There are a number of software applications that make it easy to explore Escher-esque tesselation designs, and you can find them easily using your browser search engine. The regular solids, known as polyhedra , held a special fascination for Escher.
He made them the subject of many of his works and included them as secondary elements in a great many more. There are only five polyhedra with exactly similar polygonal faces, and they are called the Platonic solids : the tetrahedron, with four triangular faces; the cube, with six square faces; the octahedron, with eight triangular faces; the dodecahedron , with twelve pentagonal faces; and the icosahedron, with twenty triangular faces.
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. 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? We found that the tessellations on a sheet of paper were based on geometric shapes like squares and triangles.
What kind of geometric shapes can you put on a sphere? When Escher made his art, abstraction was popular. Many artists, such as Picasso, transformed realistic subject-matter into abstract compositions. Escher, on the other hand, used mathematical concepts that he applied to human and animal forms in order to divide and animate the picture plane. For example, he used symmetry, geometry, tessellations, and optical illusions in his prints.
Exploring his creative universe will bring a bit of magic into your classroom. Mathematicians, scientists, and psychologists were very interested in M. Some even corresponded with the artist, letting him know how much they enjoyed his work and delving into great detail about the mathematical principles his artworks demonstrate.
Throughout his career, he experimented with and reworked his ideas until he obtained the desired effect. You are commenting using your Twitter account. You are commenting using your Facebook account. Notify me of new comments via email. Notify me of new posts via email.
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