tessella - the Scientia Review

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
2

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 interlocking
shapes like triangles and rectangles. There are many different ways to
arrange the tesserae to produce a picture.
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.
One way is to use simple rows and columns
of square or rectangular pieces.
The patterns and pictures are then created
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 patterns.
And sometimes the tesserae are
irregularly shaped and so there is not
pattern in how they are laid out. This is
called “crazy paving.”
An example of “crazy paving” by mosaic artist Sonia King
3

The Earliest Mosaics
The earliest known mosaics date back to
3000 B.C. They were made of pieces of
colored shells, stone, and ivory. Excavations
in Iran have discovered the first examples
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, mosaics
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.
Roman (left) and Islamic (right) mosaics
4
The gold leaf mosaic on the ceiling of the Florence Baptistry

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. Mosaic
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
An example of a geometric mosaic found at Antioch
digs found the incredible mosaics of
Antioch. The archeologists expected to
find great monuments and temples, but
instead discovered more than three hundred
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.
Part of one of the Antioch mosaics located at the Worcester Art Museum
5

Medieval to Modern Mosaics
This mosaic made of fur shows the emperor Franz Joseph I
of Austria
Eventually, production of mosaics halted,
until the 19th century when there was a revival
of interest in mosaics. The Art Nouveau movement
during this time period artists like Antoni
Gaudi and Josep Maria Jujol started using
After the fall of the Roman Empire, the
popularity of mosaics also began to decline.
However, during the Middle Ages the
flourishing tile industry helped keep mosaics
alive in churches and abbeys. Often
these religious buildings would be decorated
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.
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.
6
An example of curved, interlocking mosaic pieces
Part of the groundbreaking mosaic work of Gaudi and Jujol in Guell Park

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 numerous
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-
Roman mosaic found at Calleva Atrebatum in modern-
day England
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 royalty.
We only know the mosaic through
Renaissance sources because it was destroyed
in 1569. In the 6th century, after
the fall of the Western Roman Empire,
the Ostrogoths produced the mosaics
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 Byzantine
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 mosaics
in Ravenna was commissioned by bishop Reparatus between 673-79 in
the Basilica of Sant'Apollinare in Classe.
7

Middle Eastern Mosaics
Islamic mosaics inside the Dome of the Rock in Palestine
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 capital
of the Arab Caliphate. This mosque was
built in 706 A.D. and at one time there were
Golden mosaics in the dome of the Great Mosque in Corduba
(965-970)
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 animals
were prohibited after the Arab conquest
and the spread of Islam. Islamic architects
used mosaic technique to decorate
religious buildings and palaces after the
8
An example of an Islamic nature mosaic
more than 200 artisans working.
Because Islam prohibits depictions
of people and animals in
art, Islamic religious mosaics
filled mosques with their fantastic
geometric patterns. The nonreligious
buildings in the Middle
East, however, had many mosaics
showing nature scenes with
animals.

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 surface
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-
Tool table for ancient roman mosaics a Roman villa in Spain
val and Roman times used this method.
Sometimes the tesserae have fallen off
mosaics and you can see the underdrawings
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 directly
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.
A 'Direct Method' mosaic courtyard made from irregular pebbles and stone strips
9

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 adhesive
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 projects
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
10

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
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 adhesive
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 together.
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.
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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 create
her mosaics. Most often, she uses rocks and semiprecious stones in geometric
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, reflectivity
and texture. Most mosaics are
grouted, which means the spaces between
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 period,
but are still just as beautiful.
12
Nebula Aqua mosaic by Sonia King
The Nebula Chroma mosaic King created for the Children’s Medical Center of Dallas

What are Tessellations? A tessellation is a pattern of figures that fill
a plane with no spaces or gaps. All tessellations
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 regular,
semi-regular, and demi-regular. The
word itself comes from the Latin word
An example of a demi-regular tessellation
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.
Part of M. C. Escher’s Metamorphosis II tessellations
13
An example of one of Nikolas Schiller’s tessellations
made from aerial photographs

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 polygons
comprises a regular tessellation, and two types of congruent regular polygons
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 interior
angles of all the shapes at a point
types of congruent regular polygons
must add to 360 degrees. Although there
are 18 ways to fit regular polygons around a point, only 8 of these combina-
14
As shown above, semi-regular tessellations use multiple
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

Tessellations in Nature
A honeycomb is an example of a natural tessellation
Many tessellations occur all around us
in nature. Honeycombs are a great example
of a hexagonal tessellation pattern.
Fruit like raspberries, grapefruit,
oranges, pineapple skin, and limes all
have repeating patterns that can be classified
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 Giant’s
Causeway. The Bimini Wall is an underwater wall of rock with many right angles.
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 crystal
structure of the rock intact.
This is just one of the many examples
of tessellated pavement
and other types of tessellations
in nature.
15
The Bimini Wall

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 architecture
and design. After failing to become an architect, Escher decided to
study decorative arts. In 1937, he started to incorporate mathematics into his
Regular Division of the Plane III, woodcut, 1957 - 1958.
artwork which consisted mostly of
lithographs and woodcuts. From a paper
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 tessellations.
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
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.
16

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 tessellation.
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 reflection
while others use a combination of
the three. All possible tessellations belong
to one of the 17 wallpaper groups
each consisting of a different combination
of translational, rotational, and reflective
symmetries.
17
Although they appear different, both
these tessellations belong to the same
wallpaper group, p4m.

Case Study: Nikolas Schiller
Nickolas Schiller is an artist who uses digital maps as his medium. While attending
George Washington University, Schiller created a blog that showcased
his modified maps. Using existing digital photographs, Schiller creates imaginary
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 buildings
from the area around Washington D.C. The two images below are composed
of repeated images of the state house in Madison, Wisconsin and the
downtown of Montpelier.
18

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 tessellation
is com- plete.
19

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.
20

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 mosaics.
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 symmetry
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.
21

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 children
to appreciate the connection between mathematics and art.
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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
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