CHAPTER 6 TWO-COLORED WALLPAPER PATTERNS 6.0 ...

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translation which must be color-preserving (P × P = P); but the same translation is produced by combining the corresponding hidden glide reflections (G 1 followed by G 2 ) of the shown opposite vectors, hence it has to be color-reversing (P × R = R), too! The contradiction we have arrived at shows that there cannot possibly be any pm-like two-colored wallpaper patterns other than the six types already derived in 6.3.1. 6.3.3 Examples. You should pay special attention to the sixth example, which should belong to the cm family but is in fact a p b ′ 1m because its in-between glide reflection is inconsistent with color: c′ m pm′ p b ′ 1m Fig. 6.22 p′ m p b ′ g p b ′ 1m
6.3.4 Translations and hidden glide reflections revisited. The examples in 6.3.1 and 6.3.3 make ‘visually obvious’ the fact that there always exists a glide reflection employing the same axis as any given reflection. Such a ‘hidden’ glide reflection exists because a translation parallel to the reflection axis is always there (just as in the case of p1m1 and pmm2 border patterns); and it is easy to see that the hidden glide reflection’s minimal gliding vector is always equal to the minimal translation vector along the reflection axis. But why should such a parallel translation be there, after all? The double application of every glide reflection produces a parallel translation of vector twice as long as the gliding vector (2.4.2, 5.4.1), but why should a ‘vectorless’ reflection carry the obligation to produce a translation parallel to itself? This is best explained through a ‘proof without words’: Fig. 6.23 [Since every wallpaper pattern has translations in at least two directions, pick one in a direction non-perpendicular to that of the reflection axis; then subsequent application of reflection (A to B), translation (B to C), reflection (C to D), and translation (D to E) produces a parallel to the reflection translation (mapping A to E)!]
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: produced six pm-like two-colored wa
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′
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
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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
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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