CHAPTER 6 TWO-COLORED WALLPAPER PATTERNS 6.0 ...

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described in this section. Indeed if we view a cm two-colored pattern as a ‘merge’ of a pm pattern (reflections) and a pg pattern (glide reflections), we see that there are only two possibilities for each ‘partner’: only pm, pm′ for pm (6.4.4 rules out p b ′ 1m and c′ m , while 6.4.3 rules out p′m and p b ′ g ) and only pg, pg′ for pg (p b ′ 1g is ruled out by 6.4.4). But two × two = four, and we can certainly write down the new (cm) types as ‘products’ of the old ones (pm, pg): cm = pm × pg, cm′ = pm′ × pg′ , p c ′ g = pm′ × pg, p c ′ m = pm × pg′ Of course this ‘multiplication’ was first introduced in section 5.7, where we viewed pmm2s as ‘products’ of pm11s and p1m1s. 6.4.6 Further examples and symmetry plans. Fig. 6.37 cm′ p c ′ g p c ′ m cm cm′
p c ′ g p c ′ m R P R P R P R P R P R P R P Fig. 6.38 6.5 p2 types (p2, p2′ , p b ′ 2 ) 6.5.1 A good place to start! ‘Experimenting’ a bit with the p2 pattern in figure 6.5, we get a couple of ‘genuine’ two-colored ones: Fig. 6.39 p2′
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: pattern ‘look the same’. It’s
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′
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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