Hyperbolic Geometry
Hyperbolic geometry—an alternative geometry to Euclidean geometry, in which Euclid's first four axioms are true but the fifth axiom, the parallel postulate, is not. Hyperbolic geometry was discovered almost accidentally in the 19th century. Its discoverers were not looking for an alternative geometry, but instead were trying to prove that such a geometry could not exist. Ever since Euclid first set out his five basic assumptions (axioms) of geometry around 300 BC, a debate had raged over whether the fifth axiom is truly a basic assumption, or whether it can in fact be proven using the other four axioms; this would make it not an assumption, but a theorem. Many mathematicians tried to prove the fifth axiom from the other four, but sooner or later each such "proof" was found to have some hole in it.
In the middle of the 19th century the mathematicians Lobachevsky, Bolyai and Beltrami set out to prove the fifth axiom using a new technique: they tried to show that if the fifth axiom were false, all sorts of absurd and contradictory consequences would result. The fifth axiom states that, given a line and a point P that is not on the line, there is one and only one line through P that never meets the original line. So, Lobachevsky, Bolyai and Beltrami, each working separately, tried to see what would be the consequences if, given a line and a point P not on the line, there could be more than one line through P that never met the original line.
Lobachevsky, Bolyai and Beltrami expected that this assumption would lead to ridiculous results, but it did not. Instead, they got one reasonable result after another, discovering many interesting and consistent properties of this alternative geometry. They succeeded in describing the geometry of its triangles, its area measurements, and its rules of trigonometry. Eventually they decided that this new geometry was every bit as consistent as Euclidean geometry, and they gave it a name: hyperbolic geometry.
The mathematical community was not immediately ready to accept the existence of hyperbolic geometry. In 1868, however, Beltrami won over the critics by describing a concrete model for hyperbolic geometry: a new way of measuring lengths and angles in a disk (a circle together with its interior), that would have all the properties that he and his colleagues had discovered for hyperbolic geometry.
In the hyperbolic disk, the shortest path between two points is not an ordinary straight line, but rather is an arc of a semicircle whose center is on the boundary of the disk; diameters of the disk are also shortest paths, but other straight lines are not. Thus, these arcs of semicircles are referred to as the straight lines of hyperbolic geometry, and to an inhabitant of the hyperbolic disk, these arcs would appear straight, since they are the most direct paths between points.
The famous Dutch artist M. C. Escher, who is renowned for the mathematical themes of his work, has exploited the unusual properties of hyperbolic geometry in several of his etchings, including Angels and Devils (see Figure X). Using Euclidean length measurements, the angles and devils are many different sizes. Using hyperbolic measurements, however, all the angels are exactly the same size; likewise for the devils. An inhabitant of the hyperbolic disk would see a vast tiling of the disk by identical angel tiles and identical devil tiles, in much the same way that in Euclidean geometry we have tilings by identical squares. From Escher's etching it is evident that using hyperbolic measurements, the area of the disk is infinite, since it contains infinitely many tiles, and each tile is the same size. Furthermore, the distance to any point on the boundary is infinite, since to get to the boundary it is necessary to cross an infinite number of tiles.
There are many other counterintuitive aspects to hyperbolic geometry. In a triangle, for example, the measures of the interior angles always add up to less than 180 degrees, in contrast to Euclidean geometry in which the angles add up to exactly 180 degrees. The reason for this is that hyperbolic "straight lines" are arcs of semicircles, so hyperbolic triangles become very pinched at the corners. Another strange phenomenon is that as the radius of a circle grows, the area of the circle grows much faster than its Euclidean counterpart. For example, in Euclidean geometry a circle whose radius is, say, 47 units will have area roughly 22,000 square units. In Beltrami's hyperbolic disk, on the other hand, a circle whose radius is 47 units will have area roughly 1x1020, 1 followed by 20 zeros.
The hyperbolic disk is in some ways an excellent model of hyperbolic space, but it suffers from the extreme distortion that makes two angels that are the same hyperbolic size appear very different to the eye. It would be desirable to construct a model in which this distortion does not occur. In 1901, however, the great German mathematician David Hilbert demonstrated that there is no way to build a complete hyperbolic surface in ordinary three-dimensional space without some distortion; loosely speaking, the difficulty is that the area of hyperbolic circles grows so quickly that a hyperbolic surface would get very wavy around the edges as it grew, like a floppy wide-brimmed hat, and eventually it would start bumping into itself.
Although hyperbolic surfaces cannot be built in ordinary Euclidean space, hyperbolic geometry is every bit as concrete and consistent as Euclidean geometry. In fact, physicists and mathematicians speculate whether the universe might actually be governed by hyperbolic geometry, rather than Euclidean geometry. At first glance this seems nonsensical. However, on a very small scale, measurements in Euclidean and hyperbolic geometry are almost identical; it is only on a large scale that noticeable differences emerge. At present, human beings are only able to make measurements in a small portion of the universe. This opens up the possibility that the universe is hyperbolic; a question that can only be answered when human beings develop the technology to make measurements in the far reaches of the universe.
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