CHAPTER 6 TWO-COLORED WALLPAPER PATTERNS 6.0 ...

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The visual difference between the ‘two kinds’ of glide reflection axes is much more clear than the one in the example discussed in 6.2.1, so it is even less surprising that one kind of axes preserve colors while the other kind reverse colors. 6.2.3 Hunting for the third type. As we have seen in 5.2.1 and 5.5.1, it is possible to have both kinds of vertical reflection axes or half turn centers reverse colors in a two-colored border pattern. Therefore it is very reasonable to expect to have patterns in the pg family where all glide reflection axes reverse colors. Could a coloring of the familiar p4g pattern of figure 4.57 produce such an example? Well, a closer look at the NW-SE glide reflection of the p b ′ 1g pattern in figure 6.3 suggests this attempt: Fig. 6.10 Indeed all NW-SE glide reflections reverse colors. But so do the NE-SW glide reflections (which were in fact inconsistent with color in figure 6.3)! Could such a pattern ever belong to the pg family? As we will point out in sections 7.2, 7.9, and 7.10, and as you may already have observed in chapter 4, whenever a pattern has reflection and/or glide reflection in two distinct directions it must also have rotation: indeed our pattern above has color-
eversing 90 0 rotation -- not to mention its color-preserving reflections and glide reflections -- and it belongs to the p4g family (see 6.11.2). The lesson drawn out of this example is that some times we get more symmetry than desired, especially when we try to ‘hide’ a rich underlying structure by way of coloring. This is a lesson worth remembering, but what about our original quest for a pg kind of pattern with color-reversing glide reflection only? Well, perhaps it is time to be less adventurous, avoid ‘structural traps’, and look for a more down-to-earth example; not that it is the simplest way out, but, once again, a modification of the pg pattern in figure 6.4 works: Fig. 6.11 pg′ Patterns such as the one in figure 6.11 are known as pg′ . 6.2.4 Examples. These colorings should be compared to the p1 colorings employed in figure 6.7:
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: otations of both kinds, the corresp
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 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
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′
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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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