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Symmetry ElementsSymmetry Elements
Lecture 5Lecture 5
SymmetrySymmetry
MotifMotif: the fundamental part of a symmetric : the fundamental part of a symmetric design that, when repeated, creates the design that, when repeated, creates the whole patternwhole pattern
OperationOperation: some act that reproduces the : some act that reproduces the motif to create the patternmotif to create the pattern
ElementElement: an operation located at a : an operation located at a particular point in spaceparticular point in space
2-D Symmetry2-D Symmetry
Symmetry ElementsSymmetry Elements
1. Rotation1. Rotationa. Two-fold rotationa. Two-fold rotation
= 360= 360oo//22 rotation rotation
to reproduce a to reproduce a motif in a motif in a symmetrical symmetrical patternpattern
6
6
A Symmetrical PatternA Symmetrical Pattern
Symmetry ElementsSymmetry Elements
1. Rotation1. Rotationa. Two-fold rotationa. Two-fold rotation
= 360= 360oo//22 rotation rotation
to reproduce a to reproduce a motif in a motif in a symmetrical symmetrical patternpattern
= the symbol for a two-fold = the symbol for a two-fold rotationrotation
Motif
Element
OperationOperation
6
62-D Symmetry2-D Symmetry
6
6
first operation step
2-D Symmetry2-D Symmetry
Symmetry ElementsSymmetry Elements
1. Rotation1. Rotationa. Two-fold rotationa. Two-fold rotation
= 360= 360oo//22 rotation rotation
to reproduce a to reproduce a motif in a motif in a symmetrical symmetrical patternpattern
= the symbol for a two-fold = the symbol for a two-fold rotationrotation
6
6
first operation step
second operation step
2-D Symmetry2-D Symmetry
Symmetry ElementsSymmetry Elements
1. Rotation1. Rotationa. Two-fold rotationa. Two-fold rotation
= 360= 360oo//22 rotation rotation
to reproduce a to reproduce a motif in a motif in a symmetrical symmetrical patternpattern
= the symbol for a two-fold = the symbol for a two-fold rotationrotation
Symmetry ElementsSymmetry Elements
1. Rotation1. Rotationb. Three-fold rotationb. Three-fold rotation
= 360= 360oo//33 rotation rotation
to reproduce a motif to reproduce a motif in a symmetrical in a symmetrical patternpattern
66
6
2-D Symmetry2-D Symmetry
6
6
step 1
2-D Symmetry2-D Symmetry
Symmetry ElementsSymmetry Elements
1. Rotation1. Rotationb. Three-fold rotationb. Three-fold rotation
= 360= 360oo//33 rotation rotation
to reproduce a motif to reproduce a motif in a symmetrical in a symmetrical patternpattern
6
66
step 1
step 2
2-D Symmetry2-D Symmetry
Symmetry ElementsSymmetry Elements
1. Rotation1. Rotationb. Three-fold rotationb. Three-fold rotation
= 360= 360oo//33 rotation rotation
to reproduce a motif to reproduce a motif in a symmetrical in a symmetrical patternpattern
6
66
step 1
step 2
step 3
2-D Symmetry2-D Symmetry
Symmetry ElementsSymmetry Elements
1. Rotation1. Rotationb. Three-fold rotationb. Three-fold rotation
= 360= 360oo//33 rotation rotation
to reproduce a motif to reproduce a motif in a symmetrical in a symmetrical patternpattern
Symmetry ElementsSymmetry Elements
1. Rotation1. Rotation
6
6
6
6
6
6 6
6
6
6
6
6
6
6
6
6
1-fold 2-fold 3-fold 4-fold 6-fold
2-D Symmetry2-D Symmetry
Symmetry ElementsSymmetry Elements
3. Reflection (m)3. Reflection (m)
Reflection across a Reflection across a “mirror plane” “mirror plane” reproduces a motifreproduces a motif
= symbol for a mirror= symbol for a mirror
planeplane
