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 theTaj Mahal.
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Frieze patterns in the Taj Mahal The gardens and
corridors have many frieze patterns.
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Mosaics in the Taj Mahal The
ground around the Taj Mahal is laid with a tiling pattern of four-pointed stars.
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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.
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Jaisalmer in Rajasthan These are frieze
patterns appearing on the walls of the Jaisalmer Fort in Rajasthan, India.
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Friezes
We will look at the symmetries of these seven frieze patterns.
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Simplified friezes These exhibit
translational, rotational and reflective symmetries.
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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 >.
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Mosaics
A mosaic is a pattern that can be repeated to fill the plane and it is periodic along two independent directions.
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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)
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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.
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cm, p2 and pgg
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pmg, pmm and cmm
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p3, p31m, and p3m1
There are five more crystallographic groups: p4, p4g, p4m, p6 and p6m.