Friezes and Mosaics
Frieze pattern on the walls of the
Taj Mahal.
The Mathematics of Beauty
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
fourpointed
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 reflects
sunlight and at night, a single candle
is reflected multiple times enough to
illuminate 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, r v>, with vertical reflection r v.
(3) <tL, rh>, with horizontal reflection rh.
(4) <tL, tL/2rh>.
(5) <tL, rh r v >.
(6) <tL, tL/2rh , rhr v>.
(7) <tL, rh, r v >.
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.