2-D Symmetry2-D Symmetry
3-D Symmetry3-D Symmetry
New 3-D Symmetry New 3-D Symmetry ElementsElements
4. Rotoinversion4. Rotoinversion
a. 2-fold rotoinversion ( 2 )a. 2-fold rotoinversion ( 2 )
3-D Symmetry3-D Symmetry
New Symmetry New Symmetry ElementsElements
4. Rotoinversion4. Rotoinversion
b. 2-fold rotoinversion ( 2 )b. 2-fold rotoinversion ( 2 )
Step 1: rotate 360/2Step 1: rotate 360/2
Note: this is a temporary Note: this is a temporary step,step, the intermediate the intermediate
motif element does notmotif element does not
exist in the final patternexist in the final pattern
3-D Symmetry3-D Symmetry
New Symmetry ElementsNew Symmetry Elements
4. Rotoinversion4. Rotoinversion
b. 2-fold rotoinversion ( 2 )b. 2-fold rotoinversion ( 2 )
Step 1: rotate 360/2Step 1: rotate 360/2
Step 2: invertStep 2: invert
3-D Symmetry3-D Symmetry
New Symmetry ElementsNew Symmetry Elements
4. Rotoinversion4. Rotoinversion
b. 2-fold rotoinversion ( 2 )b. 2-fold rotoinversion ( 2 )
The result:The result:
3-D Symmetry3-D Symmetry
New Symmetry New Symmetry ElementsElements
4. Rotoinversion4. Rotoinversion
b. 2-fold rotoinversion ( 2 )b. 2-fold rotoinversion ( 2 )
This is the same as This is the same as mm, so not, so not
a new operationa new operation
3-D Symmetry3-D Symmetry
New Symmetry ElementsNew Symmetry Elements
4. Rotoinversion4. Rotoinversion
c. 3-fold rotoinversion ( 3 )c. 3-fold rotoinversion ( 3 )
3-D Symmetry3-D Symmetry
New Symmetry New Symmetry ElementsElements
4. Rotoinversion4. Rotoinversion
c. 3-fold rotoinversion ( 3 )c. 3-fold rotoinversion ( 3 )
Step 1: rotate 360Step 1: rotate 360oo/3 /3
Again, this is a temporary Again, this is a temporary step, the intermediate step, the intermediate motif element does not motif element does not exist in the final patternexist in the final pattern
1
3-D Symmetry3-D Symmetry
New Symmetry New Symmetry ElementsElements
4. Rotoinversion4. Rotoinversion
c. 3-fold rotoinversion ( 3 )c. 3-fold rotoinversion ( 3 )
Step 2: invert through Step 2: invert through centercenter
3-D Symmetry3-D Symmetry
New Symmetry New Symmetry ElementsElements
4. Rotoinversion4. Rotoinversion
c. 3-fold rotoinversion ( 3 )c. 3-fold rotoinversion ( 3 )
Completion of the first Completion of the first sequencesequence
1
2
3-D Symmetry3-D Symmetry
New Symmetry New Symmetry ElementsElements
4. Rotoinversion4. Rotoinversion
c. 3-fold rotoinversion ( 3 )c. 3-fold rotoinversion ( 3 )
Rotate another 360/3Rotate another 360/3
3-D Symmetry3-D Symmetry
New Symmetry New Symmetry ElementsElements
4. Rotoinversion4. Rotoinversion
c. 3-fold rotoinversion ( 3 )c. 3-fold rotoinversion ( 3 )
Invert through centerInvert through center
3-D Symmetry3-D Symmetry
New Symmetry New Symmetry ElementsElements
4. Rotoinversion4. Rotoinversion
c. 3-fold rotoinversion ( 3 )c. 3-fold rotoinversion ( 3 )
Complete second step to Complete second step to create face 3 create face 3
1
2
3
3-D Symmetry3-D Symmetry
New Symmetry New Symmetry ElementsElements
4. Rotoinversion4. Rotoinversion
c. 3-fold rotoinversion ( 3 )c. 3-fold rotoinversion ( 3 )
Third step creates face 4 Third step creates face 4
(3 (3 (1) (1) 4) 4)
1
2
3
4
3-D Symmetry3-D Symmetry
New Symmetry New Symmetry ElementsElements
4. Rotoinversion4. Rotoinversion
c. 3-fold rotoinversion ( 3 )c. 3-fold rotoinversion ( 3 )
Fourth step creates face 5 Fourth step creates face 5 (4 (4 (2) (2) 5) 5)
1
2
5
