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/2rh>. (5) <tL, rh rv >. (6) <tL, tL/2rh , rhrv>. (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.