NON EUCLIDEAN GEOMETRY ART PLUS
(You can find out more about Cantor sets in Plus articles Measure for measure and How big is the Milky It is an example of what is called mathematically a Cantor set, more colloquially, a fractal dust. The set of omitted points has interesting properties, for example it is a fractal. 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.įigure 6: A non-Euclidean tiling of the disc by a polygon with some sides at infinity. Then each of its copies meets the circle in four circular arcs. 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 inįigure 6. In the case of figure 1a this is not very interesting: the tiles simply pile up all the way round the circle. In two dimensions the analogue of this would be to study what we can see on the boundary of the disc, namely the circle. They are easier to describe because the surface of a sphere is a two-dimensional object and The patterns we get here are of amazing beauty. 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 window pane at infinityĪs you might imagine from figure 5, tilings of hyperbolic 3-space are rather hard to draw.
NON EUCLIDEAN GEOMETRY ART MOVIE
It is a still from the movie Not Knot!, published by A K Peters Ltd. Image created by Charles Gunn of the Technische Universität Berlin. and in three dimensionsįigure 5: A tiling of three-dimensional hyperbolic space. Your whole world would be contained inside the disc, with the tiles stretching out to the horizon like an infinite chess board. Since the boundary circle is infinitely far away, you would never be able to get to it or even be aware of it. If you were a two-dimensional hyperbolic being, the tiles shown in figure 1a would all seem the same to you, they would all look like regular heptagons. Same length and, as it turns out, meet at the same angle - the tiles are regular hyperbolic polygons. The tiles are bounded by straight line segments all of which have the The sides of the tiles in figure 1a are pieces of circles with precisely this property. In the hyperbolic metric, the shortest distance between two points is along the arc of a circle that meets the circle at infinity at right angles, as shown in figure 2. "straight" if it gives you the shortest distance between its end-points.
To understand this, forget about your intuitive understanding of straightness for a moment and take a slightly more abstract view: a path is Although the sides of each tile look slightly curved to us, in the hyperbolic metric they are actually straight. Measured in the hyperbolic metric, the sides of each tile in figure 1a have the same length, as do the sides of any two distinct tiles. The role of straight lines in this geometry is played by arcs of circles that meet the boundary in right angles, like the red arc shown here. If you count carefully, you will find that the number of tiles in the nth layer is exactly 7 times the 2nth Fibonacci number: nįigure 2: All of the hyperbolic plane fits into the disc bounded by the blue circle. Here the disc is filled up by tiles that are arranged in layers around the centre. The same exponential growth law is manifested in our tiling of figure 1a. In the Plus article Strange geometries.) Hyperbolic tilings in two dimensions. (You can read more about hyperbolic geometry, Once you start looking for it, you see this type of growth throughout the natural world. So to fit into ordinary Euclidean space, a big hyperbolic disc has to crinkle up round its edges like a kale leaf. If the radius is large, then this means that the circumference of a hyperbolic circle is much larger than that of a Euclidean circle. The circumference of this hyperbolic circle is proportional not to its radius, but to e radius, where e is the base of the natural logarithm and is roughlyĮqual to 2.718. In hyperbolic geometry, because distances are measured differently, the points that are equally far away from our point c still form a circle, but c is no longer at what looks like its centre. A kale leaf is crinkled up around its edge.
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