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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# Frieze Patterns and Symmetry Groups

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