Research Paper on "Art and Mathematics Are Related"

Research Paper 10 pages (2688 words) Sources: 1+

[EXCERPT] . . . .

art and mathematics are related and that this relation could be used to the advantage of educators to overcome student anxiety regarding mathematics and, in particular, difficult geometry concepts

Outline the basic topics to be covered in the study

What is hyperbolic geometry?

Who is MC Escher?

How does Escher's work relate to hyperbolic geometry?

How to design an appropriate, useful, and successful class project that integrates art and mathematics

What Is Hyperbolic Geometry?

Explain the historical development of hyperbolic geometry.

Explain what a Poincare disk is.

Explain how distance is illustrated in Euclidean geometry.

Explain how distance is handled in non-Euclidean geometry

Context: Who Is M.C. Escher?

Provide simple biographical information about M.C. Escher.

Draw a relationship between a Poincare disk and Escher's Circle Limit III

Tell us what the artistic piece can show us about non-Euclidean geometry.

Designing an Appropriate Class Project

Project phase 1: introduce hyperbolic geometry through art, using MC Escher's Circle Limit III as a basic starting point

Project phase 2: transition into a more detailed description of the mathematical concepts underlying Circle Limit III by using Poincare's disk as a segue piece

Project phase 3: allow the students to actively create their own hyperbolic shapes with c

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ut-out equilateral triangles and tape

Conclusions

Draw the final conclusion that art and mathematics are definitely related, and in fact can be considered to be two sides of the same human endeavor to understand the world around us

Conclude that MC Escher's Circle Limit III is a useful artistic example that can be used in the classroom to demonstrate and explain complex concepts in hyperbolic geometry

Using Escher to Instruct Students Regarding Hyperbolic Geometry

Introduction

Though not always apparent, there are a number of significant connections between mathematics and art. In general, these two fields of intellectual inquiry are perceived as distinct and utterly separate. However, this is entirely untrue. Artists, whether they realize it or not, employ any number of mathematical concepts and techniques in the process of creating their artistic productions. These concepts can include ideas such as sequencing, topology, perspective, and others. Obviously, one of the most important mathematical concepts that artists regularly incorporate into their work is geometry. Geometry is, in effect, the mathematical representation of the physical world around us. Therefore, when artists also make representations about the world, they very often must incorporate geometric concepts.

From this we can see that art and mathematics are not quite so separate as one might immediately think. There are connections to be found. Just because Picasso didn't make formulaic notations at the bottom of each of his paintings does not mean that aspects of his work cannot be reduced to those formulas. On the other hand, just because Newton -- the founder of modern calculus -- didn't conceptualize differential equations through sculpture doesn't mean that such a thing is impossible. It is fully within the realm of possibility, then, that mathematics and art are really two sides of the same human endeavor -- to understand and account for the nature of the world around us. One student of this endeavor may rely on numbers and equations, while another would take up a paintbrush or sketchpad. In both cases, however, the goal of conceptualizing the world is the same.

Since mathematics education produces singular anxiety for many students, this confluence with art presents significant possibilities for the imaginative educator (Granger 10). It is possible that we could, as educators, use art as a physical and visual means of explaining complex mathematical concepts in other than abstract terms. Over reliance on complex equations and difficult language can and will stymie many students. By endeavoring to ground mathematical theory in artistic reality, students can leans mathematical lessons in the process of seeing how math and art aren't really all that dissimilar.

The purpose of this study is to demonstrate how the fundamental similarity between math and art can be exploited as a means to teach difficult mathematical concepts to students. To show how this could happen, a particularly complex -- if intellectually intriguing -- mathematical concept will be explored: the concept of distance in hyperbolic geometry, specifically in a Poincare disk. While such a lofty mathematical discussion might seem well beyond the capacity of any artistic representation, in fact at least one artist has expertly demonstrated how this concept could be transformed into a work of exquisite artistic beauty. M.C. Escher's Circle Limit III -- known as "the most stunning example of a hyperbolic tessellation, even though it really isn't a hyperbolic tessellation at all" (Potter and Ribando 27) -- will be the basis of this discussion and will illustrate how art can be used to explain complicated mathematical concepts. Subsequent sections in this discussion will include an explanation of hyperbolic geometry and how distance is calculated therein as contrasted to Euclidean geometry, some background on M.C. Escher and his work, exposition on the relationship between his Escher's Circle Limit III and Poincare's disk, and the development of a basic classroom activity to further reinforce the concept for students.

Context: What Is Hyperbolic Geometry?

Hyperbolic geometry is a kind of non-Euclidean geometry. In this type of geometry, the "parallel axiom is replaced by the following axiom: through a point not on a given line, more than one line may be drawn parallel to the given line" (Corbitt). Hyperbolic geometry is also sometimes known as Lobachevskian geometry because a Russian mathematician developed it in the early 1800s by the name of Nikolai Lobachevsky (Corbitt). The hyperbolic place is the tiled representation of non-Euclidean space, just as the tiles on a bathroom floor is one for Euclidean space (Potter and Ribando 21).

