Transformations and Symmetry
Transformations
• Reflection• Translation• Glide Reflection• Rotation• Scaling (not topological)
A transformation turns a geometric figure into another by one of the actions above.
Reflection
The blue triangle is reflected across line m (called the line of reflection).
Translation
Sliding a figure. A translation could be accomplished by several reflections.
Glide Reflection
The figure reflects across the line, and slides forward.
Rotation
The blue image is rotated about the point p.
Scaling
The blue star has been scaled upward by a multiplying factor. Scaling (changing size) is
not a topological transformation.
Symmetry
•Our basic idea of symmetry is 2-fold symmetry
•A figure whose halves are mirror images of each other over a fold line
•We could think of this as being able to transform half of the image into the other half by reflection
Which are the lines of symmetry?
Only m
Which are the lines of symmetry?
L, M, N, O
Which are the lines of symmetry?
M, O
Which are the lines of symmetry?
T,R
Symmetry (II)
• More complicated types of symmetry arise if we consider objects obtained from reflection, PLUS other transformations such as rotation
• Various patterns have been characterized as symmetry groups
Conway Notation
• Is one way mathematicians use to describe various symmetry groups
• Can get very complicated!
Kali• We’re going to look at some
symmetry groups using the program Kali
• Keep an eye out for two basic notations:
– An integer (1,2,3) denotes a ROTATIONAL symmetry
– An * denotes a reflection as well
KALI
The title picture …
…was created in kali, then colored in with paint
Symmetry in Nature
Chinese Rose
Symmetry in Nature
Crab
Symmetry in Nature
Starfish
Symmetry in Art
Quilt
Symmetry in Art
Pennsylvania Dutch Hex Sign
Symmetry in Art
Persian Carpet
Symmetry in Art
Ukranian Painted Easter Eggs
Symmetry in Art
Mosaic Tile (Iran, 14th C.)
Symmetry in Art
A Kaliedotile
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