2

Table of Contents Geometry of Mosaics.......................................................................................3

The Earliest Mosaics......................................................................................4

Case Study: The Antioch Mosaics.................................................................5

Medieval to Modern Mosaics........................................................................6

Case Study: The Ravenna Mosaics...............................................................7

Middle Eastern Mosaics................................................................................8

The Direct Method........................................................................................9

The Indirect Method....................................................................................10

The Double Indirect Method........................................................................11

Case Study: Sonia King...............................................................................12

What are Tessellations?................................................................................13

Regular and Semi-Regular Tessellations.....................................................14

Tessellations in Nature................................................................................15

Case Study: M.C. Escher............................................................................16

Wallpaper Groups.......................................................................................17

Case Study: Nikolas Schiller......................................................................18

Make Your Own: The Line Method............................................................19

Make Your Own: The Slice Method...........................................................20

Glossary......................................................................................................21

About the Authors......................................................................................22

Illustration Credits......................................................................................23

3

Geometry of Mosaics

Mosaic is an art form that uses small pieces of materials placed next to each

other to create an image or pattern. The term for each piece of material is

tessera (plural: tesserae). The tesserae can be shaped like squares or be inter-

locking shapes like triangles and rectangles. There are many different ways to

arrange the tesserae to produce a picture.

One way is to use simple rows and col-

umns of square or rectangular pieces.

The patterns and pictures are then cre-

ated by using different colored pieces.

Another way of laying the tesserae out

is to have the tiles follow the outline of

a special shape, for example a central

graphic or letters. The tiles can also

overlap to form curvy, twisting pat-

terns. And sometimes the tesserae are

irregularly shaped and so there is not

pattern in how they are laid out. This is

called “crazy paving.”

Opus tessellatum mosaic (3rd century); the refers to

the mainly vertical rows in the main background

behind the animal, where tiles are not also aligned

to form horizontal rows.

An example of “crazy paving” by mosaic artist Sonia King

4

The Earliest Mosaics The earliest known mosaics date back to

3000 B.C. They were made of pieces of

colored shells, stone, and ivory. Excava-

tions in Iran have discovered the first ex-

amples of glazed tiles used in mosaics,

around 1500 B.C. These early mosaics

were made up of random placements of the

stones and some simple patterns. In Roman

times, geometric patterns became popular

and many buildings had mosaic floors.

It wasn’t until the 200 B.C. that mosaics were used to depict images. The minute

tesserae, or the small pieces of tile that make up a mosaic, were cut so small that

sometimes they were only a few millimeters in size. These pieces were so small

that artists could imitate paintings. The expansion of the Roman Empire brought

the popularity of mosaics to the corners of the globe. As empires rose and fell, mo-

saics stayed an important art form. Mosaics were used to depict religious scenes,

everyday life, and geometric patterns. West of Europe mosaics were also very

popular. In Islamic countries mosaics were not used to create images, but instead

they consisted of complex patterns and tessellations.

The gold leaf mosaic on the ceiling of the Florence Baptistry

Roman (left) and Islamic (right) mosaics

5

Case Study:

Antioch was an ancient city located near modern-day Antakya, Turkey. The

city thrived as a center of trade through the second to sixth century A.D. Mo-

saic floors were popular with wealthy merchants and because so many lived in

Antioch, hundreds of designs were constructed. Earthquakes destroyed the

prosperous city in 526 and 528 A.D. and it wasn’t until 1932 that archeological

digs found the incredible mosaics of

Antioch. The archeologists expected to

find great monuments and temples, but

instead discovered more than three hun-

dred mosaic floors. The largest of

which, measuring 20.5 by 23.3 feet, is

called The Worcester Hunt, and is

housed in the Worcester Art Museum.

The materials used by the artists who created the Antioch mosaics were mostly

colored marble and limestone. The designs of the mosaics discovered at Antioch

range from realistic images and scenery to geometric patterns.

An example of a geometric mosaic found at Antioch

Part of one of the Antioch mosaics located at the Worcester Art Museum

6

Medieval to Modern Mosaics

After the fall of the Roman Empire, the

popularity of mosaics also began to de-

cline. However, during the Middle Ages the

flourishing tile industry helped keep mosa-

ics alive in churches and abbeys. Often

these religious buildings would be deco-

rated by tiled patterns on the floors and

ceilings. The styles used started to become

less like traditional mosaics, and more

kinds and shapes of tile were used.

