by Marek14 » Mon Feb 06, 2006 7:55 am
The Euclidean space is the one which results from the ancient form of Euclid's Fifth Postulate. In Euclidean plane, for every straight line, and every point not lying on it, there is exactly one straight line which passes through the point and doesn't intersect the line. You get other two basic geometries by replacing this axiom with another.
If you claim that NO such line exists, i.e. that any two straight lines intersect, you get elliptic geometry. This is similar to geometry on the surface of the sphere, but since great circles on the sphere intersect in two points instead of just one, elliptic geometry considers any two antipodal points on the sphere identical. Sphere, basically, is made up of two identical copies of full elliptic space. We will use spherical geometry here, as it's better known and a bit simpler to think about.
If you claim that MORE THAN ONE such line exists, you get in the realm of hyperbolic geometry.
Some of the important properties of those spaces are:
1. Curvature
Euclidean space has zero curvature. Spherical space is positively curved, while hyperbolic is negatively curved. You can imagine this as taking a base straight line and constructing two perpendiculars in different points. In Euclidean space, those lines will be parallel. In spherical space, as you move along one of the lines, you'll find the other line to come closer and closer, until they intersect. In hyperbolic space, the other line starts to go away from yours, quickly disappearing in the distance. Intuitively, you might grasp the fact that spherical space is "smaller" than Euclidean (as it is finite, while Euclidean space is infinite), but also that hyperbolic space is "larger" than Euclidean.
2. Angular sum of triangles.
In Euclidean space, the sum of angles in a triangle is always pi. In spherical space, it's LARGER, in hyperbolic space it's SMALLER. But that's not the only interesting thing. The other interesting thing is that the area of the triangle is directly proportional to it's "excess" or "defect" - i.e. how much its angle sum deviates from pi. In both S and H spaces, very small triangles have the angle sum close to pi.
Maximum possible angle sum in S-space is 5*pi, at which point the triangle would comprise the whole sphere. In H-space, the minimum possible angle sum is zero. At this moment, the triangle has infinite sides, but still only a finite area. This also means that there are finite, compact areas in H-space that can't be covered with any triangle, no matter what size.
3. Demise of congruent triangles
Unlike Euclidean space where two triangles can have the same angles despite difference in side lengths, the angles determine the sides in S and H spaces. This means that many things we do daily (like painting a picture, or drawing floor plans) would be very hard to do in S-space, and almost impossible in H-space. It also means that there is "absolute scale" in these spaces.
4. Isocurves
"Isocurve" is simply any curve with constant curvature. In Euclidean space we have circle (with positive curvature) and straight line (with zero curvature). In other spaces, it is more complex.
The basic notion is that curvature of isocurve must be GREATER OR EQUAL to the curvature of the whole space. The isocurve is straight line if and only if it's curvature is EQUAL to the space curvature. This obviously holds in E-space, which has curvature zero.
Spherical space has positive curvature, so ONLY circles are accepted as isocurves. The straight lines are "great circles" with the same radius as the sphere.
In H-space, though, the space curvature is negative, and so we have three distinct classes of isocurves:
a) Circles with positive curvature
b) Horocycles with zero curvature
c) Pseudocycles with negative curvature
Straight line is then a special case of pseudocycle.
The corollary is that if we construct a two-dimensional analogue of the horocycle, the HOROSPHERE, we find that its surface has common Euclidean planar geometry. Living in H-space doesn't need you to abandon the Euclidean amenities completely!
This also ties to the "projectant's nightmare" in hyperbolic world. In spherical world, you can do your drawings in scale by drawing on surface of a smaller sphere than the all-space. In hyperbolic space, though, you would be forced to draw on pseudospherical surface, but the scale would be LARGER - in other words, there is no way to perfectly shrink a planar figure.
5. Tilings.
As you might know, Euclidean plane can be tiled with polygons in three different ways:
{3,6} - six triangles per vertex.
{4,4} - four squares per vertex.
{6,3} - three hexagons per vertex.
There are no other possibilities, since the inner angles of regular polygons are fixed. In S-space, they are not fixed, but we demand them to be larger than in E-space. Thus, we have five regular tilings, which correspond to five Platonic polyhedra:
{3,3}, {3,4}, {3,5}, {4,3}, {5,3}
In H-space we demand the inner angles to be smaller. Since there is no limit of how small they can be, there is an infinite family of regular tilings.
{3,7+}, {4,5+}, {5,4+}, {6,4+}, {7+,3+}
where n+ means "n or larger"
While 3D E-space can only be tiled with cubes (of all regular polyhedra), there are more possibilities for S and H spaces. The E-tiling is written as {4,3,4}, reading as "cubes ({4,3}), meeting by four at an edge, and by eight ({3,4}) at vertex. There are six regular tilings of S-space:
{3,3,3}, {3,3,4}, {3,3,5}, {3,4,3}, {4,3,3}, {5,3,3}
There are only four regular tilings of H-space if we demand that the cells must be finite in their extent:
{3,5,3}, {4,3,5}, {5,3,4}, {5,3,5}