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This site uses cookies to offer you an improved and personalised experience. If you continue to browse, we will assume your consent for the same. Used by permission. We explore tesselations sometimes called tilings as shown in the following examples. The natural question is "How did he do it? We look at the art which inspired M. Escher, and we discuss the techniques Escher used to create these wonderful images.
One of the projects in the class is to make your own tessellation. Further down on the page are links to student work. Students are of course NOT graded on their artistic ability!
We are looking for the ability to incorporate what was learned into a practical application. Escher also decorated spheres like the one shown on the left. What mathematics do we need to understand this type of art work? In the woodcut Four Regular Solids Escher has intersected all but one of the Platonic solids in such a way that their symmetries are aligned, and he has made them translucent so that each is discernable through the others.
Which one is missing? There are many interesting solids that may be obtained from the Platonic solids by intersecting them or stellating them. To stellate a solid means to replace each of its faces with a pyramid, that is, with a pointed solid having triangular faces; this transforms the polyhedron into a pointed, three-dimensional star.
Here the stellated figure rests within a crystalline sphere, and the austere beauty of the construction contrasts with the disordered flotsam of other items resting on the table. Notice that the source of light for the composition may be guessed, for the bright window above and to the left of the viewer is reflected in the sphere. Here are solids constructed of intersecting octahedra, tetrahedra, and cubes, among many others. One might pause to consider, that if Escher had simply drawn a bunch of mathematical shapes and left it at that, we probably would never have heard of him or of his work.
Instead, by such devices as placing the chameleons inside the polyhedron to mock and alarm us, Escher jars us out of our comfortable perceptual habits and challenges us to look with fresh eyes upon the things he has wrought.
As we will see in the next section, Escher often exploited this latter feature to achieve astonishing visual effects. Inspired by a drawing in a book by the mathematician H. To get a sense of what this space is like, imagine that you are actually in the picture itself. As you walk from the center of the picture towards its edge, you will shrink just as the fishes in the picture do, so that to actually reach the edge you have to walk a distance that, to you, seems infinite.
Indeed, to you, being inside this hyperbolic space, it would not be immediately obvious that anything was unusual about it—after all, you have to walk an infinite distance to get to the edge of ordinary Euclidean space too.
A strange place indeed! Even more unusual is the space suggested by the woodcut Snakes. Here the space heads off to infinity both towards the rim and towards the center of the circle, as suggested by the shrinking, interlocking rings. If you occupied this sort of a space, what would it be like? In addition to Euclidean and non-Euclidean geometries, Escher was very interested in visual aspects of Topology, a branch of mathematics just coming into full flower during his lifetime.
Topology concerns itself with those properties of a space which are unchanged by distortions which may stretch or bend it—but which do not tear or puncture it—and topologists were busy showing the world many strange objects. It has the curious property that it has only one side, and one edge. What do you predict will happen if you attempt to cut such a strip in two, lengthwise?
Another very remarkable lithograph, called Print Gallery, explores both the logic and the topology of space. Here a young man in an art gallery is looking at a print of a seaside town with a shop along the docks, and in the shop is an art gallery, with a young man looking at a print of a seaside town… but wait!
Somehow, Escher has turned space back into itself, so that the young man is both inside the picture and outside of it simultaneously. The secret of its making can be rendered somewhat less obscure by examining the grid-paper sketch the artist made in preparation for this lithograph.
Note how the scale of the grid grows continuously in a clockwise direction. And note especially what this trick entails: A hole in the middle. A mathematician would call this a singularity , a place where the fabric of the space no longer holds together. There is just no way to knit this bizarre space into a seamless whole, and Escher, rather than try to obscure it in some way, has put his trademark initials smack in the center of it.
All artists are concerned with the logic of space, and many have explored its rules quite deliberately. Picasso, for instance.
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