Free Essay

Tessellations: Mathematical Art

In: Other Topics

Submitted By sjcramer
Words 2820
Pages 12
Tessellations: Mathematical Art
San Juanita Cramer
Southern New Hampshire University
The Heart of Mathematics
Professor Anca Parrish

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.


Bautista, G. (2011, June 3). Regular Tessellations: Why only three of them? Retrieved January 10, 2016, from
Cartwright, M. (2013, May 14). Roman Mosaics. Retrieved December 21, 2015, from
Gelgi, F. (2010, July 1). The Fountain Magazine - Issue - The Influence of Islamic Art on M.C. Escher. Retrieved December 21, 2015, from
GEM1518K - Mathematics in Art & Architecture - Project Submission. (n.d.). Retrieved January 18, 2016, from
Harris, A. (2000). Retrieved January 10, 2016, from
History of Tessellations and Background Information - Tessellations. (n.d.). Retrieved December 21, 2015, from
Hoopes-Myers,Boise, M. (2010). Tour of Tessellations. Retrieved December 21, 2015, from
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
Real Life Applications of Tessellations. (n.d.). Retrieved January 18, 2016, from
"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
The Tessellated Pavements of Tasmania. (n.d.). Retrieved January 18, 2016, from
Tessellations in the World. (n.d.). Retrieved December 21, 2015, from Tessellations - The Beginnings. (n.d.). Retrieved December 21, 2015, from
Warf, B. (2006). Encyclopedia of human geography. Thousand Oaks, Calif.: Sage Publications
What Is a Tessellation? (n.d.). Retrieved December 13, 2015, from
York, N. (2004). Islamic art and geometric design: Activities for learning. New York, N.Y.: Metropolitan Museum of Art.…...

Similar Documents

Free Essay

Shortcut in Mathematical Calculations

...5 → Final Answer ● Difference of Two Square Numbers Given: 16² - 25² Steps: - Get the sum of the two base numbers 25 + 16 = 41 - Difference of the two base numbers 25 – 16 = 9 - Multiply the sum and difference 41 × 9 = 369 ↓ Final Answer Examples: 1. Given: 17² - 20² - 20 + 17 = 37 - 20 – 17 = 3 - 37 × 3 = 111 → Final Answer 2. Given: 58² - 69² - 69 + 58 = 127 - 69 – 58 = 11 - 127 × 11 = 1397 → Final Answer 3. Given: 45² - 51² - 51 + 45 = 96 - 51 – 45 = 6 - 96 × 6 = 576 → Final Answer SHORTCUTS IN MATHEMATICAL CALCULATIONS (Project in Number Theory 8) Phoebe Kyle Nadine B. Malig-on 8 – Archimedes Mdme. Marichu S. Gajardo (subject teacher)...

Words: 535 - Pages: 3

Premium Essay

Mathematical Competence of Grade 7 Students

...mathematics should be taught. The principles that underlie this approach are strongly influenced by Hans Freudenthal's concept of 'mathematics as a human activity'. He felt that students should not be considered as passive recipients of ready-made mathematics, but rather that education should guide the students towards using opportunities to reinvent mathematics by doing it themselves.Study situations can represent many problems that the students experience as meaningful and these form the key resources for learning; the accompanying mathematics arises by the process of mathematization. Starting with context-linked solutions, the students gradually develop mathematical tools and understanding at a more formal level. Models that emerge from the students' activities, supported by classroom interaction, lead to higher levels of mathematical thinking.For more information about RME in the Netherlands, see: * Mathematics education in The Netherlands, a guided tour (pdf). * Realistic Mathematics Education: work in progress.Other relevant research publi | One of the key issues to look at when examining any Learning Theory is Transfer of Learning. Indeed, this is such an important idea, that it is a field of research in its own right. Researchers and practitioners in this field work to understand how to increase transfer of learning -- how to teach for transfer.IntroductionConstructivismSituated LearningTransfer of LearningGeneral Learning Theory ReferencesTop of......

Words: 5288 - Pages: 22

Free Essay

The Art

...126 Nptes." The Appoggiatura. " The Acciacoatura. Chapter XVI Musical Form." -Thematic 129 Figures and their Treatment." IPhrasing. 136 Development. Chapter XVII The Suite." bande." The The Old Dances." The Chaconne." The SaraThe Courante." The uet." Passacaglia." The MinThe Gavotte." The Bourree." The Pavane." The AUemande." The Eigaudon." Gigue. Chapter XVIII The Sonata." Sonatar-movement. 143 Chapter XIX The Slow Movement." Bondo-f orms Minuet-form." Finale ."Song-"orm." The The . 155 Scherzo." Chapter XX Other Sonata Forms." 167 Overture." Concerto." Sonatina. "Symphony. Chapter The XXI Vocal Forms." Cavatina." The Mass." Ariar-form.Vocal The Art-song. Bondo. 172 Strophe-form." Chapter XXII Contrapuntal Forms." Imitations." 178 Monophony." Canon. Homophony." phony." Poly186 Chapter XXIII TheFugue." tion." Answer." Eepercussion and Subject." Coda. Counter-subject." Episodes." Stretto." ExposlOrgan 195 Point." Chapter XXIV Modern Waltz. Polonaise." Mazurka." Galop." March. nade. SereCavatma." Eomanza " Bhapsodie. pourri." PotBarcarolle. Pastorale. Tarantella. Albumleaf." Ballade." Berceuse. Nocturne." Poem. "Symphonic Dance "ThePolka." Beverie." " " Fonns.-Drawing-room-Music.-The " " " " Chapter XXV Conclusion. 204 THE Theory of Music CHAPTER ACOUSTICS.......

Words: 33733 - Pages: 135

Premium Essay


...primarily as a painter. Matisse is commonly regarded, along with Picasso and Marcel Duchamp, as one of the three artists who helped to define the revolutionary developments in the plastic arts in the opening decades of the 20th century, responsible for significant developments in painting and sculpture. Although he was initially labeled a Fauve (wild beast), by the 1920s he was increasingly hailed as an upholder of the classical tradition in French painting. His mastery of the expressive language of colour and drawing, displayed in a body of work spanning over a half-century, won him recognition as a leading figure in modern art. Henri Matisse uses Fauvism as a style began around 1900 and continued beyond 1910. The movement as such lasted only a few years, 1904–1908, and had three exhibitions. (Fauvism is the style of les Fauves (French for "the wild beasts"), a loose group of early twentieth-century Modern artists whose works emphasized painterly qualities and strong color over the representational or realistic values retained by Impressionisms. Famous of works: * Woman Reading (1894), Musée National d'Art Moderne Paris * Le Mur Rose (1898), Musée National d'Art Moderne * "Canal du Midi" (1898), Thyssen-Bornemisza Museum * Notre-Dame, une fin d'après-midi(1902), Albright-Knox Art Gallery,Buffalo, New York * "Luxe, Calme, et Volupté" (1904), Musée National d'Art Moderne * Green Stripe (1905) * The Open Window (1905) * Woman with a......

Words: 781 - Pages: 4

Premium Essay


...pastels, and oils on posters. National Museum of Art, Washington D.C. And Taipei Fine Arts Museum, Taiwan Yayoi has a wide array of art that catches my eye. With then use of polka dots and motifs in her paintings to make wonderful works of art. Watercolors are also a bonus when looking at an art display as it can be very vivid and in high school I enjoyed doing thing that's involved the same things that she does. Pastels and Oils were also fun to use and were some personal favorites that make art a fun thing to do that someone who may not be interested in doing will like it. In one of her paintings from her exhibit, Infinite Obsession, she uses a wide variety of colors and polka dots. There are many different things that one could decipher about this painting but it almost looks like it could be a city or galaxy. The colors that stand out the most are green, blue, and red. It features an abundance of varying sights and colors that will wow a person with it being fluid and precise as it doesn't look like someone just threw paint on a board but it looks like someone did the work intentionally with thought and reasoning behind it. Faith Ranggold, Born1930, Harlem, New York, Page 17[->1] Mixed media on Painted Quilts. Norton Museum of Art,Florida and National Museum of American Art.,Washington D.C. Being black during a time with racism and segregation and still being able to make it in the art industry makes Faith a really......