The Poincare model of hyperbolic geometry translates this non-Euclidean sense into a curved, hyperbolic space. Jules Henri Poincare was a French mathematician whose model of hyperbolic geometry placed the whole of an infinite flat plane within the confines of a large finite circle. No points can exist outside of or on the edge of this circle, and the interior would be akin to a curved surface as an infinite number of points are compressed into a finite space within the circle (Ernst 108). In other words, the number of points along the inner edge of the circle would continue to approach infinity, just as points along a parabola will approach infinity as the line closes in on its asymptote. From the point-of-view of anyone looking down on a Poincare disk, it would appear that distances between two points become compressed down to almost zero as objects approach the edge of the circle. Whereas the distance between two points in Euclidean geometry will always be equidistant, in a Poincare disk the equal distances will actually appear to get smaller as they approach the edge of the disk. It is not surprising that this concept is difficult for students to master. Finding a way to incorporate art into the discussion would be invaluable toward easing untrained minds into the rigors of hyperbolic geometry without excessive use of mathematical proofs and equations.

Context: Who Is M.C. Escher?

M.C. Escher was born in the Netherlands in 1898, the son of an engineer. Pushed into architecture by his father, Escher eventually made his way into graphic design and art where his true talents were seen to lie (Ernst 7). Escher traveled extensively throughout Europe, a fact that heavily influenced his art. His art has befuddled many critics, who have difficulty sometimes in interpreting it. Nonetheless, we can divide his work into a series of periods, which roughly correspond to the type of art that he was producing during those eras.

From 1922-1937, Escher produced landscapes, many from small towns in Italy. 1937-1945 marked a metamorphosis period or images that transformed into other images. Following that from 1946-1956, Escher studied perspective, creating the masterpieces that still confuse and amaze. From 1956 until 1970, Escher produced so-called impossible pieces of artwork that could not exist in the real world, as his work approached infinity (Ernst 22-23). Escher was well-known to use mathematical principles, especially in these later productions, in order to create the highly complex pieces of art for which he has become famous (Smit and Lentra 446).

It was during this latter period that Escher produced the Coexter Prints, among which was included Circle Limit III, a stunning example of art paralleling geometry, in this case the Poincare disk. Circle Limit III, produced in 1959, is a five-color woodcut. Escher never revealed exactly how he managed to produce it (Dunham 24). It is circular and apparently shows groupings of fish getting smaller as they approach the edge of the circle. However, from our previous discussion of hyperbolic geometry, we know this to be an incorrect perception. In fact, this woodcut is essentially a Poincare disk, in which all of the fish are the same size, but appear smaller as they approach infinity at the edge of the hyperbolic shape, i.e. The edge of the circle. To the right, observe Escher's Circle Limit… READ MORE

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Quoted Instructions for "Art and Mathematics Are Related" Assignment:

“

**Need one to two page outline and at least 6 references which are not internet sites.**

***Please make a better topic for the paper after read the direction following:

In this paper, you will want to explore is some topic of mathematics in relation to your field or hobby. This paper is also your required capstone paper. Details for the process of producing the paper will be outlined on this discussion group page. The key components of your paper must be:

1. Evidence of appropriate methods of scholarly research. You must have at least five references from books or peer reviewed journals. You should incorporate this research with appropriate citations (include page numbers).

2. Clear and effective exposition of your thesis, organized in well-developed paragraphs and transitions between them.

3. Evidence of your selection and understanding a specific mathematical topic, at a level appropriate for this upper division mathematics class.

4. For the capstone papers, a lesson teaching a specific topic in mathematics.

As you complete your work for this class, it is a good idea to be thinking of possible topics for your expository research paper.

You should be exploring mathematical fields and "objects" that may have some relevance to your final paper. For example, does the appearance of the Fibonacci sequence in pine cones give any "evidence" that mathematics is discovered rather than invented? Since fractal geometry describes our mountain ranges and group theory our DNA, is this further evidence? The mathematical object (Fibonacci sequences) and the mathematical fields (fractal geometry and group theory) have interesting applications to music, literature and the arts. Can they be used to teach mathematics? Etc.

To grade the paper with the following in mind, in order of importance:

1. Your understanding and explanation of the mathematics.

2. The clarity of your thesis and goal for the project and the effectiveness of your arguments to support it.

3. The quality of your research, i.e., how effectively you have used your sources. This would take into account the choice of sources, your citations and how you demonstrated your understanding of what you have read, etc.

4.The format of your paper (well-developed paragraphs and competent sentence structures, mastery of the mechanics of standard written English, etc.)

