![]() ![]() Non-Euclidean Geometry and Curved Space.The results of these two types of non-Euclidean geometry are identical with those of Euclidean geometry in every respect except those propositions involving parallel lines, either explicitly or implicitly (as in the theorem for the sum of the angles of a triangle). The second alternative, which allows no parallels through any external point, leads to the elliptic geometry developed by the German Bernhard Riemann in 1854. Lobachevsky in 1826 and independently by the Hungarian Janos Bolyai in 1832. Allowing two parallels through any external point, the first alternative to Euclid's fifth postulate, leads to the hyperbolic geometry developed by the Russian N. ![]() Oxford University Press.Non-Euclidean geometry, branch of geometry in which the fifth postulate of Euclidean geometry, which allows one and only one line parallel to a given line through a given external point, is replaced by one of two alternative postulates. Escher and Bruce Brooks Pfeiffer (Editor). The Thirteen Books of Euclid’s Elements, translation and commentaries by Thomas Heath, (1956) Dover Publications. We have been fooled by our Euclidean intuition! It may seem to us that the devils reside inside a circle, but for them there is no ‘outside’, as the circle is infinitely large from a hyperbolic perspective. All the devils, even those which appear close to the boundary from the Euclidean perspective, are actually infinitely far from the boundary circle. The devils close to the boundary circle in Figure 5 appear smaller than those near the centre of the disc, but the hyperbolic area and height of each devil from head to tail are the same when hyperbolic distance, rather than Euclidean distance, is used. It is therefore impossible to deduce Euclid’s parallel axiom from the first four, for if it could be deduced, then it would also hold in hyperbolic geometry, which is not true. ![]() This shows that hyperbolic geometry, while obeying the first four of Euclid’s axioms, violates the fifth one. These hyperbolic lines are therefore parallel to l, and some of them are shown on Figure 6. Given a hyperbolic line l, and a point P inside the disc, and not on this line, there are infinitely many hyperbolic lines through P which do not intersect l. The arcs and diameters are called the hyperbolic lines, and any two points inside the disc can be connected by a unique hyperbolic line. Six such arcs have been drawn on Figure 5. The analogues of straight lines are diameters of the bounding circle, and circular arcs intersecting this boundary at right angles. Points on the boundary are infinitely far apart - the notion of the distance is distorted in hyperbolic geometry. ![]() Escher represented this geometry on such a disc tiled with infinitely many angels and devils. The arena for hyperbolic geometry is a disc: the interior of a circle. It states that for any right-angled triangle the square of the hypotenuse c is equal to the sum of the squares of the other two sides a and b (Figure 2). The Pythagorean theorem is a good example. It stands out from other areas of mathematics in that a proof of a theorem can be given in pictorial terms, without a need for algebra or mathematical symbols, and without sacrificing rigour. Geometry - in the ancient Greek geo means earth, and metron means measurement - is a branch of mathematics concerned with distances, shapes and areas. Geometry is also moving forward, but once a correct proof of a theorem has been presented, it becomes timeless. This is because the science is moving forward. In these fields, anything written more than 50 years ago is regarded as outdated, and often wrong. Imagine teaching biology or physics from a source of similar age. Although the text goes back to the 3rd century BCE, it has until relatively recently been used for teaching geometry at high school and university levels. The Elements is regarded as one of the most influential books of all times, and is surpassed only by the Bible in the number of editions published. The Fellows’ Library holds several copies of the Elements by Euclid of Alexandria, including the 1620 edition donated by Robert Greene*, a Fellow of the College from 1703 to 1730. ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |