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1 Colored symmetry Applications in: crystal surfaces & crystal growth strained materials twinning magnetic structures ferroelectrics other physical props textures (see Colored Symmetry by Shubnikov & Belov

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2 Colored symmetry Applications in: crystal surfaces & crystal growth strained materials twinning magnetic structures ferroelectrics other physical props textures Spin moment structure in MnO determined by neutron diffrn - only Mn +2 ions shown (see Colored Symmetry by Shubnikov & Belov

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3 Colored symmetry Applications in: crystal surfaces & crystal growth strained materials twinning magnetic structures ferroelectrics other physical props textures Will consider here only two colors - black & white (see Colored Symmetry by Shubnikov & Belov

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4 Colored symmetry Anti-equality – Pair of gloves which are, say, black on outside & white on inside if both left-handed, can get: congruent equality - do nothing mirror anti-equality - turn one inside-out

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5 Colored symmetry Anti-equality – Pair of gloves which are, say, black on outside & white on inside if both left-handed, can get: congruent equality - do nothing mirror anti-equality - turn one inside-out if right- & left-handed, can get: mirror equality - nothing congruent anti-equality - turn one inside-out

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6 Colored symmetry New operations – anti-rotations (rotation + "sign change") 1' 2' 3' 4' 6' mirror anti-rotations (mirror rotation + "sign change") 1' 2' 3' 4' 6'

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7 Colored symmetry New operations – anti-rotations (rotation + "sign change") 1' 2' 3' 4' 6' mirror anti-rotations (mirror rotation + "sign change") 1' 2' 3' 4' 6' c 11 c 12 c 13 0 c 21 c 22 c 23 0 c 31 c 32 c 33 0 0 0 0 ±1

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8 Colored symmetry Polar & neutral figures – polar white outside, black inside black outside, white inside neutralblack inside & outside or white inside & outside

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9 Colored symmetry Operations – mirror in xy plane - anti-symmetric 2' anti-reflection 1 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 -1

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10 Colored symmetry Operations – anti-rotation of 180° along x-axis 2' 1 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 -1

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11 Colored symmetry Operations – anti-inversion 1' -1 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 -1

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12 Colored symmetry Operations – 4'

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13 Colored symmetry Operations – 4' 6' black underneath

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14 Colored symmetry Operations – 2/m' black underneath

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15 Colored symmetry Operations – 2/m' 2'/m white underneath black underneath

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16 Colored symmetry Operations – 2'/m' white underneath black underneath

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17 Colored symmetry Operations – 2'/m' 2'm'm white underneath black underneath

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18 Colored symmetry Operations – 2m'm'

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19 Colored symmetry Operations – 2m'm' 2'2'2

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20 Colored symmetry Operations – 2/mm'm' white underneath black underneath

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21 Colored symmetry Glides – g g' m=a' a=m'

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22 The 36 lattice translations for colored groups

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23 The 36 lattice translations for colored groups

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24 The 46 dichromatic plane groups

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25 The 46 dichromatic plane groups

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26 The 46 dichromatic plane groups

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27 The 46 dichromatic plane groups

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28 Comments 1. Mirror translation ( ) – gives another mirror of same character if uncolored. If colored, new mirror is colored 2. Mirror translation (||) – colored translation gives colored glide

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29 Comments 1. Mirror translation ( ) – gives another mirror of same character if uncolored. If colored, new mirror is colored 2. Mirror translation (||) – colored translation gives colored glide 3. Axis translation ( ) – new axis has same charater if uncolored. If colored, new axis is colored or uncolored, depending on the axis. Character is unchanged by colored translation 4. Axis translation (||) – colored translation makes axis colored screw or uncolored, depending on the axis

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30 Comments 4.Axis translation (||) – colored translation makes axis colored screw or uncolored, depending on the axis 2 t || ' ––> 2 & 2 1 ' 2 1 t || ' ––> 2 1 & 2'

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31 Comments 4.Axis translation (||) – colored translation makes axis colored screw or uncolored, depending on the axis 2 t || ' ––> 2 & 2 1 ' 2 1 t || ' ––> 2 1 & 2' 4 t || ' ––> 4 & 4 2 ' 4 1 t || ' ––> 4 1 & 4 3 '

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32 Comments 5. n = odd axes cannot be black-white 6. Two intersecting 2-fold axes: 30° ––> new 645° ––> new 460° ––> new 3etc. new axis uncolored if 2-folds are same color…….otherwise, new axis is colored uncolored 4 colored 4

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17-plane groups When the three symmetry elements, mirrors, rotation axis and glide planes are shown on the five nets, 17-plane groups are derived.

17-plane groups When the three symmetry elements, mirrors, rotation axis and glide planes are shown on the five nets, 17-plane groups are derived.

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