SUGGESTED TOPICS

I. For prospective teachers: You will want to investigate a mathematical topic with the idea of deepening your understanding of it so to better teach it. You’ll need to make connections to mathematics appropriate to this course. Here are some possible topics to explore.

a. Investigations into different types of averages. Suppose you were teaching students to find an average of a set of student scores. You might generalize the problem by trying to estimate the highest score the student can attain if there were going to be two more exams. You might represent this situation geometrically. You might want to explore real life applications of using averages. For example in sports basketball players vie for the season's individual scoring title (which is won by the player with the highest

average number of points per game). You might want to consider how to

estimate who will win if you are two games away from the end of the

season.

b. The importance of definitions. For students to learn mathematics, they must fully understand its language. You might explore some definition in mathematics and examine its different conceptions. For example, consider parallelism. Are there different ways to define it? Certainly. Take a look at some books. Are there different ways the word parallel is used in modern parlance? Certainly. Think about parallel rays from the sun; parallel electrical circuits; parallel processing (with computers); parallel bars in gymnastics; etc. Or, what about about absolute value? This is a topic that has many applications. It can be defined both algebraically and geometrically. It can be generalized to complex numbers. Think about how it is applied to situations outside of mathematics. Or, what about distance? Is there only one way to measure it? You might

want to learn about taxicab distance. It is a wonderful way to explore the Cartesian plane as a grid of streets along every horizontal and vertical line. What would a taxicab circle look like? You might be surprised. There is also another kind of distance called a Hamming distance, which is used in communication. Sometimes when teaching a concept, it is important to know that there are other ways to interpret it in different contexts. This is the idea of betting knowing something by knowing what it is not.

c. Special mathematical “objects” or “relations”. You might want to explore some special numbers in mathematics, such as pi, e, phi. These numbers have a rich history and applications in different areas of mathematics. They can be used to introduce probability problems and problems in geometry and arithmetic. Zero is always a mysterious to students. It too has a wonderful history. It has different functions in mathematics. Or, what is a function? What idea does it express? What was its historical evolution? How many different kinds of functions are there? Are all functions like the ones you studied in calculus? What are pathological functions? How are functions represented? There are many interesting questions to explore with this topic that could motivate the introduction of functions to students.

d. The magic of geometry. You might want to examine the idea of congruence and symmetry. A nice motivating tool for students to learn about this is Frieze patterns. There are only seven essentially different frieze patterns and they have an interesting history. Also, important sets of problems in mathematics deal with points that satisfy certain conditions regarding their position in a plane or in space. Historically these are called locus problems. You might want to explore these problems (there's a famous one called "dog on a leash") and think about how you would teach them to your students. Or, you might want to think about how you would introduce the idea of similarity to students, at different educational levels. What does it mean for two figures to be similar? How could we describe this geometrically and algebraically. Can we talk about similar graphs of

functions? Can arcs of circles be similar? Or, you could explore the concept of area. Can it be defined in the same way for all figures? Can you find the area of a triangle if you don't know the altitude? Investigate Hero's formula. Archimedes discovered many important facts about area including its connection to circumference in circles. Learn about this. Think about the area formulas of elementary and high school, and those that we learn in calculus (integrals).

e. You might want to examine different branches of mathematics and trace their history. When did projective geometry emerge on the scene and why? The idea of perspective is used by artists. Did they motivate the mathematicians to discover or invent projective geometry? Yes, to a certain extent. What is the historical and conceptual evolution of trigonometry? This is a field rich with applications to sound, music, design of machinery, etc. Space shuttle orbits are described by sine curves. Think about how motivating this would be if you want to teach these curves to students.

f. Mathematics has applications is a wide variety of areas: art, music, sports, communication, literature, etc. Illustrate such applications with a view toward showing how a specific mathematical topic can be taught via its applications in these fields.

II. For other majors. You will want to explore is some topic of mathematics in relation to your field or some area that interests you. It could be a hobby like photography or magic, or a "vice" (smile), like gambling. Here are some examples:

a. In the last half-century there has been what we can call a " mathematization" of many sports. Scientific studies of physical activities (anatomic conditions, nutrition, training methods) are conducted with the aim of finding better ways for improving performance. The complex processes regulating physical behavior are quantified High performance and winning are measured and recorded. Statistics now determine strategies and salaries, as well as betting odds in sports pools. We see the effects of mathematization in sports in the day-to-day activities of participants. For example, players historically have been trained using demonstrations and apprenticeships with coaches to learn how movements should be accomplished. Now coaching is approached scientifically. Players are exposed to a type of cybernetic feedback which compares their own performances against some external standard or goal. This is achieved and even permanently archived with mathematical precision by joining video/film and computer technologies. Your final paper could investigate mathematization of some sport, or sports in general.