Eventually, production of mosaics halted,

until the 19th century when there was a revival

of interest in mosaics. The Art Nouveau move-

ment during this time period artists like Antoni

Gaudi and Josep Maria Jujol started using

This mosaic made of fur shows the emperor Franz Joseph I

of Austria

found objects to create mosaics. Their most popular work is in Guell Park, where

they used broken tiles and ceramics to cover buildings. Modern mosaics make use

of found objects and recycled pieces of pottery and ceramics to make interesting

patterns and tessellations.

An example of curved, interlocking mosaic pieces

Part of the groundbreaking mosaic work of Gaudi and Jujol in Guell Park

7

Case Study:

Ravenna is a city in modern-day Italy that was the capital city of the

Western Roman Empire from 402 until 476. During the height of the Roman

Empire, Ravenna was a bustling city filled with trade and fine arts. It’s numer-

ous churches and public buildings became the center of late Roman mosaic art.

An example of a great mosaic in Ravenna is the Church of San Giovanni Evan-

gelista, which was commissioned by

her in order to fulfill a promise she had

made having lived through a deadly

storm at sea. The mosaic depicted the

great storm along with portraits of roy-

alty. We only know the mosaic through

Renaissance sources because it was de-

stroyed in 1569. In the 6th century, af-

ter the fall of the Western Roman Em-

pire, the Ostrogoths produced the mosa-

ics in the Arian Baptistry, Baptistry of

Neon, and the Archiepiscopal Chapel as

well as many more. When Ravenna was

conquered in 539 A.D. by the Byzan-

tine Empire it became the center of

great Christian mosaic works. The mo-

saics in the Basilica of San Vitale and the Basilica of Sant'Apollinare Nuovo

are outstanding examples of Byzantine mosaics. The last of the Byzantine mo-

saics in Ravenna was commissioned by bishop Reparatus between 673-79 in

the Basilica of Sant'Apollinare in Classe.

Roman mosaic found at Calleva Atrebatum in modern-

day England

8

Middle Eastern Mosaics

In modern-day Southern Arabia, the

earliest mosaics date back to the late 3rd

century. The pre-Islamic cultures in these

areas created mosaics depicted scenes of

animals and people, but designs that

showed representations of people or ani-

mals were prohibited after the Arab con-

quest and the spread of Islam. Islamic ar-

chitects used mosaic technique to decorate

religious buildings and palaces after the

Muslim conquests of the eastern provinces of

the Byzantine Empire. The design of mosaics

began to include complex geometric patterns

or twisting vines and trees. The most important

early Islamic mosaic work is the decoration of

the Umayyad Mosque in Damascus, then capi-

tal of the Arab Caliphate. This mosque was

built in 706 A.D. and at one time there were

Islamic mosaics inside the Dome of the Rock in Palestine

more than 200 artisans working.

Because Islam prohibits depic-

tions of people and animals in

art, Islamic religious mosaics

filled mosques with their fantas-

tic geometric patterns. The non-

religious buildings in the Middle

East, however, had many mosa-

ics showing nature scenes with

animals.

An example of an Islamic nature mosaic

Golden mosaics in the dome of the Great Mosque in Corduba

(965-970)

9

The Direct Method There are three main techniques for laying out mosaics: the direct

method, the indirect method and the double indirect method. The direct method

of mosaic construction involves directly gluing the tesserae onto whatever sur-

face the mosaic will be on. This technique is useful for placing mosaics on

three-dimensional surfaces such as vases. Most mosaics created in the medie-

val and Roman times used this method.

Sometimes the tesserae have fallen off

mosaics and you can see the under-

drawings which are the drawings made

on the surface before the tiles are

added. The disadvantage of the direct

method is that you have to work di-

rectly on the surface, which could mean

sitting for days on a floor. The direct

method is not suitable for long-term or

large scale projects but is advantageous

for small projects because it allows a

lot of flexibility in design changes for

the artist.

Tool table for ancient roman mosaics a Roman villa in Spain

A 'Direct Method' mosaic courtyard made from irregular pebbles and stone strips

10

Indirect Method The indirect method is used for large mosaic projects where it is impractical to

work on-site. The tesserae are placed face-up on a mesh or sheet with an adhe-

sive backing in the pattern they will appear. The mosaic is then transferred

onto the wall, floor, or other surface. This method is useful for very large pro-

jects like murals. Many artists use this method because it allows them to work

in their own studios and rework necessary parts without altering the entire

work. Also, using the indirect method it is possible to maintain a more even

mosaic than using the direct method.