3-D Symmetry3-D Symmetry
New Symmetry New Symmetry ElementsElements
4. Rotoinversion4. Rotoinversion
c. 3-fold rotoinversion ( 3 )c. 3-fold rotoinversion ( 3 )
Fifth step creates face 6Fifth step creates face 6
(5 (5 (3) (3) 6) 6)
Sixth step returns to face 1Sixth step returns to face 1
1
6
5
3-D Symmetry3-D Symmetry
New Symmetry New Symmetry ElementsElements
4. Rotoinversion4. Rotoinversion
c. 3-fold rotoinversion ( 3 )c. 3-fold rotoinversion ( 3 )
This is uniqueThis is unique1
6
5
2
3
4
3-D Symmetry3-D Symmetry
New Symmetry New Symmetry ElementsElements
4. Rotoinversion4. Rotoinversion
d. 4-fold rotoinversion ( 4 )d. 4-fold rotoinversion ( 4 )
3-D Symmetry3-D Symmetry
New Symmetry New Symmetry ElementsElements
4. Rotoinversion4. Rotoinversion
d. 4-fold rotoinversion ( 4 )d. 4-fold rotoinversion ( 4 )
3-D Symmetry3-D Symmetry
New Symmetry New Symmetry ElementsElements
4. Rotoinversion4. Rotoinversion
d. 4-fold rotoinversion ( 4 )d. 4-fold rotoinversion ( 4 )
1: Rotate 360/41: Rotate 360/4
3-D Symmetry3-D Symmetry
New Symmetry New Symmetry ElementsElements
4. Rotoinversion4. Rotoinversion
d. 4-fold rotoinversion ( 4 )d. 4-fold rotoinversion ( 4 )
1: Rotate 360/41: Rotate 360/4
2: Invert2: Invert
3-D Symmetry3-D Symmetry
New Symmetry New Symmetry ElementsElements
4. Rotoinversion4. Rotoinversion
d. 4-fold rotoinversion ( 4 )d. 4-fold rotoinversion ( 4 )
1: Rotate 360/41: Rotate 360/4
2: Invert2: Invert
3-D Symmetry3-D Symmetry
New Symmetry New Symmetry ElementsElements
4. Rotoinversion4. Rotoinversion
d. 4-fold rotoinversion ( 4 )d. 4-fold rotoinversion ( 4 )
3: Rotate 360/43: Rotate 360/4
3-D Symmetry3-D Symmetry
New Symmetry New Symmetry ElementsElements
4. Rotoinversion4. Rotoinversion
d. 4-fold rotoinversion ( 4 )d. 4-fold rotoinversion ( 4 )
3: Rotate 360/43: Rotate 360/4
4: Invert4: Invert
3-D Symmetry3-D Symmetry
New Symmetry New Symmetry ElementsElements
4. Rotoinversion4. Rotoinversion
d. 4-fold rotoinversion ( 4 )d. 4-fold rotoinversion ( 4 )
3: Rotate 360/43: Rotate 360/4
4: Invert4: Invert
3-D Symmetry3-D Symmetry
New Symmetry New Symmetry ElementsElements
4. Rotoinversion4. Rotoinversion
d. 4-fold rotoinversion ( 4 )d. 4-fold rotoinversion ( 4 )
5: Rotate 360/45: Rotate 360/4
3-D Symmetry3-D Symmetry
New Symmetry New Symmetry ElementsElements
4. Rotoinversion4. Rotoinversion
d. 4-fold rotoinversion ( 4 )d. 4-fold rotoinversion ( 4 )
5: Rotate 360/45: Rotate 360/4
6: Invert6: Invert
3-D Symmetry3-D Symmetry
New Symmetry New Symmetry ElementsElements
4. Rotoinversion4. Rotoinversion
d. 4-fold rotoinversion ( 4 )d. 4-fold rotoinversion ( 4 )
This is also a unique operationThis is also a unique operation
3-D Symmetry3-D Symmetry
New Symmetry New Symmetry ElementsElements
4. Rotoinversion4. Rotoinversion
d. 4-fold rotoinversion ( 4 )d. 4-fold rotoinversion ( 4 )
A more fundamental A more fundamental representative of the patternrepresentative of the pattern
3-D Symmetry3-D SymmetryWe now have We now have 88 unique unique 3D3D symmetry operations: symmetry operations:
1 2 3 4 6 m1 2 3 4 6 m 3 4 3 4
CombinationsCombinations of these elements are also possible of these elements are also possible
A A completecomplete analysis of analysis of symmetry about a point in spacesymmetry about a point in space requires that we try all possible combinations of these requires that we try all possible combinations of these symmetry elementssymmetry elements