Lec 5 - Symmetry 1academic.brooklyn.cuny.edu/geology/powell/courses...1 Symmetry (Part 1) Lecture 5...
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1 Symmetry (Part 1) Symmetry (Part 1) Lecture 5 Lecture 5 Symmetry Symmetry Motif Motif : 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 pattern whole pattern Operation Operation : some act that reproduces the : some act that reproduces the motif to create the pattern motif to create the pattern Element Element : an operation located at a : an operation located at a particular point in space particular point in space 2 2 - - D Symmetry D Symmetry Symmetry Elements Symmetry Elements 1. Rotation 1. Rotation a. Two a. Two - - fold rotation fold rotation = 360 = 360 o o / / 2 2 rotation rotation to reproduce a to reproduce a motif in a motif in a symmetrical symmetrical pattern pattern 6 6 A Symmetrical Pattern A Symmetrical Pattern
Transcript of Lec 5 - Symmetry 1academic.brooklyn.cuny.edu/geology/powell/courses...1 Symmetry (Part 1) Lecture 5...
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Symmetry (Part 1)
Symmetry (Part 1)
Lecture 5
Lecture 5
Symmetry
Symmetry
Motif
Motif : 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 pattern
whole pattern
Operation
Operation: some act that reproduces the
: some act that reproduces the
motif to create the pattern
motif to create the pattern
Element
Element : an operation located at a
: an operation located at a
particular point in space
particular point in space
22-- D Symmetry
D Symmetry
Symmetry Elements
Symmetry Elements
1. Rotation
1. Rotation
a. Two
a. Two-- fold rotation
fold rotation
= 360
= 360oo// 22rotation
rotation
to reproduce a
to reproduce a
motif in a
motif in a
symmetrical
symmetrical
pattern
pattern
6
6
A Symmetrical Pattern
A Symmetrical Pattern
2
Symmetry Elements
Symmetry Elements
1. Rotation
1. Rotation
a. Two
a. Two-- fold rotation
fold rotation
= 360
= 360oo// 22rotation
rotation
to reproduce a
to reproduce a
motif in a
motif in a
symmetrical
symmetrical
pattern
pattern
= the symbol for a two
= the symbol for a two-- fold
fold
rotation
rotation
Motif
Element
Operation
Operation
6
6
22-- D Symmetry
D Symmetry
6
6
first
operation
step
22-- D Symmetry
D Symmetry
Symmetry Elements
Symmetry Elements
1. Rotation
1. Rotation
a. Two
a. Two-- fold rotation
fold rotation
= 360
= 360oo// 22rotation
rotation
to reproduce a
to reproduce a
motif in a
motif in a
symmetrical
symmetrical
pattern
pattern
= the symbol for a two
= the symbol for a two-- fold
fold
rotation
rotation
6
6
first
operation
step
second
operation
step
22-- D Symmetry
D Symmetry
Symmetry Elements
Symmetry Elements
1. Rotation
1. Rotation
a. Two
a. Two-- fold rotation
fold rotation
= 360
= 360oo// 22rotation
rotation
to reproduce a
to reproduce a
motif in a
motif in a
symmetrical
symmetrical
pattern
pattern
= the symbol for a two
= the symbol for a two-- fold
fold
rotation
rotation
Symmetry Elements
Symmetry Elements
1. Rotation
1. Rotation
b. Three
b. Three-- fold rotation
fold rotation
= 360
= 360oo// 33rotation
rotation
to reproduce a motif
to reproduce a motif
in a symmetrical
in a symmetrical
pattern
pattern
6
66
22-- D Symmetry
D Symmetry
3
6
6
step 1
22-- D Symmetry
D Symmetry
Symmetry Elements
Symmetry Elements
1. Rotation
1. Rotation
b. Three
b. Three-- fold rotation
fold rotation
= 360
= 360oo// 33rotation
rotation
to reproduce a motif
to reproduce a motif
in a symmetrical
in a symmetrical
pattern
pattern
6
6
6
step 1
step 2
22-- D Symmetry
D Symmetry
Symmetry Elements
Symmetry Elements
1. Rotation
1. Rotation
b. Three
b. Three-- fold rotation
fold rotation
= 360
= 360oo// 33rotation
rotation
to reproduce a motif
to reproduce a motif
in a symmetrical
in a symmetrical
pattern
pattern
6
6
6
step 1
step 2
step 3
22-- D Symmetry
D Symmetry
Symmetry Elements
Symmetry Elements
1. Rotation
1. Rotation
b. Three