b. Learn about a new branch of mathematics called fractal geometry. It combines mathematical theory with beautiful graphics and has many applications in our world. Explore the fruitful connections between fractals and music, between fractals and cell phones, between fractals and nature, between fractals and art, etc. There is interesting literature that involves the idea of fractals and the related chaos theory. The sciences of chaos and fractal geometry also apply to several different areas of social science, including, but not limited to urban planning, anthropology, sociology and psychology. Chaos theory concerns randomness, which anthropologists postulate, and which urban planners and psychologists and sociologists study. For example, psychologists use fractal geometry to understand how complex thought arises from trillions of simple neurons.

c. The development of communication techniques in a human being is called literacy, and the techniques used are those of language. However, the art of communicating historically has taken different forms in different cultures. For example in the world of the Deaf and hard-of-hearing, communication is achieved through signs. In early cultures, communication often took the form of pictures. In today's world we focus of language as we communicate orally or through letters, email, etc. Regardless of the medium, communication involved the setting of contexts, the conveying of meaning, the expression of emotions, etc. Mathematics literacy in a similar way involves the development of communication techniques, and the techniques are those that combine ordinary language with the language of mathematics (definitions, symbols, etc.). The art of communicating in mathematics is often focused on understanding, explaining, coping with and managing the reality of the world in which we live, but not always. The art of communicating in mathematics takes different forms -- theorems, proofs, problems, research papers, expository articles, informal and formal talks. Mathematics uses language, and is a language in itself. Investigate mathematics as a language. You could focus on mathematics as a universal language, and compare the art of communicating in mathematics with the art of communicating in your field. Perhaps, you can talk about good practices in communicating in your fields as compared with good practices in communicating in mathematics. Or you could investigate mathematics as a language and relate it to how languages and other forms of communication are used in different cultures. You would need to illustrate with specific topics in mathematics.

d. Coding theory is always a fun topic to explore, not only for communications majors, but for those interested in the humanities. Think about recent literature and theater (The DaVinci Code, Breaking the Code, etc.)

e. Artists have been called “unconscious mathematicians”. You could investigate the mathematics in work by of Leonardo da Vinci, Georges Braque, M.C. Escher, Pier Mondrian, Salvador Dali, Crockett Johnson, or other artists of your choice, and discuss how they behaved like “unconscious mathematicians”.

f. The country is divided about what to do about such issues as the social security system and the electoral college Investigate the mathematics of social security. Review newspaper articles to see what kind of statistical and probablistic arguments are being made to defend different types of amendments or changes to the system as opposed to the status quo. Or investigate the mathematics of voting, to see if there is a way to determine a fair voting system.

This assignment asks you to secure references. Once you do,

list them with complete bibliographic data.

Books should include title, author, city of publication, publisher, and date. Journal and magazine articles should include title of article, title of journal, number of journal, date, and page numbers. Be sure to take careful notes on your references. When you cite or rephrase a passage from a resource, be sure to indicate author and page numbers.

You may NOT use internet sites as resources for your paper, unless I have suggested them (like MATHWORLD).

It is important that you secure at least 6 references. This will enable you can get good ideas for the paper.

****

Please use this as a model of detailed outline:

Goal of the paper: To explore how to teach mathematics using art.

Research question: How to use Escher's paintings to teach students about distance in hyperbolic geometry.

Outline:

I. Introduction: discussion of the connections between art and mathematics.

II. Setting the context: What is hyperbolic geometry

A. Brief history of the discovery/invention of hyperbolic geometry.

B. Discussion of some important characteristics of the Poincare model hyperbolic geometry with respect to both what it shares in common with the Cartesian model of Euclidean geometry and how it differs from the Cartesian model of Euclidean geometry.

C. Explanation of the concept of distance in Euclidean geometry.

D. Explanation of the concept of distance in hyperbolic geometry.

III. Introducing the artist: M.C. Escher

A. Brief biography of M.C. Escher.

B. Discussion of Escher's disk full of flying fish and how it can be viewed as Poincare's model of hyperbolic.

C. Discussion of what the painting tells us about the way we

measure distance in hyperbolic geometry.

a. These fish appear to us to be getting smaller and smaller as they approach the edges of the painting. But in reality, because of the way we measure distance in hyperbolic geometry, these fish are really all the same size.

b. How we can see the "lines" (which are sometimes curved) of hyperbolic geometry in Escher's painting by following the fishes' backbones. By counting fish along one of these "lines" we can see how it will take infinitely many of the fish to get to the boundary, up to the limitations of the physical models.

IV. Applications to teaching: a class project.

Description of the construction of a model of

hyperbolic geometry based on Poincare's model and Escher's work: Ask the students to tape together a

lot of equilateral triangles so that seven of their 60

degree angles meet at each vertex. The more triangles they use, the "floppier" their paper will get.

V. Conclusion

VI. Works Cited List of at least six references (NOT WEBSITES) with complete bibliographic data.

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