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

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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 cut-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 III followed by a Poincare disk. Note the distinct similarities.

An examination of Escher’s Circle Limit III can thus tell us much about distance in hyperbolic geometry. In both Escher’s woodcut and the Poincare disk, the images showcased appear smaller as one’s eye moves toward the edge of the circle. However, this is an illusion created by our traditional, Euclidean perceptions. Because of the way that distance is measured in a hyperbolic space, all of the objects shown in the circle are actually the same size. As we follow the backbones of the fish in Escher’s representation, we can see, then, that the lines separating one fish from the next are actually all the same distance even though they appear to grow shorter. This is because, as already noted, the hyperbolic space stretches to infinity at its edges. There is no end. Therefore, the perception that the lines are getting smaller toward the edges is, in fact, a result of two-dimensional perspective drawing attempting to illustrate the nature of an infinite hyperbolic space.

Put another way, hyperbolic lines are represented by circular arcs perpendicular to the bounding circle of the disk, shown by the spines on the fish in Circle Limit III. Ever-decreasing Euclidean distances represent equal distances in this hyperbolic space as the eye approaches the edge of the disk (Dunham 23). The objects along the edge of the circle are the same size, thus, as those on the interior and the distances are equal. In theory, the fish should continue to exist ad infinitum, although there were no doubt physical limitations on what Escher’s hand could manage.

Developing an Appropriate Class Project simple, yet appropriate, classroom project that would synthesize the various elements of art and geometry that have been discussed up to this point is easy to devise at this point. To begin with, the students must be treated as active participants in the endeavor, given the opportunity to act as artists themselves and in the process develop the mathematical principles that Escher himself epitomized when he created the woodcut Circle Limit III in 1959 (Ernst 109). The surest way to engage the intellectual processes of students — especially when the demanding concepts of hyperbolic geometry are concerned — is to allow them to be active participants in their own education. Art is the physical representation of the theories behind the geometry; therefore, it seems that the best classroom project would allow the students to make real the theories discussed.

This project would begin with a brief grounding in theory. Before the students can be allowed to begin, they should have in their minds — even if only way in the back — a sense of the mathematical ideal to which they will be striving. The class project will begin with an opening discussion of M.C. Escher and his work, with a particular focus on Circle Limit III. This piece, as we have already seen, is a five color woodcut that embodies the principles laid out in the Poincare disk and the basic concepts of hyperbolic geometry, in particular distance. Using the art as the starting point for the discussion will engage the students immediately and distract them from the fact that they are still sitting in a geometry classroom.

Following that introduction, the teacher should transition into a brief examination of hyperbolic geometry by using the Poincare disk as a talking point to bridge the gap between the art of Escher and the principles of non-Euclidean mathematicians. The discussion can gradually move into greater and greater theory, repeatedly referring back to both Circle Limit III and the Poincare disk to show how the non-Euclidean concept of distance is embodied in both art and geometry. By the end of this part of the classroom activity, the students should have a basic grasp of how Escher’s work embodies geometric concepts and how practical art can be a medium through which geometry can be fully understood.

With that information in hand, the students can be encouraged to actively engage these concepts and attempt the creation of their own non-Euclidean, hyperbolic shapes. If the students are asked to tape together a series of equilateral triangles such that seven of the angles meet at their vertices, some of the nature of a hyperbolic shape will be illustrated. The more triangles that the students are able to successfully join will result in a “floppier” paper, but one that also more closely approximates the curved nature of a hyperbolic shape. The students will struggle and play with the shapes and in the process will create objects which are based upon the underlying principles of hyperbolic geometry as embodied in M.C. Escher’s Circle Limit III.

Conclusions

Clearly, then, it is evident that useful connections can be drawn between art and mathematics for the purposes of improving pedagogical designs. Mathematics can be difficult for many students to master. Geometry is no exception to this difficulty. When we add the increased complexity of non-Euclidean hyperbolic geometry to the educational context, the situation becomes all the more frustrating. Students are liable to stare blankly when hyperbolic concepts are discussed in abstract terms. Some educators may fail to discern useful methods for demonstrating the underlying concepts that form the basis of this type of geometry.

By combining artistic representations with mathematics, this difficulty can be lessened somewhat. We have seen that many pieces of art are infused with mathematical concepts and ideas. Perspective, for instance, is one of the simplest mathematical concepts that regularly figures into artistic representations. However, understanding perspective is a far cry from comprehending distance and other concepts in hyperbolic geometry. Nonetheless, some artistic representations effectively embody these difficult concepts. In fact, M.C. Escher’s flying fish is an excellent example of Poincare’s disk, a hyperbolic representation of distance and lineality in non-Euclidean space.

By using Escher’s very apt flying fish image, students can visually understand the complex nature of hyperbolic geometry and gain a basic grasp of the mechanics of a non-Euclidean space. A further development of this understanding can come through an interactive classroom experience that requires students to engage with a hyperbolic shape of their own making. In this way, we can best hope to educate about geometric concepts that strain the ability of the mind to easily grasp. Our usual world exists only in Euclidean terms. However, mathematics — and geometry in particular — encompasses so much more than just Euclidean representations. Understanding these additional concepts is important for the student and can be difficult for the educator to demonstrate. By combining hyperbolic geometry with art — namely Escher’s flying fish — art becomes the medium through which better mathematical understanding is fostered.

Works Cited

Corbitt, Mary Kay. “Geometry.” World Book Multimedia Encyclopedia. World Book, Inc., 2003.

Dunham, Douglas. “A Tale Both Shocking and Hyperbolic.” Math Horizons Apr. 2003: 22-26.

Ernst, Bruno. The Magic Mirror of M.C. Escher. NY: Barnes and Noble Books, 1994.

Granger, Tim. “Math Is Art.” Teaching Children Mathematics 7.1 (Sept. 2000): 10.

Potter, Melissa and Ribando, Jason M. “Isometrics, Tessellations and Escher, Oh My!” American Journal of Undergraduate Research 3.4 (2005): 21-28.

Smit, B. de and Lenstra, H.W. “The Mathematical Structure of Escher’s Print Gallery.” Notices of the AMS 50.4 (Apr. 2003): 446-457.

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