The Four Types of Symmetry in
A pattern is symmetric if there is at least one
(rotation, translation, reflection, glide reflection)
that leaves the pattern unchanged.
Symmetries create patterns that help us organize our world conceptually.
Symmetric patterns occur in nature, and are invented by artists, craftspeople,
musicians, choreographers, and mathematicians.
In mathematics, the idea of symmetry gives us a precise way to think
about this subject. We will talk about plane symmetries, those that
take place on a flat plane, but the ideas generalize to spatial
Plane symmetry involves moving all points around the plane so that their
positions relative to each other remain the same, although their absolute
positions may change. Symmetries preserve distances, angles, sizes, and
- For example, rotation by 90 degrees about a fixed point is an
example of a plane symmetry.
- Another basic type of symmetry is a reflection. The reflection of
a figure in the plane about a line moves its reflected image to where it
would appear if you viewed it using a mirror placed on the line. Another
way to make a reflection is to fold a piece of paper and trace the figure
onto the other side of the fold.
- A third type of symmetry is translation. Translating an object
means moving it without rotating or reflecting it. You can describe a
translation by stating how far it moves an object, and in what direction.
- The fourth (and last) type of symmetry is a glide reflection. A
glide reflection combines a reflection with a translation along the
direction of the mirror line.
To rotate an object means to turn it around. Every rotation has a center
and an angle.
To translate an object means to move it without rotating or reflecting it.
Every translation has a direction and a distance.
To reflect an object means to produce its mirror image. Every reflection
has a mirror line. A reflection of an "R" is a backwards
A glide reflection combines a reflection with a translation along the
direction of the mirror line. Glide reflections are the only type of symmetry
that involve more than one step.
Just remember the golden rule: A fiigure,
picture, or pattern is said to be symmetric if there is at least one symmetry
that leaves the figure unchanged
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