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Words 2820

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San Juanita Cramer

Southern New Hampshire University

The Heart of Mathematics

Professor Anca Parrish

Abstract

This paper discusses the historical background of tessellations, the mathematics of tessellations, and the applications of tessellations in the real world. Tessellations are found everywhere. M.C. Escher is the father of tessellations and his style and examples are discussed as well as the Islamic tessellations. There is an overview of the mathematics that is involved in tessellations and the polygons that can be tessellated and those that can’t. Finally, tessellations are used in real world applications. Examples are given of tessellated buildings and tessellations found in nature.

Tessellations: Mathematical Art What is a term used for the tiling a surface without gaps or overlaps? The term is Tessellation. The Math Forum states that “ a tessellation is created when a shape is repeated over and over again covering a plane without any gaps or overlaps” (“What is a Tessellation?”, n.d) Early cultures used tessellations to cover the floors and ceilings of buildings, many of its artistic elements can be found in many early cultures (Hoopes-Myers, 2010). Tessellations are also found in the nature. A perfect example of nature’s tessellation is the honeycomb of the honeybee; there are no gaps or overlaps in its hexagonal shapes. In Ireland, a volcanic episode created tessellations in the landscape of The Giant’s Causeway (“Giant’s Causeway” n.d.). Artists like M. C Escher use tessellations to create fascinating works of art. In his works Escher concentrated on tessellations and repeated forms. Mathematicians and scientists embraced Escher’s works because they involved the concepts of “geometry, logic, space and infinity” (M.C. Escher Biography, n.d.). One doesn’t have to be a mathematician to tessellate but knowing how shapes will fit together helps in creating beautiful tiled images. Tessellations had functionality in ancient cultures, in mathematics, and in today’s real world it is considered a form of art.

Tessellation History Historically the art of tessellation can be traced back as far as 4000 BC. The Sumerians decorated their homes and temples with pieces of hardened clay known as tessellations. The artistic elements of tessellations can be found in some ancient cultures like, the Arabic, the Egyptians, Greek, Japanese, Persians, and Romans just to name a few. Islam, which prohibited the use of living objects in its art, used only geometric tessellations. The Romans on the other hand, included detailed tessellated images of humans and nature (Hoopes-Myers, 2010). The ancient Greeks and Romans decorated their homes with tessellated mosaics. The mosaics were created without using the mathematical tessellations; however they often showed the repeated pattern of geometric shapes that tessellated to form a background or border (“Tessellations”n.d.). Tessera means small cube in Latin. Small stone cubes were used to make up “tessellata”, the mosaics on the floor and tiling of Roman buildings. Records show many sophisticated tilings were used to decorate flat surfaces by ancient craftsman. This type of tile can be seen in ancient designs but there is nothing written about this technique. Only regular polygons can be used to individually tile a plane but they can be used to create others, if the tiling is based on a square or triangle it can be modified to create irregular polygons that can tile a plane (History of Tessellations, n.d.). Mark Cartwright (2013) writes about the technique of creating the mosaics, also called “opus tessellatum”, using “small black, white and coloured squares typically measuring between 0.5 and 1.5cm…in size”. Materials used to make the “…squares (tesserae or tessellae) were cut from … marble, tile, glass, smalto (glass paste), pottery, stone and even shells”. Dating back to the 5th century the first pebbled floor was found in Greece. During the Bronze Age, the Minoan civilization in Crete and the Mycenaean civilization in Greece set their floors with the small pebbles. In the 8th century, matching patterns were used in the near East (Cartwright, 2013). There are two forms of tessellations, Escher style and the Islamic tessellation. The Islamic tessellation is much older than the Escher style. The Escher style is named after the Dutch artist Maurits C. Escher. Escher was famous for his tessellations; he created some beautiful work using “real things” in a repeating pattern (“Tessellations – The Beginnings”, n.d.). The online magazine, The Fountain (2010) writes that “in Islamic art, the spiritual world is regarded as being reflected in nature through geometry and rhythm. Hence, Islamic artists used geometry as an aid to raise their spiritual understanding as well as the viewer’s…” (Gelgi, 2010). Both forms are beautifully crafted. Maurits C. Escher was a graphic artist that mastered the art of tessellation. In the early 1920s, while in Spain he visited the Alhambra Palace. The tilings he found there were created in the 14th century by Moors and Christian artisans. They created beautiful, symmetrical and geometrical patterns using colored tiles. Some of the tilings in the Alhambra Palace are not considered tessellations because they are not repetitive but many found in the palace are true tessellations. This trip would have a great impact on his career as a graphic artist. His tessellations included shapes of animals, people, and everyday objects. He used “real things” in a repeating pattern. The Alhambra is now a tourist attraction, “theologians talk about the awe and balance of math as a way to feel a religious kind of awe” (“Tessellations-The Beginning”, n.d.). The Islamic tessellations are geometric forms created with only a compass and a ruler. The circle is the foundation for the Islamic pattern and “is often an organizing element…and it structures all the complex Islamic patterns using geometric shapes (York, 2004, p. 11). Three basic characteristics make up this pattern:

