Mathematics of Symmetry
(Part 2)

Symmetric Patterns
A plane figure has
symmetry
if there is a non-trivial transformation that maps
the
figure
onto
itself.
A
trivial
transformation
refers
to
the
identity
transformation which maps every point in the plane onto itself.
Symmetry lines of a square (reflection)

Symmetric Patterns
If a figure can be rotated less than
360
0
about a point so that the image and
the pre-image are
indistinguishable
, then the figure has
rotational symmetry
.
Rotational Symmetries of a square (
0
0
/360
0
,
90
0
,
180
0
, and
270
0
)
The square has rotational symmetry of
order 4
because there are four rotations of less than
360
0
that produce an image indistinguishable
from the original. The rotational symmetry has
a
magnitude
of
90
0
=
360
0
4
.
Some mathematicians refer to
180
0
rotational
symmetry about a point as a
point symmetry
.

Symmetric Patterns

Design and Pattern
A
design
is a figure with at least one non-trivial symmetry. A
pattern
is a design
that has a translation symmetry. A plane pattern has symmetry if there is an
isometry of the plane that preserves it.
Arts and mathematics intersect in the concept of symmetry. In art, symmetry is
a basic design element, something that many people consider pleasing to look
at. In mathematics, symmetry can be defined and verified by finding motions
that leave a design unchanged. These motions can be combined and analyzed
in much the same way numbers are.

Tessellations
A
tessellation
is a repeating pattern of figures

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- Spring '14
- Rotational symmetry, tessellations, Mathematics of Symmetry