CHAPTER 6 TWO-COLORED WALLPAPER PATTERNS 6.0 ...

More documents

Recommendations

Info

Reflections reverse colors, hidden glide reflections preserve colors (g); there exists color-reversing translation along the reflection axes (p b ′ ). Such patterns are known as p b ′ g . Fig. 6.17 pg′ → p ′ m Once again we get color-reversing translation along the reflection axes (p′), all of which preserve colors (m): the new pattern is known as p′m . Comparing the two patterns in figures 6.15 & 6.17 we see that they are similar not only in name, but in structure as well; in fact the only thing that makes them distinct is that the p′ m has colorreversing translation while the pm doesn’t. But didn’t we promise back in 6.2.5 that “in the presence of (glide) reflection translation will play no role in the classification process”? Well, there is indeed another, more subtle way of distinguishing between pm and p′ m , and that is their hidden (‘old’) glide reflection, which of course preserves colors in the case of the pm (an ‘offspring’ of pg) but reverses colors in the case of the p′m (an ‘offspring’ of pg′ )!
Fig. 6.18 pg′ → pm′ All reflections and hidden glide reflections in this pm′ pattern do reverse colors (m′ ). Notice the absence of color-reversing translation. Fig. 6.19 p b ′ 1g → c′ m
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: Fig. 6.15 pg → pm All reflections
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′
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
show all

CHAPTER 6 TWO-COLORED WALLPAPER PATTERNS 6.0 ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?