b. Three-- fold rotation
fold rotation
= 360
= 360oo// 33rotation
rotation
to reproduce a motif
to reproduce a motif
in a symmetrical
in a symmetrical
pattern
pattern
Symmetry Elements
Symmetry Elements
1. Rotation
1. Rotation
6
6
6
6
6
66
6
6
6
6
6
6
66
6
1-fold
2-fold
3-fold
4-fold
6-fold
22-- D Symmetry
D Symmetry
4
Symmetry Elements
Symmetry Elements
3. Reflection (m)
3. Reflection (m)
Reflection across a
Reflection across a
““ mirror plane
mirror plane””
reproduces a motif
reproduces a motif
= symbol for a mirror
= symbol for a mirror
plane
plane
22-- D Symmetry
D Symmetry
33-- D Symmetry
D Symmetry
New 3
New 3-- D Symmetry
D Symmetry
Elements
Elements
4. Rotoinversion
4. Rotoinversion
a. 2
a. 2-- fold rotoinversion ( 2 )
fold rotoinversion ( 2 )
33-- D Symmetry
D Symmetry
New Symmetry
New Symmetry
Elements
Elements
4. Rotoinversion
4. Rotoinversion
b. 2
b. 2-- fold rotoinversion ( 2 )
fold rotoinversion ( 2 )
Step 1: rotate 360/2
Step 1: rotate 360/2
Note: this is a temporary
Note: this is a temporary
step,
step,the intermediate
the intermediate
motif element does not
motif element does not
exist in the final pattern
exist in the final pattern
33-- D Symmetry
D Symmetry
New Symmetry Elements
New Symmetry Elements
4. Rotoinversion
4. Rotoinversion
b. 2
b. 2-- fold rotoinversion ( 2 )
fold rotoinversion ( 2 )
Step 1: rotate 360/2
Step 1: rotate 360/2
Step 2: invert
Step 2: invert
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33-- D Symmetry
D Symmetry
New Symmetry Elements
New Symmetry Elements
4. Rotoinversion
4. Rotoinversion
b. 2
b. 2-- fold rotoinversion ( 2 )
fold rotoinversion ( 2 )
The result:
The result:
33-- D Symmetry
D Symmetry
New Symmetry
New Symmetry
Elements
Elements
4. Rotoinversion
4. Rotoinversion
b. 2
b. 2-- fold rotoinversion ( 2 )
fold rotoinversion ( 2 )
This is the same as
This is the same as mm, so not
, so not
a new operation
a new operation
33-- D Symmetry
D Symmetry
New Symmetry Elements
New Symmetry Elements
4. Rotoinversion
4. Rotoinversion
c. 3
c. 3-- fold rotoinversion ( 3 )
fold rotoinversion ( 3 )
33-- D Symmetry
D Symmetry
New Symmetry
New Symmetry
Elements
Elements
4. Rotoinversion
4. Rotoinversion
c. 3
c. 3-- fold rotoinversion ( 3 )
fold rotoinversion ( 3 )
Step 1: rotate 360
Step 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 pattern
exist in the final pattern
1
6
33-- D Symmetry
D Symmetry
New Symmetry
New Symmetry
Elements
Elements
4. Rotoinversion
4. Rotoinversion
c. 3
c. 3-- fold rotoinversion ( 3 )
fold rotoinversion ( 3 )
Step 2: invert through
Step 2: invert through
center
center
33-- D Symmetry
D Symmetry
New Symmetry
New Symmetry
Elements
Elements
4. Rotoinversion
4. Rotoinversion
c. 3
c. 3-- fold rotoinversion ( 3 )
fold rotoinversion ( 3 )
Completion of the first
Completion of the first
sequence
sequence
1 2
33-- D Symmetry
D Symmetry
New Symmetry
New Symmetry
Elements
Elements
4. Rotoinversion
4. Rotoinversion
c. 3
c. 3-- fold rotoinversion ( 3 )
fold rotoinversion ( 3 )
Rotate another 360/3
Rotate another 360/3
33-- D Symmetry
D Symmetry
New Symmetry
New Symmetry
Elements
Elements
4. Rotoinversion
4. Rotoinversion
c. 3
c. 3-- fold rotoinversion ( 3 )
fold rotoinversion ( 3 )
Invert through center
Invert through center
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33-- D Symmetry
D Symmetry
New Symmetry
New Symmetry
Elements
Elements
4. Rotoinversion
4. Rotoinversion
c. 3
c. 3-- fold rotoinversion ( 3 )
fold rotoinversion ( 3 )
Complete second step to
Complete second step to
create face 3
create face 3
1
2
3
33-- D Symmetry
D Symmetry
New Symmetry
New Symmetry
Elements
Elements
4. Rotoinversion
4. Rotoinversion
c. 3
c. 3-- fold rotoinversion ( 3 )
fold rotoinversion ( 3 )
Third step creates face 4
Third step creates face 4
(3
(3 →→
(1)
(1) →→4)4)
1 2
3
4
33-- D Symmetry
D Symmetry
New Symmetry
New Symmetry
Elements
Elements
