geometry adventures

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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.

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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. Momo-saics 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

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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

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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

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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 Wesaf-tern 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

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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)

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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 under-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

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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.

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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

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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 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

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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-reguregu-lar, 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

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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 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

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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

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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.

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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 combinareflec-tion of the three. All possible tessellations be-long to one of the 17 wallpaper groups each consisting of a different combina-tion of translacombina-tional, rotacombina-tional, and re-flective symmetries.

Although they appear different, both these tessellations belong to the same wallpaper group, p4m.

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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.

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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 tessellation 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

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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

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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.

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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.

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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

geometry adventures

Table of Contents

Geometry of Mosaics

The Earliest Mosaics

Case Study:

Medieval to Modern Mosaics

Case Study:

Middle Eastern Mosaics

The Direct Method

Indirect Method

Double Indirect Method

Case Study:

What are Tessellations?

Regular and Semi-Regular Tessellations

Tessellations in Nature

Case Study:

Wallpaper Groups

Case Study: Nikolas Schiller

Make Your Own: The Line Method

Make Your Own: The Slice Method

Glossary

About the Authors

Illustration Credits