.48 pg → pgg Fig. 6.49 pg
Page 41 and 42: So far so good: we obtained four tw
Page 43: This is a demonstration of a signif
Page 47 and 48: You should compare these pgg symmet
Page 49 and 50: Fig. 6.61 pg → pmg Fig. 6.62 pg
Page 51 and 52: last two types, offspring of p b
Page 53 and 54: Fig. 6.68 p b ′ mg 6.7.4 Symmetry
Page 55 and 56: Fig. 6.70 pmg → pmm Somewhat conf
Page 57 and 58: The two patterns in figures 6.72 &
Page 59 and 60: Fig. 6.77 p b ′ mm → c′ mm Wh
Page 61 and 62: p b ′ gm p b ′ mm p b ′ gm Fi
Page 63 and 64: Fig. 6.80 pmg → pmm → cmm Fig.
Page 65 and 66: Fig. 6.84 p b ′ gg → p b ′ gm
Page 67 and 68: Due to not-that-obvious color incon
Page 69 and 70: p c ′ mm p c ′ mg cm′ m ′ F
Page 71 and 72: Fig. 6.93 pm′ Fig. 6.94 cmm′
Page 73 and 74: Fig. 6.97 p b ′ gm You should be
Page 75 and 76: 6.10 p4 types (p4, p4′ , p c ′
Page 77 and 78: Fig. 6.101 p4′ Fig. 6.102 p c ′
Page 79 and 80: 6.10.5 Symmetry plans. We use ‘st
Page 81 and 82: twofold rotation at C, must both be
Page 83 and 84: p4g p4′ g ′ m p4g′ m ′ p4
Page 85 and 86: Fig. 6.111 We see that the p4m may
Page 87 and 88: Fig. 6.113 pm′ m ′ 2 → p4m′
Page 89 and 90: 6.12.3 Further examples. Our ‘tri
Page 91 and 92: consistent with color! As a consequ
Page 93 and 94: Predictably, fourfold centers prese
Page 95 and 96:
must have the same effect on color
Page 97 and 98:
Fig. 6.124 p31m′ 6.15 p3m1 types
Page 99 and 100:
Fig. 6.126 p3m′
Page 101 and 102:
p3m1 types. They are not that cruci
Page 103 and 104:
and dots, respectively): Fig. 6.129
Page 105 and 106:
6.17 p6m types (p6m, p6′ mm′ ,
Page 107 and 108:
at the intersection of two reflecti
Page 109 and 110:
Fig. 6.136 p6m′ m ′ In this exa
Page 111 and 112:
Finally, some triangles inside the
Page 113 and 114:
6.18 All sixty three types together
Page 115 and 116:
cm cm′ p c ′ g p c ′ m (II) 1
Page 117 and 118:
cmm cm′ m ′ cmm′ p c ′ mm p
Page 119 and 120:
p4m p4′ mm′ p4′ m ′ m p4m
Page 121:
p6m p6′ m ′ m p6m′ m ′ p6
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CHAPTER 6 TWO-COLORED WALLPAPER PATTERNS 6.0 ...

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