1. They are made up of a small number of repeated geometric elements. The basis for the pattern is the circle, square and straight line and are combined, repeated, intertwined and arranged in detailed combinations. The equilateral triangle and other square are the basis for two types of grid. There is a third grid of hexagons; “regular tessellations” is the mathematical term for these grids.

2. They are two-dimensional. Islamic designs often have a background and foreground pattern… Some geometric designs are created by fitting all the polygonal shapes together like the pieces of a puzzle, leaving no gaps and, therefore, requiring no spatial interplay between foreground and background. The mathematical term for this type of construction is “tessellation.”

3. They are not designed to fit within a frame (York, 2004, p. 11).

The circle which has no beginning or end is considered by Islamic artists as the most perfect geometric form. It is infinite (2004). Tessellation is a form of art that has been around for centuries. In its earliest form it was simple square tiles used to cover walls and floors. Artists like M.C. Escher have taken tessellation to a whole other level. There are different types of tessellations; however most follow the rule of a repeating design that has no gaps.

Do the Math Laying down a floor in a private home or public building may seem easy but it does require some mathematical knowledge. Knowing how to calculate the number of tiles required to lay down a simple floor is necessary. What happens when the design of the floor requires laying down different tiles in different patterns? Not all shapes can be used in tessellations. Guillermo Bautista (2011) states in his blog, Regular Tessellations: Why Only Three of Them? when tiling a floor one thing to remember is that the shapes used should cover the floor without an gaps or overlapping. This is a tessellated floor. Squares, equilateral triangles, or regular hexagons are regular polygons that are easily used in tiling (Figure 1).

Figure 1: Regular Polygons that tessellate.

Figure 1: Regular Polygons that tessellate. Note that there are no gaps or overlapping at the points where the panes meet. Squares, equilateral triangles and hexagons tessellate. This is because when the sums of their interior angles are added they add up to 360 degrees. Not all regular polygons can tessellate (Figure 2). Here, the pentagons, heptagons, and octagons will not tessellate Note that there are gaps where the points meet (Bautista 2011). These points are called the vertex. Figure 2: Regular Polygons that do not tessellate.

Figure 2: Regular Polygons that do not tessellate.

Why do some shapes tessellate and others do not? In order to answer this question Andrew Harris (2000) states, it should be understood that: * a whole turn around any point on a surface is 360° * the sum of the angles of any triangle = 180° * the sum of the angles of any quadrilateral = 360° * how to calculate or measure the interior angles of polygons

Regular polygons’ interior angles can be calculated in two ways 1. To find the size of one exterior angle, take a whole turn of 360° and divide it by the number of exterior angles (= the number of sides). Then use the exterior angle = the corresponding interior angle + 180° (because angles on a straight line add up to 180°) to find the interior angle. e.g. for a regular pentagon (5 sides, so it has 5 exterior angles) the exterior angle = 360° ÷ 5 = 72° so the interior angle = 180° - 72° = 108°” (Harris 2000).

Another way to work it out is: 2. The sum of the interior angles of a n-sided regular polygon = (n - 2) × 180°.

Once the total has been calculated in this way, the size of one of the interior angles can be found by dividing by the number of interior angles (= n).

e.g. for a regular pentagon (5 sides, so has 5 interior angles) the sum of the interior angles = (5 - 2) × 180° = 3 × 180° = 540° so one interior angle = 540° ÷ 5 = 108°” (Harris 2000).