Words: 466 - Pages: 2

Free Essay


...As a lover of art I tend to find, beauty in everything some way. Whether it be a simple flower out in a field of brown grass, or a pattern that everyone thinks looks too busy. I didn’t realize how the ‘Beauty is all around you’ quote would fit me until I actually started to study it. No, I’m not art major because I can’t draw to save my life, but I found that studying it for a hobby is more relaxing than I thought it would have been. Living in New York City, I was able to go to an art museum every weekend. Look at the different paintings on the wall and feel a different emotion for each. If was funny how an 18x24 painting could make me feel so different about certain things. All a painting is and ever will be, and lines and different stroke one a white canvas if you break it down bit by bit. But together those line and strokes make something beautiful. When I moved from New York I thought I would have to give up that passion for a whole year. I, personally knew that I could do it but then I would be bored on my weekends, something I didn’t want to happen ever especially since I felt I was moving to the middle of nowhere. But I was mistaken when I found out that El Paso had an art museum. That was very shocking news to me, I know not everywhere has an art museum or a museum at all. Recently I was able to visit, I know it’s not good to compare two things together all the time, of course the El Paso art museum isn’t, The Metropolitan Museum of Art or the Whitney Museum of......

Words: 733 - Pages: 3

Premium Essay

Mathematical Techniques for Economists 112ecn

...Mathematical Techniques for Economists 112ECN INVESTMENT APPRAISAL USING A SPREADSHEET As the mathematics involved in calculating the NPV of a project can be quite time consuming, a spreadsheet program can be a great help. Although Excel has a built in NPV formula, this does not take the initial outlay into account and so care has to be taken when using it. Example An investment requires an initial outlay of £25,000 with the following expected returns: £5,000 at the end of year 1 £6,000 at the end of year 2 £10,000 at the end of year 3 £10,000 at the end of year 4 £10,000 at the end of year 5 Is this a viable investment project if money can be invested elsewhere at 15%? Solution Follow the instructions set out below: CELL Enter Explanation A1 NPV Example Label to remind you what example this is A3 YEAR Column heading label B3 RETURN  Column heading label C3 PV Column heading label C1 Interest rate = Label to tell you interest rate goes in next cell.  D1 15% Value of interest rate. (NB Excel automatically treats this % format as 0.15 in any calculations.) A4 to A9 Enter numbers 0 to 5 These are the time periods B4 -25000  Initial outlay (negative because it is a cost) B5 5000 Returns at end of years 1 to 5  B6 6000 B7 10000 B8 10000 B9 10000 C4 =B4/(1+$D$1)^A4 Formula calculates PV corresponding to return in cell B4, time period in cell A4 and interest rate in cell D1. Note the $ to anchor cell D1. C5 to C9 Copy cell C4......

Words: 1248 - Pages: 5

Premium Essay

Measles Mathematical Model

...UCI Program in Public Health Seminar Series Andrew Noymer, Ph.D. and Katelyn C. Corey “I’m going to Disneyland”, Or: what levels of vaccination are necessary for measles control and eradication?” Monday, May 4, 2015 12:00 PM - 1:00 PM  Calit2 Auditorium * Presentation is based on a mathematical model of measles transmission in developing countries * Rescheduled seminar due to Stéphane Helleringer, Ph.D inability to attend * Corey UCI public health major, fall graduate school in UCLA * Noymer Ph.D in sociology from UCBerkeley * M.Sc., London School of Hygiene & Tropical Medicine * Is now an associate professor in population health and disease prevention in public health school UCI * Ross Model of malaria, 1911 * Dm/dt = bf’ (1-m) vm * Where m is malaria rate in humans, b is mosquito bite rate, f = Pr (infectious|infected) f’ for mosquitos, density of mosquitos: a: overall u: infected, v is the recovery rate, h is the hatch (birth) rate of mosquitos * 2 equation ODE model * Feedback loop * This equation had remained influential to this day * You do not get measles twice * Kermack-mckendrick model * Measles is a viral disease of humans caused buy the measles virus * Highly contagious * Family of paramyxoviridae * Prodrome period * Said it came 10,000 years ago when we tried to domesticate wild dogs * Vaccine preventable * R0 measures how......

