CHAPTER 6 TWO-COLORED WALLPAPER PATTERNS 6.0 ...

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p b ′ 1g pg′ p b ′ 1g Fig. 6.12 6.2.5 Are there any more pg types? The three look-alikes in figures 6.4, 6.9, and 6.11 represent the three types (pg, p b ′ 1g, pg′ , respectively) discussed so far in this section: they all have glide reflection in a single direction, and what makes them distinct is the effect of their glide reflections on color. Could there be other such types? Well, in the absence of rotations and reflections, the only other isometry that could split the three types into subtypes is translation. At first it looks like we could have three × two = six cases: three possibilities for glide reflection (color-preserving only (PP) or both color-preserving and color-reversing (PR) or color-reversing only (RR)), and two possibilities for translation (color-preserving only (PP) or both color-preserving and colorreversing (PR), see section 6.1). But a pattern’s glide reflections and translations are not independent of each other: as we will prove in section 7.4, and as you can see in figure 6.13 right below, a glide reflection (mapping A to B) and a translation (mapping B to C) combined produce another glide reflection (mapping A to C) parallel to the first one (G × T = G); moreover (see figure 6.21 further below), the combination of two parallel glide reflections of opposite vectors is a translation perpendicular to their axes (G × G = T).
Fig. 6.13 In view of these facts, the multiplication rules of 5.6.2 analyse the six cases mentioned above as follows: G(PP) × T(PP) = G(PP), G(PP) × G(PP) = T(PP): pg G(PP) × T(PR) = G(PR), G(PP) × G(PP) = T(PP): impossible G(PR) × T(PP) = G(PR), G(PR) × G(PR) = T(PR): impossible G(PR) × T(PR) = G(PR), G(PR) × G(PR) = T(PR): p′ b 1g G(RR) × T(PP) = G(RR), G(RR) × G(RR) = T(PP): pg′ G(RR) × T(PR) = G(PR), G(RR) × G(RR) = T(PP): impossible So, there are no more types in the pg family after all. Using the examples of this section you can certainly confirm that the only member of the pg family that has both kinds of translations (colorpreserving and color-reversing) is the one that has both kinds of glide reflections (p b ′ 1g), just as the above equations indicate. And an important byproduct of the entire discussion, quite useful to remember throughout this chapter, is this: in the presence of (glide) reflections, translations play no role at all when it comes to classifying two-colored wallpaper patterns; indeed a pattern with (glide) reflection has translations of both kinds if and only if it has both kinds of (glide) reflections! (Recall (1.4.8) that every reflection may be viewed as a special case of glide reflection.) 6.2.6 Symmetry plans. We capture the structure of the three pg types as follows:
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: eversing 90 0 rotation -- not to me
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 and 44: This is a demonstration of a signif
Page 45 and 46: pg′ g ′ pgg′ pgg′ Fig. 6.56
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
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Due to not-that-obvious color incon
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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
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twofold rotation at C, must both be
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p4g p4′ g ′ m p4g′ m ′ p4
Page 85 and 86:
Fig. 6.111 We see that the p4m may
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Fig. 6.113 pm′ m ′ 2 → p4m′
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6.12.3 Further examples. Our ‘tri
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consistent with color! As a consequ
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Predictably, fourfold centers prese
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must have the same effect on color
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Fig. 6.124 p31m′ 6.15 p3m1 types
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Fig. 6.126 p3m′
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p3m1 types. They are not that cruci
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and dots, respectively): Fig. 6.129
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6.17 p6m types (p6m, p6′ mm′ ,
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at the intersection of two reflecti
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Fig. 6.136 p6m′ m ′ In this exa
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Finally, some triangles inside the
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6.18 All sixty three types together
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cm cm′ p c ′ g p c ′ m (II) 1
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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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