symmetry elements lecture 5. symmetry motif: the fundamental part of a symmetric design that, when...

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


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





= the symbol for a two-fold = the symbol for a two-fold rotationrotation

Motif

Element

OperationOperation

6


6

6

first operation step







6

6

first operation step

second operation step








1. Rotation1. Rotationb. Three-fold rotationb. Three-fold rotation


to reproduce a motif to reproduce a motif in a symmetrical in a symmetrical patternpattern

66

6


6

6

step 1






6

66

step 1

step 2






6

66

step 1

step 2

step 3







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



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



New 3-D Symmetry New 3-D Symmetry ElementsElements

4. Rotoinversion4. Rotoinversion

a. 2-fold rotoinversion ( 2 )a. 2-fold rotoinversion ( 2 )


New Symmetry New Symmetry ElementsElements


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


New Symmetry ElementsNew Symmetry Elements



Step 1: rotate 360/2Step 1: rotate 360/2

Step 2: invertStep 2: invert





The result:The result:





This is the same as This is the same as mm, so not, so not

a new operationa new operation




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





Step 2: invert through Step 2: invert through centercenter





Completion of the first Completion of the first sequencesequence

1

2





Rotate another 360/3Rotate another 360/3





Invert through centerInvert through center





Complete second step to Complete second step to create face 3 create face 3

1

2

3





Third step creates face 4 Third step creates face 4

(3 (3 (1) (1) 4) 4)

1

2

3

4





Fourth step creates face 5 Fourth step creates face 5 (4 (4 (2) (2) 5) 5)

1

2

5





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





This is uniqueThis is unique1

6

5

2

3

4




d. 4-fold rotoinversion ( 4 )d. 4-fold rotoinversion ( 4 )





1: Rotate 360/41: Rotate 360/4






2: Invert2: Invert






4: Invert4: Invert






6: Invert6: Invert





This is also a unique operationThis is also a unique operation





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

symmetry elements lecture 5. symmetry motif: the fundamental part of a symmetric design that, when...

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