An example of a mosaic created using the indirect method. This mosaic is in the mausoleum of Galla Placidia

11

The double indirect method is when tesserae are placed face-up on a sheet of

paper (often adhesive-backed paper, sticky plastic or soft lime or putty) as it

will appear when installed. When the mosaic is complete, another sheet of ad-

hesive paper is placed on top of it. The piece is then turned over, the original

paper is carefully removed, and the piece is installed as in the indirect method

described above. This allows the artist to see the work as it is being put to-

gether. However, this method is very complex and requires great skill on the

part of the artist to avoid damaging the work. Its greatest advantage lies in the

possibility of the operator directly controlling the final result of the work,

which is important when the human figure is involved.

Double Indirect Method

An example of a mosaic floor created using the double indirect method. This specific mosaic was discovered in Israel and

covers more than 600 square feet

12

Case Study:

Sonia King was born in 1953 and is a well-known mosaic artist. Her work is

displayed across America and the world. She uses a variety of materials to cre-

ate her mosaics. Most often, she uses rocks and semiprecious stones in geomet-

ric patterns. King’s artwork does not depict images or real life scenes. Instead,

she uses patterns and natural materials to create abstract landscapes.

King creates contemporary, abstract

mosaic art with a complex variety of

tesserae, working with spacing, reflec-

tivity and texture. Most mosaics are

grouted, which means the spaces be-

tween the tiles are filled with a type of

cement, but King prefers not to use

grout and instead emphasize shadows

and negative space. Her abstract, mod-

ern mosaics are very different from the religious mosaics of the Byzantine pe-

riod, but are still just as beautiful.

Nebula Aqua mosaic by Sonia King

The Nebula Chroma mosaic King created for the Children’s Medical Center of Dallas

13

What are Tessellations? A tessellation is a pattern of figures that fill

a plane with no spaces or gaps. All tessel-

lations use translational symmetry while

some patterns use translational, rotational,

and reflective symmetry. In total there are

17 different combinations of symmetries

that can be used to create tessellations.

Subcategories of tessellations include regu-

lar, semi-regular, and demi-regular. The

word itself comes from the Latin word

tessella which means “small stone” and also

refers to the tiny bits of stone, clay, or glass

that make up mosaics. Used since ancient

times, tessellations remain an important artistic

tool to this day. Modern Artists such as M. C.

Escher and Nikolas Schiller make extensive

use of tessellations in their artwork.

An example of a demi-regular tessellation

An example of one of Nikolas Schiller’s tessellations

made from aerial photographs

Part of M. C. Escher’s Metamorphosis II tessellations

14

Regular and Semi-Regular Tessellations

The two major types of tessellations are regular and semi-regular tessellations

both of which use only regular polygons. A series of congruent regular poly-

gons comprises a regular tessellation, and two types of congruent regular poly-

gons comprise a semi-regular tessellation. All regular tessellations use only

equilateral triangles, squares, or regular hexagons. These shapes can tessellate

by themselves because 360, the number

of degrees around a point, is a multiple

of their interior angle (refer the box at

the bottom of the page) . Because semi

-regular tessellations can use more than

one type of regular polygon there are

eight possible combinations of shapes.

As with regular tessellations, the inte-

rior angles of all the shapes at a point

must add to 360 degrees. Although there

are 18 ways to fit regular polygons around a point, only 8 of these combina-

As shown above, semi-regular tessellations use multiple

types of congruent regular polygons

As shown above, equilateral triangles, squares, and regular hexagons are the only three

polygons whose interior angles can add up to 360 degrees. Because of this, they are the

only three shapes that can be used to make regular tessellations

15

Tessellations in Nature Many tessellations occur all around us

in nature. Honeycombs are a great ex-

ample of a hexagonal tessellation pat-

tern. Fruit like raspberries, grapefruit,

oranges, pineapple skin, and limes all

have repeating patterns that can be clas-

sified as tessellations. Animals can also

sport tessellations: snakeskin, tortoise

shells, and fish scales are all interesting

examples of tessellations in nature.