Words: 368 - Pages: 2

Free Essay


...INTRODUCTION In all of human history, art has mirrored life in the community, society, and the world in all its colors, lines, shapes, and forms. The same has been true in the last two centuries, with world events and global trends being reflected in the art movements. The decades from 1900 to the present have seen the human race living in an ever shrinking planet. The 20th century saw a boom in the interchange of ideas, beliefs, values, and lifestyles that continues to bring the citizens of the world closer together. Technological breakthroughs From the Industrial Revolution of the late 1800s, the world zoomed into the Electronic. Age in the mid-1900s, then into the present Cyberspace Age. In just over 100 years, humans went from hand-cranked telephones to hands-free mobile phones, from the first automobiles to inter-planetary space vehicles, from local radio broadcasting to international news coverage via satellite, from vaccinations against polio and smallpox to laser surgery. Social, political, and environmental changes There has been migration across the globe, allowing different cultures, languages, skills, and even physical characteristics of different races to intermingle like never before. The 20th century also suffered through two World Wars, and several regional wars in Asia, Africa, and the Middle East. There was the Great Depression of the 1930s, and the Asian economic crisis of the 1990s. Considered the modern-day plague, AIDS has......

Words: 2738 - Pages: 11

Free Essay

Mathematical Circles

...Sadovskii & AL Sadovskii Intuitive Topology: V. V. Prasolov Groups and Symmetry: A Guide to Discovering Mathematics: David W. Farmer Knots and Surfaces: A Guide to Discovering Mathematics: David W. Farmer & Theodore B. Stanford Mathematical Circles (Russian Experience): Dmitri Fomin, Sergey Genkin & Ilia Itellberg A Primer of Mathematical Writing: Steven G. Krantz Techniques of Problem Solving: Steven G. Krantz Solutions Manual for Techniques of Problem Solving: Luis Fernandez & Haedeh Gooransarab Mathematical World Mathematical Circles (Russian Experience) Dmitri Fomin Sergey Genkin Ilia Itenberg Translated from the Russian by Mark Saul Universities Press Universities Press (India) Private Limited Registered Office 3-5-819 Hyderguda, Hyderabad 500 029 (A.P), India Distribllted by Orient Longman Private Limited Regisfered Office 3-6-752 Himayatnagar, Hyderabad 500 029 (A.P), India Other Office.r BangalorelBhopaVBhubaneshwar/Chennai Emakulam/Guwahati/KolkatalHyderabad/Jaipur LucknowlMumbailNew Delhi/Patna ® 1996 by the American Mathematical Society First published in India by Universities Press (India) Private Limited 1998 Reprinted 2002, 2003 ISBN 81 7371 115 I This edition has been authorized by the American Mathematical Society for sale in India, Bangladesh, Bhutan, Nepal, Sri Lanka, and the Maldives only. Not for export therefrom. Printed in India at OriO!'l Print:rs, Hyderabad 500 004 Published by Universities Press (India) Private Limited......

Words: 86787 - Pages: 348

Premium Essay

Mathematical Economics and Finance

...Mathematical economics and finance: good or bad? Pareto and Walras were the first to use mathematics in economics and finance at the end of the nineteenth century. They created classical models of the free markets and explained these mathematically. After these models were created, other famous economists came up with mathematical economical ideas, such as Schumpeter and Keynes. Mathematics was used to simplify and clarify various complicated theories. This use resulted in both advantages and disadvantages. This essay will evaluate the uncritical use of mathematics in economics and finance on multiple aspects. Firstly, if one uses mathematics in economics and finance uncritically, one needs to know all the numerical values and other variables to actually explain and predict certain events and its mutual interdependencies. This knowledge is never entirely available, because the values depend on too many particular circumstances. What’s more, the succession of events in the history of the economics does not show any internal coherence, which makes predicting harder. Moreover, there is no controlled experiment in the economic research field and no falsifiable hypotheses are made (Von Hayek, 1989). Furthermore, different economic models have different implications, thus are applicable to according situations. Hence, not one mathematical model could be implemented (Rodrik, 2015). In addition, human beings and their behaviour should be included in the theories of economics.......