These two methods work for regular polygons. Furthermore, Bautista (2011) demonstrates (Figure 3) the regular polygons that tessellate the plane must have the interior angles that meet at a common place sum up to 360 degrees.

Figure 3. Interior angles of polygons that tessellate the plane add up to 360 degrees.

Figure 3. Interior angles of polygons that tessellate the plane add up to 360 degrees.

Figure 4. Polygons that do not tessellate

Figure 4. Polygons that do not tessellate We must remember that equilateral triangles, squares, and regular hexagons will tessellate and are called regular tessellations. Not all the regular polygons will tessellate this way. The regular polygons that will not tessellate are pentagons, regular heptagons, and regular octagons.

Tessellations Today

Today tessellations can be found almost everywhere. Walk into a building and more than likely the floor is made of tile and laid out in the form of tessellations. Look up, and there is art on the wall that may have a tessellated pattern to it. The function and the beauty of tessellations can be seen in many applications today such as architecture, landscaping and art.

Because the use of images portraying humans or animals is prohibited on the buildings of the Islamic culture they used “geometric, floral, arabesque, and calligraphic primary forms, which are often interwoven into the architecture” (“Real Life Applications of Tessellations”, n.d.). A building with a plain façade can be boring but add tessellations to the façade of a building will add beauty and interest to it. Elegant shapes of glass or granite will stunningly cover the basic shape of a building. This can be seen in the tiling pattern on the façade of a building found in the Federation Square in Melbourne (Figure 5) (GEM1518K).

Figure 4.

Figure 4.

One example of tessellations used in landscaping is city zoning. A city is usually tessellated in zones to distinguish certain electrical districts, water districts, voting districts, school districts, and so on. In the Encyclopedia of Human Geometry, Barry Warf (Warf, 2006, p. 482) writes “…a metropolitan area may be divided into school districts, health districts, police districts, and so forth” (2006). Another example is the interest in the stimulation of landscapes in the fields of ecology and agronomy. Geometrical tessellations are proposed to promote new growth on patchy landscapes. These landscape models have been used to assist in forestry management. Traditional ecological neutral landscape models cannot be stimulated because they are typically geometric (Le Ber et all, 2009, p. 3536). Finally, natural tessellations can be seen in the landscape of Tasmania. On the flat rock of Eaglehawk Neck, there exists what is called Tessellated Pavement. It gets its name because the rocks have fractured into polygonal blocks (Figure 6) (Tessellated Pavements n.d.).

Figure 6. Tessellated Pavement

Figure 6. Tessellated Pavement

Tessellations have been used to cover the walls and floors of buildings for centuries but tessellations can also create wonderful works of art. M.C. Escher mastered the art of tessellations. In 1922 he produced 8 heads. Four different heads can be seen at first but the other four heads are not revealed until the picture is seen upside down (Figure 7). Notice that there are no gaps or spaces between the heads. (M.C. Escher, nd.)

Figure 7. 8 heads

Figure 7. 8 heads

Bruce Bilney is an artist who is known for his Australian themed tessellations. His work features animals in comfortable poses instead of the bulky shapes most often used by other artists. He created Sea Turtle Hexagon Tessellation for a tile floor company. The hexagonal piece will fit with other hexagonal pieces and when matched with another tile of the same design it will create whole turtles (Figure 8) (“Sea Turtle”, n.d.).

Figure 8. Sea Turtles

Figure 8. Sea Turtles

From the plain square tiles on a floor or wall to wonderful artwork, tessellations can be found and enjoyed by everyone everywhere. Escher used tessellations to create some very fascinating thought provoking art. Tessellations can be manmade but they are also found in nature. They are found in the pattern of a snakes skin, honeycomb, and in the cracked surface of dry earth. In ancient cultures tessellations were used as flooring and wall coverings but while this was done more for functionality it created beautiful artwork. Yes, math is needed to tessellate but you don’t have to be a math genius, just know basic math. Much of this work can still be seen today. Many homes and buildings are now using tessellations to cover floors and walls but are doing so in more appealing ways than the ancients did all the while creating art.

References

Bautista, G. (2011, June 3). Regular Tessellations: Why only three of them? Retrieved January 10, 2016, from http://mathandmultimedia.com/2011/06/04/regular-tessellations/.