But perhaps the most interesting examples are found in the Bimini Wall and the Gi-

ant’s Causeway. The Bimini Wall is an underwater wall of rock with many right an-

gles. For many years, it was thought to be part of the ancient city of Atlantis. The

Giant’s Causeway is located in Northern Ireland and consists of many hexagonal

columns made of basalt, or hardened lava. Both of these geological phenomena are

called “tessellated pavement.” These fascinating rock formations look like they

have been cut into regular hexagons and rectangles, but they are actually naturally

formed, either from volcanoes or eroded bedrock. The right angles of the Bimini

Wall were created when water

and sand eroded surrounding

rock but left the inherent crys-

tal structure of the rock intact.

This is just one of the many ex-

amples of tessellated pavement

and other types of tessellations

in nature.

A honeycomb is an example of a natural tessellation

The Bimini Wall

16

Case Study:

M. C. Escher was a Dutch graphic artist that was famous for the tessellations in

his art. Born in 1898, his family lived in Leeuwarden, the Netherlands before

moving to Arnhem. After high school he attended the Haarlem School of ar-

chitecture and design. After failing to become an architect, Escher decided to

study decorative arts. In 1937, he started to incorporate mathematics into his

artwork which consisted mostly of

lithographs and woodcuts. From a pa-

per by George Polya, Escher learned

about the different symmetries used to

create tessellations. His 1936 Regular

Division of the Plane work and his

1937 work Metamorphosis I were some

of his first works to make use of tessel-

lations. In 1958, he published a book

called Regular Division of the Plane

that was composed of his previous

works that made use of tessellations.

His final work to use tessellations was

his Metamorphosis III which he made

Regular Division of the Plane III, woodcut, 1957 - 1958.

The concept of this work (Metamorphosis I) is to morph one image into a tessellated pattern, then gradually to alter the

outlines of that pattern to become an altogether different image. From left to right, the image begins with a depiction of

the coastal Italian town of Atrani. The outlines of the architecture then morph to a pattern of three dimensional blocks.

These blocks then slowly become a tessellated pattern of cartoon like figures in oriental attire.

17

Wallpaper Groups Two tessellations are in the same wallpaper group if they have the exact same

symmetries. This means two tessellations that have the same symmetries can

be translated, rotated, and reflected in the same way and still produce a tessel-

lation. Symmetries aren’t always easy to spot because two tessellations can

look different and still be in the same wallpaper group.

Yevgraf Fyodorov proved that only 17

wallpaper groups existed in his 1891

book, The symmetry of regular systems

of figures. Some wallpaper groups use

only translations, rotations, and reflec-

tion while others use a combination of

the three. All possible tessellations be-

long to one of the 17 wallpaper groups

each consisting of a different combina-

tion of translational, rotational, and re-

flective symmetries.

Although they appear different, both

these tessellations belong to the same

wallpaper group, p4m.

18

Case Study: Nikolas Schiller Nickolas Schiller is an artist who uses digital maps as his medium. While at-

tending George Washington University, Schiller created a blog that showcased

his modified maps. Using existing digital photographs, Schiller creates imagi-

nary maps that contain tessellations and other types of mathematically inspired

designs. His tessellated works take their inspiration from Arab mosaics such as

Great Mosque in southern Spain. Many of his images are of government build-

ings from the area around Washington D.C. The two images below are com-

posed of repeated images of the state house in Madison, Wisconsin and the

downtown of Montpelier.

19

Make Your Own: The Line Method

1. First draw an equilateral triangle and draw a wavy line down one of the

sides.

2. Copy this wavy line and rotate it 60 degrees. You should now have two out

of the three of the sides of the tes- sellation drawn.

3. On the final side drawn a wavy line from one vertex to the midpoint.

4. Copy this line and rotate it 180 degrees around the midpoint. Your tessella-

tion is com- plete.

20

Make Your Own: The Slice Method

1. Start with a square or equilateral triangle

2. Cut out a piece out of one side of the shape

3 . Paste this piece onto the opposite side of

the figure.

21

Glossary Archaeological digs: excavation sites where remnants of civilizations ancient

Limestone: a sedimentary rock composed of calcium carbonate

Ivory: The dentine from the teeth or tusks of elephants used for carvings and mosa-

ics.

Found objects: an object that is used in art that is intended for some other use.

Tesserae: The individual tiles that make up a mosaics

Underdrawings: drawings done on a surface before the mosaic is laid down. This

ensure that the tiles are placed properly

Grout: The substance used to fill in the cracks between the tiles of the mosaics

Ostrogoths: An Eastern Germanic tribe that played an important role in the fall of

the Roman Empire.