Words: 790 - Pages: 4

Premium Essay

Mathematical Happenings Paper

...Mathematical Happenings Egypt is one of the advanced civilizations in the ancient world. If it was not for them and their advancements in mathematics, the world we live in now would be a very different place. They paved the way for the Greeks and other ancient civilizations to continue improving not only the world of math but also most other industries. One of the first people to start writing down anything were the Ancient Egyptians. They needed to keep track of the days for planting and harvesting, they needed geometry to build things, and arithmetic for trading purposes. It paved the way for the barter system. They needed a way to figure out how much of something else they would get for their product. The members of the Egyptian society that were in charge of numbers and keeping track of the surplus of good were priests. Their jobs besides their religious duties were in charge of creating a writing system, keeping records, create a calendar, watch the sky for astrological events and other intellectual endeavors. The number system for Ancient Egypt was called hieroglyphics. The system is based on groupings of 10. The numbers each have a name. The number 1 is called the stick, the number 10 is called the heel bone, the number 100 is called the scroll, the number 1,000 is called the lotus flower, the number 10,000 is called the bent finger or snake, the number 100,000 is called the burbot fish or tadpole and the number 1,000,000 is called the astonished man.......

Words: 736 - Pages: 3

Premium Essay

Mathematical Optimization

...Mathematical Optimization: Models, Methods and Applications Final Assignment 06-11-2015 Rasmus pages / 13.137 characters (including spaces) | Part 1 General about part 1 The purpose with this part is to analyze a Single-Sourcing Problem (SSP). A Single-Sourcing Problem of course both has benefits and risks, but I will discuss that furthermore through the assignment. During the assignment I will try to discuss and comment on everything that I do. My code and the answers I receive from can be seen in my appendices. (i) In the first question in part 1, I am asked to solve the SSP using the data in Figure 1. We have 4 facilities and 30 customers. In Figure 1 the demand of each customer is also given, and of course I will have to satisfy this. Therefore this will become one of my constraints. It is also known that each facility has a capacity, and of course this will become a constraint as well. Because it is a SSP problem, we are also given the information that each customer has to be served by exactly one facility. When a facility delivers one unit to a customer it faces a cost. The purpose with the first question is to minimize the cost that the facility faces delivering the units. I will now show what the problem looks like: Minimizexi=1mj=1nai,j dj xi,j subject to j=1ndj xi,j≤ci , i=1,…,m i=1m xi,j=1 , j=1,…,n x∈0,1, i=1,…,m , j=1,…, n Now I have formulated the problem, and I will now use a Mixed Integer Linear Programming solver...

Words: 5806 - Pages: 24

Free Essay

Mathematical Economics

...imputations that satisfy individual rationality and group rationality for all S. Marginal contribution of player i in a coalition S ∪ i: v(S ∪ i) − v(S) Shapley value of player i is an weighted average of all marginal contributions |S|!(n − |S| − 1)! [v(S ∪ i) − v(S)]. n! πi = S⊂N Example: v(φ) = v(1) = v(2) = v(3) = 0, v(12) = v(13) = v(23) = 0.5, v(123) = 1. C = {(x1 , x2 , x3 ), xi ≥ 0, xi + xj ≥ 0.5, x1 + x2 + x3 = 1}. Both (0.3, 0.3, 0.4) and (0.2, 0.4, 0.4) are in C. The Shapley values are (π1 , π2 , π3 ) = ( 1 , 1 , 1 ). 3 3 3 Remark 1: The core of a game can be empty. However, the Shapley values are uniquely determined. Remark 2: Another related concept is the von-Neumann Morgenstern solution. See CH 6 of Intriligator’s Mathematical Optimization and Economic Theory for the motivations of these concepts. 10.15 The Nash bargaining solution for a nontransferable 2-person cooperative game In a nontransferable cooperative game, after-play redistributions of payoffs are impossible and therefore the concepts of core and Shapley values are not suitable. For the case of 2-person games, the concept of Nash bargaining solutions are useful. Let F ⊂ R2 be the feasible set of payoffs if the two players can reach an agreement and Ti the payoff of player i if the negotiation breaks down. Ti is called the threat point of player i. The Nash bargaining solution (x∗ , x∗ ) is defined to be the solution 1 2 to the following problem: 102 x2 T (x1 ,x2 )∈F max (x1 − T1......

Words: 33068 - Pages: 133

Free Essay

The Art

...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...

Words: 508 - Pages: 3