4. Rotoinversion
4. Rotoinversion
c. 3
c. 3-- fold rotoinversion ( 3 )
fold rotoinversion ( 3 )
Fourth step creates face 5
Fourth step creates face 5
(4
(4 →→
(2)
(2) →→5)5)
1
2
5
33-- D Symmetry
D Symmetry
New Symmetry
New Symmetry
Elements
Elements
4. Rotoinversion
4. Rotoinversion
c. 3
c. 3-- fold rotoinversion ( 3 )
fold rotoinversion ( 3 )
Fifth step creates face 6
Fifth step creates face 6
(5
(5 →→
(3)
(3) →→6)6)
Sixth step returns to face 1
Sixth step returns to face 1
1
6
5
8
33-- D Symmetry
D Symmetry
New Symmetry
New Symmetry
Elements
Elements
4. Rotoinversion
4. Rotoinversion
c. 3
c. 3-- fold rotoinversion ( 3 )
fold rotoinversion ( 3 )
This is unique
This is unique
1
6
5
2
3
4
33-- D Symmetry
D Symmetry
New Symmetry
New Symmetry
Elements
Elements
4. Rotoinversion
4. Rotoinversion
d. 4
d. 4-- fold rotoinversion ( 4 )
fold rotoinversion ( 4 )
33-- D Symmetry
D Symmetry
New Symmetry
New Symmetry
Elements
Elements
4. Rotoinversion
4. Rotoinversion
d. 4
d. 4-- fold rotoinversion ( 4 )
fold rotoinversion ( 4 )
33-- D Symmetry
D Symmetry
New Symmetry
New Symmetry
Elements
Elements
4. Rotoinversion
4. Rotoinversion
d. 4
d. 4-- fold rotoinversion ( 4 )
fold rotoinversion ( 4 )
1: Rotate 360/4
1: Rotate 360/4
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33-- D Symmetry
D Symmetry
New Symmetry
New Symmetry
Elements
Elements
4. Rotoinversion
4. Rotoinversion
d. 4
d. 4-- fold rotoinversion ( 4 )
fold rotoinversion ( 4 )
1: Rotate 360/4
1: Rotate 360/4
2: Invert
2: Invert
33-- D Symmetry
D Symmetry
New Symmetry
New Symmetry
Elements
Elements
4. Rotoinversion
4. Rotoinversion
d. 4
d. 4-- fold rotoinversion ( 4 )
fold rotoinversion ( 4 )
1: Rotate 360/4
1: Rotate 360/4
2: Invert
2: Invert
33-- D Symmetry
D Symmetry
New Symmetry
New Symmetry
Elements
Elements
4. Rotoinversion
4. Rotoinversion
d. 4
d. 4-- fold rotoinversion ( 4 )
fold rotoinversion ( 4 )
3: Rotate 360/4
3: Rotate 360/4
33-- D Symmetry
D Symmetry
New Symmetry
New Symmetry
Elements
Elements
4. Rotoinversion
4. Rotoinversion
d. 4
d. 4-- fold rotoinversion ( 4 )
fold rotoinversion ( 4 )
3: Rotate 360/4
3: Rotate 360/4
4: Invert
4: Invert
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33-- D Symmetry
D Symmetry
New Symmetry
New Symmetry
Elements
Elements
4. Rotoinversion
4. Rotoinversion
d. 4
d. 4-- fold rotoinversion ( 4 )
fold rotoinversion ( 4 )
3: Rotate 360/4
3: Rotate 360/4
4: Invert
4: Invert
33-- D Symmetry
D Symmetry
New Symmetry
New Symmetry
Elements
Elements
4. Rotoinversion
4. Rotoinversion
d. 4
d. 4-- fold rotoinversion ( 4 )
fold rotoinversion ( 4 )
5: Rotate 360/4
5: Rotate 360/4
33-- D Symmetry
D Symmetry
New Symmetry
New Symmetry
Elements
Elements
4. Rotoinversion
4. Rotoinversion
d. 4
d. 4-- fold rotoinversion ( 4 )
fold rotoinversion ( 4 )
5: Rotate 360/4
5: Rotate 360/4
6: Invert
6: Invert
33-- D Symmetry
D Symmetry
New Symmetry
New Symmetry
Elements
Elements
4. Rotoinversion
4. Rotoinversion
d. 4
d. 4-- fold rotoinversion ( 4 )
fold rotoinversion ( 4 )
This is also a unique operation
This is also a unique operation
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33-- D Symmetry
D Symmetry
New Symmetry
New Symmetry
Elements
Elements
4. Rotoinversion
4. Rotoinversion
d. 4
d. 4-- fold rotoinversion ( 4 )
fold rotoinversion ( 4 )
A more fundamental
A more fundamental
representative of the pattern
representative of the pattern
33-- D Symmetry
D Symmetry
We now have
We now have 88unique
unique 3D3Dsymmetry operations:
symmetry operations:
1 2 3 4 6 m
1 2 3 4 6 m
3 4
3 4
Combinations
Combinationsof these elements are also possible
of these elements are also possible
A
A complete
completeanalysis of
analysis of symmetry about a point in space
symmetry about a point in space
requires that we try all possible combinations of these
requires that we try all possible combinations of these
symmetry elements
symmetry elements
