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Friezes and Mosaics The Mathematics of Beauty Frieze pattern on the walls of the Taj Mahal.

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Friezes and Mosaics The Mathematics of Beauty Frieze pattern on the walls of Taj Mahal.
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Page 1: Friezes and Mosaics The Mathematics of Beauty Frieze pattern on the walls of the Taj Mahal.

Friezes and Mosaics

The Mathematics of Beauty

Frieze pattern on the walls of theTaj Mahal.

Page 2: Friezes and Mosaics The Mathematics of Beauty Frieze pattern on the walls of the Taj Mahal.

Frieze patterns in the Taj Mahal The gardens and

corridors have many frieze patterns.

Page 3: Friezes and Mosaics The Mathematics of Beauty Frieze pattern on the walls of the Taj Mahal.

Mosaics in the Taj Mahal The

ground around the Taj Mahal is laid with a tiling pattern of four-pointed stars.

Page 4: Friezes and Mosaics The Mathematics of Beauty Frieze pattern on the walls of the Taj Mahal.

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.

Page 5: Friezes and Mosaics The Mathematics of Beauty Frieze pattern on the walls of the Taj Mahal.

Jaisalmer in Rajasthan These are frieze

patterns appearing on the walls of the Jaisalmer Fort in Rajasthan, India.

Page 6: Friezes and Mosaics The Mathematics of Beauty Frieze pattern on the walls of the Taj Mahal.

Friezes

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

Page 7: Friezes and Mosaics The Mathematics of Beauty Frieze pattern on the walls of the Taj Mahal.

Simplified friezes These exhibit

translational, rotational and reflective symmetries.

Page 8: Friezes and Mosaics The Mathematics of Beauty Frieze pattern on the walls of the Taj Mahal.

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 >.

Page 9: Friezes and Mosaics The Mathematics of Beauty Frieze pattern on the walls of the Taj Mahal.

Mosaics

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

Page 10: Friezes and Mosaics The Mathematics of Beauty Frieze pattern on the walls of the Taj Mahal.

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)

Page 11: Friezes and Mosaics The Mathematics of Beauty Frieze pattern on the walls of the Taj Mahal.

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.

Page 12: Friezes and Mosaics The Mathematics of Beauty Frieze pattern on the walls of the Taj Mahal.

cm, p2 and pgg

Page 13: Friezes and Mosaics The Mathematics of Beauty Frieze pattern on the walls of the Taj Mahal.

pmg, pmm and cmm

Page 14: Friezes and Mosaics The Mathematics of Beauty Frieze pattern on the walls of the Taj Mahal.

p3, p31m, and p3m1

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


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