| From: bart | 19/02/00
20:23:26 |
| Subject: straight lines | post id:
40063 |
| If I traveled in a straight line
forever would I come back to where I
was? and if I do come back to the
place I start, do straight lines exist? | |
| From: Chris Mann | 19/02/00
20:30:17 |
| Subject: re: straight lines | post id:
40070 |
| mathematically, no. The
mathematic definition for a straight line (in Euclidean space) is a circle
of infinite radius. R = inf therefore Circumferece = inf: Infinite length.
| |
| From: Grant¹ | 19/02/00
20:31:29 |
| Subject: re: straight lines | post id:
40071 |
My guess is no. While you're travelling in a straight line, the earth is moving around the sun, moving around the solar system, moving around the galaxy, moving around the universe etc. | |
| From: Robert | 19/02/00
23:41:12 |
| Subject: re: straight lines | post id:
40268 |
| I think this is more what Bart is
getting at. Read the part in bold and the rest for context. On the surface of a sphere, such as a geographical globe or the Earth itself, the shortest distance between two points is an arc of a great circle. It is natural for navigators to regard such circles as the "lines" of a special kind of two-dimensional geometry, namely spherical geometry: the geometry of figures drawn on the surface of a sphere of radius 1. It will be seen later that this is almost the same as plane elliptic geometry. Spherical geometry was studied by Menelaus of Alexandria about AD 100 and by the Arabs about 1000. Its most famous theorem (discovered by Albert Girard, a French mathematician of the early 17th century) states that the three angles of a spherical triangle (in radian measure) satisfy the inequality A + B + C > and that the area of a triangle is A + B + C - . The gigantic step of extending this geometry from two dimensions to three (or more) was taken simultaneously (in the latter half of the 19th century) by Ludwig Schläfli in Switzerland and Bernhard Riemann in Germany. Schläfli regarded spherical three-dimensional space as the "surface" of the "sphere" in Euclidean four-dimensional space (that is, the hypersurface on which the four coordinates satisfy the equation x12 + x22 + x32 + x42 = 1). If this three-dimensional continuum represents the astronomical space in which man lives, the unit of measurement (radius of the universe) must be very large; but in terms of this unit the total length of a line is 2 . This means, as Riemann remarked, that the unboundedness of space does not necessarily imply infinitely long lines. A sufficiently powerful telescope could theoretically enable an astronomer to observe the back of his own head, apart from the fact that the light reflected from his head would require thousands of millions of years to reach his eye. This idea, that space could be unbounded without being infinite, was adopted by Einstein in his general theory of relativity (see below Riemannian geometry ). (see also Index: multidimensional space) Another German, Felix Klein, at the turn of the 20th century, first saw how to remedy the awkward situation in spherical geometry that two lines through any one point, being two great circles on a sphere, meet again in an antipodal point (or point on the surface of the sphere that is furthest from the first point). He realized that, because every point determines a unique antipodal point, nothing would be lost and much would be gained by abstractly identifying each pair of antipodal points--that is, by changing the meaning of the term "point" so as to call such a pair one point. He gave the name elliptic geometry to this modification of spherical geometry and the name hyperbolic geometry to the subject created by Bolyai and Lobachevsky. Copyright 1994-1999 Encyclopædia Britannica | |
| From: Robert | 19/02/00
23:51:39 |
| Subject: re: straight lines | post id:
40270 |
| Those equations don't work above
- I'll rewrite them if necessary. In a closed universe, if you went far enough, you would indeed end up back at the same point. In the other models this wouldn't happen. What would happen exactly I'm not sure, at the moment I can only comprehend closed-universe geometry. :-) | |
| From: James Richmond (Avatar) | 20/02/00
14:23:21 |
| Subject: re: straight lines | post id:
40338 |
| A straight line is usually
defined as the shortest distance between two points. This definition works
in any number of dimensions. The only catch is that to apply the
definition there must be some way of measuring distances. In geometry, the
thing which allows us to measure distances is something called a
metric. Spaces with different geometries have different metrics. For example, consider a two dimensional surface. This might be flat, like a desktop, or curved, like the surface of a sphere. Each of these configurations of the 2D surface has an associated metric which allows us to measure distances between points along the surface. Using the flat metric, the shortest distance between two points gives us what we would normally call a straight line. On a sphere, the shortest distance between two points turns out to be along a great circle path. From a 3D point of view, the great circle path is curved, but it is the nearest thing we can get to a straight line in the curved 2D space. We are not yet sure of the global geometry of our universe. It could have geometry similar to a 4D sphere, in which case straight lines would fall along 4D "great circle" paths. travelling along such a path would eventually bring us back to our starting point (think of what happens as you travel along a line of longitude on the Earth's surface). The universe is not necessarily closed. It could be flat, in which case straight lines will continue forever, as on a flat paper sheet. On the other hand, it might be hyperbolic, in which case the shortest distance between two points would not be the "straight line" distance as we normally think of it, but nevertheless straight lines would never loop back on themselves. JR | |