Symmetry in the Plane
There are 17 different ways a plane pattern can be symmetrical. Each
symmetry involves a combination of transformations
of the plane. The symmetries are called by these cryptic names:
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p1, Two translations
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p2, Three half turns
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p3, Two 120 degree rotations
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p4, A half turn and a quarter turn
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p6, A half turn and a 120 degree rotation
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pm, Two reflections and a translation
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pmm, Four reflections along the sides of
a rectangle
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pmg, A reflection and two half turns
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cmm, Two perpendicular reflections and a
half turn
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p31m, A reflection and a 120 degree rotation
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p3m1, Three reflections along the sides
of an equilateral triangle
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p4g, A reflection and a quarter turn
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p4m, Reflections in three sides of a 45-45-90
degree triangle
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p6m, Three reflections along the sides
of a 30-60-90 degree triangle
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cm, A reflection and a parallel glide reflection
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pg, Two parallel glide reflections
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pgg, Two perpendicular glide reflections
Each link goes to a more detailed description of the symmetry, with examples
and links to art works by M. C. Escher possessing the symmetry in question.
See also:
David Joyce's
Wallpaper patterns.
Draw your own symmetry patterns with Kali.
How to identify a plane symmetry
This classification tool was restructured based on Table 5.1 in Washburn
and Crowe's Symmetries of Culture. A great book. Check it out.
Start: Does the pattern reflect in at least one direction?
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Yes: Does the pattern rotate?
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Yes: What is the smallest rotation?
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180 degrees: Are there reflections in two directions?
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Yes: Are all the centres of rotation on mirror lines?
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No=pmg
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120 degrees: Are all the centres of rotation on mirror lines?
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90 degrees: Are there mirror lines at 45 deg angles?
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60 degrees=p6m]
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No: Is there a glide reflection which doesn't lie on a mirror line?
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No: Does the pattern rotate?
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Yes: What is the smallest rotation?
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180 degrees: Is there a glide reflection?
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120 degrees=p3
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90 degrees=p4
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60 degrees=p6
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No: Is there a glide reflection?
[Transformations] [Plane
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This page maintained by David A Reid
Email: david.reid@acadiau.ca