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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.
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.
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Smit, B. de and…[continue]
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