It is a still from the movie Not Knot! As you might imagine from figure 5, tilings of hyperbolic 3-space are rather hard to draw. As an easier substitute, mathematicians usually study what they see on the boundary of hyperbolic space, that is on the sphere that defines the hyperbolic universe.
The patterns we get here are of amazing beauty. They are easier to describe because the surface of a sphere is a two-dimensional object and can be projected onto a flat plane. In two dimensions the analogue of this would be to study what we can see on the boundary of the disc, namely the circle. In the case of figure 1a this is not very interesting: the tiles simply pile up all the way round the circle. Now suppose that the original tile is still a polygon with a finite number of sides, but that it stretches all the way out to the boundary, like the yellow region in figure 6.
Then each of its copies meets the circle in four circular arcs. Although the tiles fill up all of two-dimensional hyperbolic space the interior of the disc , the totality of the arcs do not fill up the whole circle.
Figure 6: A non-Euclidean tiling of the disc by a polygon with some sides at infinity. If you were a two-dimensional being living in the same plane as the disc, but outside it, then what you would see of each hyperbolic polygon would be the circular arc in which it meets the boundary circle.
In three dimensions the analogue of a polygon with a finite number of sides is a polyhedron with a finite number of faces. Outside observers will see these faces as shadows where the polyhedron meets the sphere. Figure 7a below shows the sphere with four-sided shapes which are faces of polyhedra that tile its inside.
They pile up in remarkable patterns on the boundary of the sphere, like noses pressed against a window pane. Figure 7b shows the patterns on the sphere projected onto the plane; this time with a different colour scheme. Figure 7a: Faces of polyhedra in a 3-dimensional hyperbolic tiling pressed against the sphere at infinity.
Patterns like this were studied by the German mathematician Felix Klein - In the s David Mumford realised that they were a natural target for computer exploration. With David Wright, he embarked on a systematic study which eventually resulted not only in inspiring new mathematics, but also in the book Indra's Pearls.
The book shows off some of the remarkable patterns that arise on the window pane at infinity. It explains these patterns with the minimum of mathematical baggage, but with enough detail for the mathematically inclined to follow the reasoning and for the computationally inclined to make their own pictures.
To create a two-dimensional tiling, whether it's Euclidean like the one of figure 1b, or hyperbolic, you need a way of creating identical copies of an initial tile and placing them alongside each other. Mathematically, this job is done by reflections, rotations and translations: you get to each tile either by shifting your initial tile into a given direction by a given distance — this is known as a translation — by reflecting it in an axis, or by rotating it through a given angle around a fixed point.
Or, indeed, by a combination of any of these three movements. To describe a tiling, all you need is a description of the initial tile together with a list of symmetries that generate the tiling. Although there are infinitely many tiles, the list of symmetries can be finite, because it may be possible to get to all the tiles by performing the same symmetries over and over again.
The same is true of three-dimensional Euclidean space. To tile it by polyhedra, for example by cubes, you start with a central polyhedron and repeatedly perform a number of symmetries, only this time you allow three-dimensional rotations, translations and reflections.
Tilings of three-dimensional hyperbolic space are generated in the same way: you start with a hyperbolic polyhedron and repeatedly perform a number of hyperbolic symmetries until you have filled up the whole universe.
Without thinking too hard about what hyperbolic symmetries might look like, remember that we are interested in what happens on the sphere that bounds our hyperbolic universe. This means that we need to start with a face of the polyhedron that presses against the bounding sphere and repeatedly perform a number of movements, watching copies of the face pile up on the sphere as they do in figure 7.
But how can we describe these movements? Remember that our symmetries leave hyperbolic distance intact. But we are working on the bounding sphere at infinity, where we can no longer measure distance in the same way. In fact, on the window pane at infinity it is impossible to find a way of measuring distance that is preserved.
But all is not lost: it turns out that the transformations we are looking for do leave something intact. They transform circles into circles, with changes of radius being allowed. To describe these transformations, mathematicians project the sphere onto a flat plane using stereographic projection : place a light bulb on the North pole of the sphere so that each point of the sphere has its unique and very own shadow on the plane below.
Every point except the North pole that is, but mathematicians have a way of dealing with this and we can safely ignore the North pole here. Figure 8: Stereographic projection - draw a line from the North pole through a given point on the sphere.
The point's shadow is the place where the line hits the plane below. If you'd like to find out more about complex numbers, read this short introduction or the Plus article Curious quaternions.
For example, we can start with the two transformations Then take a basic shape, say a stick man, and apply both of these transformations to the figure over and over again. As you can see in figure 9, the images get small and are distorted as the level of the repetition increases.
If you choose the starting transformations cleverly, the patterns which emerge when the images pile up are amazingly beautiful. Click on the image to see it evolving.
Image and movie created by David Wright. To see more clearly what is going on, we often drop the original shape altogether and just look at the region where its smaller and smaller images pile up. This is called the limit set or chaotic set of the iteration, because in this part of the pictures the symmetries act in a chaotic way. Though to a mathematician, the chaos is very controlled. The limit set is shown in figure By choosing the initial symmetries with enough care, we can create limit sets with intricate patterns of tangent circles.
The two transformations written down above produce the famous Apollonian Gasket , which is shown at the beginning of this article. The concept of length, begins to make little sense.
Lewis Fry Richardson first noted the regularity between the length of national boundaries and scale size. As shown next, the relation between length estimate and length of scale is linear on a log-log plot. The coastline of South Africa is very smooth, virtually an arc of a circle. The slope estimated above is very near zero. This makes sense because the coastline is very nearly a regular Euclidean object, a line, which has dimensionality of one.
In general, the "rougher' the line, the steeper the slope, the larger the fractal dimension. We begin with a straight line of length 1, called the initiator. This new form is called the generator , because it specifies a rule that is used to generate a new form.
The rule says to take each line and replace it with four lines, each one-third the length of the original. We do this iteratively The Koch Curve. Fractal tilings in the plane. Mathematics Magazine , Vol. Davis, D. The Nature and Power of Mathematics. Writing for the liberal arts student, Davis provides substantial introductions to non-Euclidean geometry, number theory, and fractals.
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Scientific American , August 60— CrossRef Google Scholar. Lauwerier, H. Lorenz E. Mandelbrot, B. The Fractal Geometry of Nature. New York: W. On fractal geometry and a few of the mathematical questions it has raised. Proceedings of the International Congress of Mathematicians , August 16—24 — Warsaw: Polish Scientific Publishers. Peitgen, H. The Science of Fractal Images.
New York: Springer-Verlag. Contains most of the two-volume text Fractals for the Classroom. Peterson, I. Ants in labyrinths and other fractal excursions.
Science News 42— Richardson, L. Weather Prediction by Numerical Process. Republished by Dover Publications Robinson, C. Rodriguez-Iturbe, I. Cambridge: Cambridge University Press. Ruelle, D.
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