Friezes and Mosaics

The Mathematics of Beauty

Frieze pattern on the walls of theTaj

Mahal.

Frieze patterns in the Taj

Mahal

The gardens and corridors have many frieze patterns.

Mosaics in the Taj

Mahal

The ground around the Taj

Mahal

is

laid with a tiling pattern of four-

pointed stars.

Palace of mirrors in Jaipur, India

The palace complex at Amer

Fort near Jaipur has a hall of

mirrors.

During the day, the chamber reflectssunlight and at night, a single candleis reflected multiple times enough toilluminate the room.

Jaisalmer

in Rajasthan

These are frieze patterns appearing on the walls of the Jaisalmer

Fort in

Rajasthan, India.

Friezes

We will look at the symmetries of these seven frieze patterns.

Simplified friezes

These exhibit translational, rotational and reflective symmetries.

Main theorem for symmetry groups of friezes

There are only 7 possible symmetry groups for any frieze pattern.

They are listed as: (1) <tL

>, group generated by a translation of length L.

(2) <tL

, rv

>, with vertical reflection rv

.

(3) <tL

, rh

>, with horizontal reflection rh

.

(4) <tL

, tL/2

rh

>.

(5) <tL

, rh

rv

>.

(6) <tL

, tL/2

rh , rh

rv

>.

(7) <tL

, rh

, rv

>.

Mosaics

A mosaic is a pattern that can be repeated to fill the plane and it is periodic along two independent directions.

Main theorem for symmetry groups of mosaics

There are only 17 symmetry groups and these can be listed.

The simplest is the group generated by a single translation.(p1)

The groups pg and pm

The group pg contains glide reflections only and their axes are parallel.

The group pm has no rotations and only reflection axes which are parallel.

cm, p2 and pgg

pmg, pmm

and cmm

p3, p31m, and p3m1

There are five more crystallographic groups: p4, p4g, p4m, p6 and p6m.

Five more …