Artisans: Skilled craftsmen who create art such as sculptures and mosaics.

Tessellation: a pattern of figures that fill a plane with no spaces or gaps,

Rotational symmetry: The ability of an object to be rotated a certain amount and

still look the same.

Translational symmetry: The ability of an object to be translated a certain amount

and still look the same

Reflective symmetry: The ability of an object to be reflected across an axis of sym-

metry and still appear the same.

Regular polygon: A polygon that has angles that are all the same measure and sides

that are all the same length.

Interior angle: an angle found inside of a polygon.

Wallpaper group: A classification of a tessellation based on its symmetries. There

are 17 wallpaper groups.

Semi-regular tessellation: A tessellation that uses two or more types of regular

polygons.

Regular tessellation: A tessellation that uses a single type of regular polygon.

Lithograph: A printing method that uses a completely smooth stone or metal plate.

22

About the Authors

George Slavin grew up in Douglas, Massachusetts and he is currently a student

at the Mass Academy of Math and Science. Outside of school he likes to sing,

play piano, and ride his bicycle.

Anna Brill has lived in Maine, and Rhode Island, but currently resides in

Worcester Massachusetts. She is a student at the Mass Academy of Math and

Science. Outside of school she likes to sew, knit, bake, and read books.

This is their first book collaboration and they hope this volume inspires chil-

dren to appreciate the connection between mathematics and art.

23

Illustration Credits pg1: http://fineartamerica.com/images-medium/1-traditional-islamic-zeliji-around

-a-water-fountain-ralph-ledergerber.jpg

pg 2: http://en.wikipedia.org/wiki/File:Escher,_Metamorphosis_II.jpg

pg 4: http://en.wikipedia.org/wiki/File:Mosa%C3%AFque_d%27Ulysse_et_les_sir%C3%A8nes.jpg

http://en.wikipedia.org/wiki/File:Florenca133b.jpg

http://www.thejoyofshards.co.uk/history/index.shtml (multiple images)

pg 5: http://en.wikipedia.org/wiki/File:Antakya_Arkeoloji_Muzesi_1250287_nevit.jpg

http://www.thejoyofshards.co.uk/history/modern.shtml (multiple images)

pg 6: http://en.wikipedia.org/wiki/File:Fur_mosaic_Emperior_Franz_Josef.jpg

pg 7: http://en.wikipedia.org/wiki/File:Silchester_mosaic.jpg

http://en.wikipedia.org/wiki/File:Mosaics,_Worcester_Art_Museum_-_IMG_7457.JPG

pg 8: http://en.wikipedia.org/wiki/File:Cordoba_moschee_innen5_dome.jpg

http://en.wikipedia.org/wiki/File:Arabischer_Mosaizist_um_735_001.jpg

http://en.wikipedia.org/wiki/File:Arabischer_Maler_um_690_002.jpg

pg 9: http://en.wikipedia.org/wiki/File:Ancient_Roman_Mosaics_Villa_Romana_La_Olmeda_021_

Pedrosa_De_La_Vega_-_Salda%C3%B1a_(Palencia).JPG

http://en.wikipedia.org/wiki/File:Li_Jiang_Guesthouse.jpg

pg 10: http://farm1.static.flickr.com/89/237212229_7b1d7f02d9.jpg

pg 11: http://assets.nydailynews.com/img/2009/07/02/alg_mosaic.jpg

pg 12: http://www.solo-mosaico.org/2009/sonia-king/?lang=en (two images)

pg 13: http://en.wikipedia.org/wiki/File:Tiling_Semiregular_3-3-4-3-4_Snub_Square.svg

http://en.wikipedia.org/wiki/File:Tiling_Dual_Semiregular_V3-12-12_Triakis_Triangular.svg

pg 14: http://library.thinkquest.org/16661/simple.of.regular.polygons/regular.1.html

pg 15: http://en.wikipedia.org/wiki/File:Buckfast_bee.jpg

http://cltblog.com/media/2008/10/charlotte_spheres2-zoom.jpg

http://en.wikipedia.org/wiki/File:Escher,_Metamorphosis_II.jpg

pg 16: http://en.wikipedia.org/wiki/Regular_Division_of_the_Plane

http://en.wikipedia.org/wiki/Metamorphosis_I

pg 17: http://en.wikipedia.org/wiki/Wallpaper_group

pg 18: www.nikolasschiller.com

pg 19: George Slavin, 2011

pg 20: George Slavin, 2011