Types of Symmetry

How do you determine the symmetry type of a pattern? Easy: by answering at most four questions, starting with this one: Will the pattern coincide with itself if it is rotated around some center?

Is there a reflection? Is there a glide-reflection?
Simple shift
(p1)

Glide reflection
(pg)

Is there a glide-reflection in an axis that is not a reflection axis?
Mirror
(pm)

Mirror & glide
(cm)

Is there a reflection? Is there a glide-reflection?
Half-turn
(p2)

Double glide
(pgg)

Are there reflections in two directions?
Parallel mirrors &
glide (pmg)

Are all rotation centers on reflection axes?
Perpendicular mirrors
& glide (cmm)

Double mirror
(pmm)

Is there a reflection?
Three rotations
(p3)

Are there reflections in two directions?
Three rotations &
mirrors (p31m)

Three mirrors
(p3m1)

Is there a reflection?
Pinwheel
(p4)

Are there four reflection axes?
Quarter-turns &
rotated mirrors (p4g)

Quarter-turns &
mirrors (p4m)

Is there a reflection?
Six rotations
(p6)

Kaleidoscope
(p6m)

Adopted from Ray McLenaghan and Silvio Levy, Geometry, in CRC Standard Mathematical Tables and Formulae, D. Zwillinger (ed.), CRC Press, 1996, Boca Raton, New York, London, Tokyo, p. 264.

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