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At Right Angles | Vol. 3, No. 1, March 2014 5

ecently, my daughter and I happened to take a trip to the

historical places of Delhi in the famous Hop-on-Hop-off

bus. In our visit to various historical places we not only

admired the marvels of the civil engineering of the past, we were

also deeply impressed by the complexity and beauty with which

our ancestors had blended geometry and art.

Among the interesting patterns that left us really in awe were

the intricate designs we found on many pavements, ceilings

and walls. We noticed that in most of these patterns a group of

motifs were joined together such that they covered the entire

plane. It was interesting to see how a single block of motif (s)

fitted so well with each other without any gaps so as to extend in

all the directions of the plane. We noticed several such designs

in almost all the historical places that we had visited. I have

attached some pictures that we took.

feature

Covering the Plane

with Repeated

Patterns -

“Hey mummy, look there. It is so beautiful!”

“Yes, my dear, it is indeed. It’s the epitome of Indian architecture.

It has tessellations and many symmetrical patterns”

Architecture speaks of its time and place, but yearns for

timelessness. Through this two part article, I will dwell on a topic

that closely links mathematics with art and culture.

H����� G�����

Keywords: Pattern, tessellation, quadrilateral, triangle, architecture, tile, Mughal, Escher

Part I

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6 At Right Angles | Vol. 3, No. 1, March 2014

In this article I wish to share the mathematical

ideas behind these perfectly fitting motifs. We will

see how these complicated patterns can be broken

into simpler components. This exercise can be

taken up with middle grade students to encourage

them to integrate mathematics with culture and

art. To start with, a quick look at these beautiful

images:

A partition at Qutub Minar Delhi Metro Station, Vishwavidayala Ceiling at Red Fort

Net Partitions at Red Fort

Partition at Qutub Minar Net partition at

VC office, DU

Wall at Qutub Minar

A house in Chandni Chowk Carpet in Bangla Sahib Gurudwara Jama Masjid

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At Right Angles | Vol. 3, No. 1, March 2014 7

In all these pictures, you will notice that there is

either one shape or a collection of shapes being

repeated on a planar surface; the shapes fill the

plane, with no gaps and no overlaps. The pattern

can be extended in all directions of the plane. Such

designs are known as tilings or tessellations

(Greek: ‘tessell’ or ‘tesera’ which means ‘tile’).

The defining features of a tessellation are: (a) an

infinite collection of congruent shapes, (b) the

shapes fit together to fill a two-dimensional plane

with no gaps and no overlaps.

The study of tessellations cuts across mathe- matics, art, architecture, culture and history.

Tessellations can be found not only in man-made

monuments but also the natural world – in the

honeycombs of bees, the skins of snakes and the

shells of tortoises. Today, the topic has relevance

in scattered fields such as x-ray crystallography,

quantum mechanics, cryptology and minimization

of waste material in the cutting of metal sheets.

One of the first mathematical studies of

tessellations was conducted by Johannes Kepler

in 1619, emanating perhaps from his study of

snowflakes. More than two centuries later, in

1891, Russian crystallographer E. S. Fedorov

gave the connection between isometries and

tessellations. Maurits Cornelius Escher (1898-

1972) was amongst the pioneers to look into

tessellating patterns in detail; he created many

masterpieces of his own.

A house in Chandni Chowk

Wall in Qutub Minar

Wall partition at Taj Mahal, Agra

Glimpse of Purani Dilli

Wall in Red Fort

Jahangir Mahal, Agra Fort