Cartwright, M. (2013, May 14). Roman Mosaics. Retrieved December 21, 2015, from http://www.ancient.eu/article/498/

Gelgi, F. (2010, July 1). The Fountain Magazine - Issue - The Influence of Islamic Art on M.C. Escher. Retrieved December 21, 2015, from http://www.fountainmagazine.com/Issue/detail/The-Influence-of-Islamic-Art-on-MC-Escher

GEM1518K - Mathematics in Art & Architecture - Project Submission. (n.d.). Retrieved January 18, 2016, from http://www.math.nus.edu.sg/aslaksen/gem-projects/maa/0203-2-03-Escher/main5.html

Harris, A. (2000). Retrieved January 10, 2016, from http://ictedusrv.cumbria.ac.uk/maths/pgdl/unit9/Tessellation.pdf.

History of Tessellations and Background Information - Tessellations. (n.d.). Retrieved December 21, 2015, from http://sites.google.com/site/samanthakeasts/home/history-of-tessellations

Hoopes-Myers,Boise, M. (2010). Tour of Tessellations. Retrieved December 21, 2015, from http://edtech2.boisestate.edu/meganhoopesmyers/502/virtualtour/history.html

Le Ber, F., Lavigne, C., Adamczyk, K., Angevin, F., Colbach, N., Mari, J. -., & Monod, H. (2009). Neutral modelling of agricultural landscapes by tessellation methods—Application for gene flow simulation. Ecological Modelling, 220(24), 3536-3545. doi:10.1016/j.ecolmodel.2009.06.019

M. C. Escher Biography. (n.d.). Retrieved January 31, 2016, from http://www.biography.com/people/mc-escher-39783

Real Life Applications of Tessellations. (n.d.). Retrieved January 18, 2016, from http://www2.gvsu.edu/oxfordj/angie.html

"SEA TURTLE HEXAGON TESSELLATION" an artistic hexagonal tessellation of turtles from well-known tessellation artist Bruce Bilney of Australia. (n.d.). Retrieved January 18, 2016, from http://www.tessellations.org/bruce-tessellation-turtle-motif-01.shtml

The Tessellated Pavements of Tasmania. (n.d.). Retrieved January 18, 2016, from http://www.amusingplanet.com/2014/01/the-tessellated-pavements-of-tasmania.html.

Tessellations in the World. (n.d.). Retrieved December 21, 2015, from http://www.shodor.org/interactivate/discussions/TessellationsWorld/ Tessellations - The Beginnings. (n.d.). Retrieved December 21, 2015, from http://www.tessellations.org/tess-beginnings.shtml

Warf, B. (2006). Encyclopedia of human geography. Thousand Oaks, Calif.: Sage Publications

What Is a Tessellation? (n.d.). Retrieved December 13, 2015, from http://mathforum.org/sum95/suzanne/whattess.html

York, N. (2004). Islamic art and geometric design: Activities for learning. New York, N.Y.: Metropolitan Museum of Art.…...

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...According to the article Art, the definition of art is “the concept that any form of creativity should be valued for its own merits alone, rather than measured against some fixed set of criteria that is laid down by the art establishment.” The second article the definition of Art “expresses human imagination, not least when it engages with humanity's destiny.” An example of painting "The Mona Lisa" by Leonardo da Vinci the painting is an Image of a lady smiling because she is pregnant according to the article The Art Newspaper. This painting goes with the second definition of art because it expresses how Mona Lisa feels about being pregnant. An example of sculpture is by Auguste Rodin “The Thinker Statue” the purpose of The Thinker statue was to represent the artist as himself at the top of the door reflecting, The Thinker is a man in sober meditation battling with a powerful internal struggle. The sculpture is a man with a pose with hand to the chin, right elbow to the left knee, and crouching position. An example of Architecture is by Gregory Ain he is known for “Dunsmuir Flats, designed in 1937, brought in Neutra's influence in greater measure, but also displayed Ain's own ideas, limiting building costs while combining both privacy and exterior light.” An example of photography according to worlds famous photos is by Arthur Sasse, he took a photo of Albert Einstein is one of the most popular figures he is considered a genius because he created the Theory of Relativity...

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