Geetha Venkataraman
Ambedkar University, Delhi
Symmetry of finite figures
!
Symmetry of infinite patterns
!
!
Strip Pattern 1
!
Wallpaper Pattern 1
Strip Pattern 2
!
Symmetry and patterns
in
Buildings/monuments/ornamental art
Symmetry and patterns
in
Escher’s art
Symmetry of finite figures
Definition (intuitive) symmetry: A symmetry of a finite figure is the
action you can perform on the finite figure such that if someone has
closed their eyes during the performance they feel as though nothing
has happened to the figure. In other words, it is in the exact same
place and looks like it has not been touched.
90 degree rotation
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B
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D
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Types of symmetries for finite figures with finitely many symmetries:
ONLY TWO TYPES
Reflections or Rotations
If a planar figures has only finitely many symmetries then the
symmetries consist either of only rotations or an equal number of
rotations and reflections.
For a finite figure X, if Sym(X), the set of symmetries, is finite, then
one of two cases occurs:
(a) Sym(X) has an equal number of rotations and reflections, say n of
each. In this case we say that the finite figure X has symmetry
type Dn. Some examples:
If X is an equilateral triangle, then it has symmetry type D3 .
If X is a square, then it has symmetry type D4 .
(b) Sym(X) has only rotations. If it has exactly n rotations then we say
that the figure X has symmetry type Cn.
Some Examples of type Cn
N4
N3
N2
Types of Symmetries of strip patterns
!
!
!
t
two `types’ of vertical mirror symmetries and translation
A strip pattern consists of a basic motif repeated along a line at equal intervals.
A strip pattern will always have translation symmetry.
A strip pattern may or may not have any of the following type of symmetries:
Reflections about vertical lines
Reflections about horizontal lines
Rotations. OK($-17-&%$"-%$("#$5+F/+("*/0$)+(-4+/6P
Examples of strip patterns
!
!
!
t
180 degree rotation about two types of rotocenters and translation
!
!
!
t
horizontal mirror symmetry and translation
!
!
t
horizontal mirror symmetry, two types of vertical mirror symmetries,
180 degree rotations about two types of rotocenters and translation
t
only translation
Symmetries of wallpaper patterns
!
!
!
s
t
translation in two directions
!
!"
!
!"
!"
The above basic strip pattern has
two types of rotocenters (denoted by red circles and yellow diamonds)
two types of vertical mirror symmetries (green and purple line)
one horizontal mirror symmetry
and translation symmetry in one direction.
Symmetries of wallpaper patterns
!
!
#"
!"
#"
!"
The above wallpaper pattern has
three types of rotocenters (denoted by red circles, yellow
diamonds and green circles)
two types of vertical mirror symmetries (green and purple lines)
two types of horizontal mirror symmetry (red and yellow lines)
Any other lines of mirror symmetry? (blue and ??)
and translation symmetry in two directions (denoted by s and t).
Symmetry and patterns
in
the Taj
!
floor plan of the Taj Mahal
Symmetry and patterns
in
the Taj
Symmetry and patterns
in
Escher’s art
M C Escher
1898-1972
Self-portrait
(c. MCE)
Symmetry and patterns
in
Escher’s art
0 degree
120 degree
240 degree
Escher’s art
0 degree
90 degree
180 degree
270 degree
Symmetry around us
Symmetry around us
Symmetry around us
Symmetry around us
Some References:
1.
David W Farmer, Groups and symmetry, Universities
Press, India, 1998.
2.
Hermann Weyl, Symmetry, Princeton University Press,
USA, 1982.
3.
For animated tessellations: http://clowder.net/hop/
17walppr/17walppr.html
4.
http://euler.slu.edu/escher/index.php/Main_Page
5.
http://mathforum.org/~sanders/connectinggeometry/
Quilt.html.
6.
http://www.taj-mahal.net/
7.
For a lecture on symmetry by Ian Stewart: http://
www2.warwick.ac.uk/newsandevents/audio/more/
symmetry/
8.
www.wikipedia.org