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Frieze Patterns and Symmetry Groups

explanation, examples, etc.
by

Elizabeth LaRosa

on 1 June 2011

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Transcript of Frieze Patterns and Symmetry Groups

Frieze Patterns & Symmetry Groups What Exactly is a Frieze Pattern? A frieze is a pattern that repeats in one direction. It is sometimes called a border
pattern, or an infinite strip pattern. They are often found in architecture. There are seven different types of frieze patterns. Types of Frieze Patterns The types of frieze patterns include translation, glide
reflection, two parallel reflections, two half turns, a
reflection and a half turn, horizontal reflection, and
three reflections. Translations When using a translation in a frieze pattern,
the pattern is dragged horizontally or vertically
to create the same pattern over and over again. There is a translation in every frieze pattern. Glide Reflection In using glide reflection in a frieze pattern, the pattern or image in the pattern is reflected and translated. Translation Glide Reflection Two Parallel Reflections Two Half Turns A Reflection and a Half Turn Horizontal Reflection Three Reflections What is a Symmetry Group? A symmetry group is a subgroup of the isometry group of the space concerned (image, signal, etc.)
There are 17 different symmetry groups. Frieze Patterns and Symmetry Groups in History... Symmetry Group #1 This symmetry group is only translations and the two translation axes
may be inclined at any angle toward each other. It has a parallelogramatic
lattice, so a fundamental region for this group is the same for the parallelogram
translation group. Symmetry Group #2 This group has both translations and 180 degree rotations, or half turns.
There are translations, but no reflections or glide reflections. The lattice is
parallelogramtic and the fundamental region for the symmetry group is half
of a parallelogram Symmetry Group # 3 This group contains reflections and the axes of reflection are parallel
to the axis of translation. There aren't any rotations or glide reflections
in this group. The fundamental region is a rectangle that is split in half
by the axis of reflection. Symmetry Group # 4 This group uses glide reflections. In the glide reflections the direction of
the glide is parallel to one axis of translation and perpendicular to the other.
There are no rotations or regular reflections. The lattice is rectangular and the
fundamental region for the translation is a rectangle split by an axis of a glide
reflection. Symmetry Group #5 This group contains reflections, glide reflections and translations. The
translations can be inclined at any angle to each other. A fundamental
region for this symmetry group is a rhombus. Symmetry Group # 6 This group contains reflections on perpendicular axes, rotations and translations.
The lattice is rectangular and the fundamental region is a rectangle. Symmetry Group # 7 Symmetry group number 7 has reflections and half turns
which is not on the line of reflection. The lattice is rectangular
and the fundamental group is rectangular as well. Symmetry Group # 8 This group consists of glide reflections and half turns.
The points for the half turns do not lie on the perpendicular
axes for the glide reflections. The lattice is rectangular as is
the fundamental region for the symmetry group. Symmetry Group # 9 Symmetry group 9 has two perpendicular axes of reflection
and the points of the half turns are on the intersection of the
two and on the intersection of the glide reflection axes.
The lattice is a rhombus as is the fundamental region. Symmetry Group # 10 This group has 90 degree rotaions and half turns with
a square lattice and a square fundamental region. Symmetry Group # 11 This group has 90 degree rotations, half turns, and reflections.
The axes of reflection are inclined toward each other 45 degrees
and the rotation axes all lie on the reflections axes. The lattice is
square and the fundamental region is a triangle. Symmetry Group # 12 This also consists of reflections, half turns, and 90 degree rotations.
However, the axes of reflection are perpendicular and the reflection
points are on the axes. The lattice is square, as is the fundamental region. Symmetry Group # 13 Symmetry group 13 has only a 120 degree rotation and a
hexagonal lattice. A fundamental region is a triangle. Symmetry Group # 14 This group contains 120 degree rotations and reflections
whose axes are inclined to each other 60 degrees. The
lattice is hexagonal and a fundamental region is a triangle. Symmetry Group # 15 This group has 120 degree rotations, reflections and glide
reflections and the center of the rotation is on the axes of the
reflections. The axes are inclined 60 degrees toward each
other. The lattice is hexagonal and the fundamental region is
a triangle. Symmetry Group # 16 Symmetry group 16 contains half turns, 60 degree rotations,
and 120 degree rotations. The lattice is hexagonal and the
fundamental region is triangular. Symmetry Group # 17 This last group has half turns, reflections, glide reflections,
60 degree turns, and 120 degree turns. The axes are inclined
30 degrees to each other. The lattice is hexagonal and the
fundamental region is triangular. Symmetry Group # 2 Symmetry Group # 8 Symmetry Group # 12 Frieze patterns were first used in art in Hungary
during the middle ages and they can be seen in
Avar - Onogurian artifacts. Frieze patterns are also
found in clothing and pottery of indigenous peoples
of North America. All seven patterns are seen in
ancient Mayan architecture. There was one classified
as only having translations and vertical reflections Cave paintings have been found that display symmetry groups.

Ancient Greeks used symmetry groups to design their temples.

M.C. Escher used symmetry groups in his artwork. and Today... There are symmetry groups in nature
in objects such as shells, flowers, and leaves.

What symmetry groups do you see in the flower? There are frieze patterns in art. They can
be seen by looking to see if any tranlation
or reflection exists. Guess the Frieze Pattern ??? Glide Reflection ??? Horizontal Reflection ??? Two Parallel Reflections
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