The world of manifolds is extremely rich and diverse, especially when higher dimensions are considered. A multitude of objects can be explored, most of them fundamentally different from one another, and we are still very far away from identifying all of them, even in relatively low dimensions.
Note: Before exploring manifolds, it is useful to define "dimension" and what it means in terms of a manifold. Recall the definition of a manifold: any surface that appears "flat" at sufficiently small scale. However, "flat" is another word for Euclidean (or plane) geometry, and we can then define the dimension of a manifold as follows:
If a manifold resembles n-dimensional Euclidean space at sufficiently small scale, then it is an n-dimensional manifold.
As an example of this, consider a typical sphere, such as the Earth (the Earth is not exactly spherical, but is often used to represent the abstract mathematical sphere). To someone standing on the Earth, its surface appears to be a flat plane, i.e. two dimensional Euclidean geometry! Therefore, we conclude that the sphere is a two dimensional manifold, or 2-manifold.
Note: The term "sphere" only includes the surface of the sphere (or the Earth) and does not include the inside of the sphere. The region of three dimensional space bounded by the two dimensional sphere is known as the ball, and is a 3-manifold.
With the above clarification of terms, one notices that is easy to define the Earth as a 2-manifold. However, when one attempts to map the surface of the Earth on a flat surface, there are inevitable distortions.
As an example, consider the Mercator projection of the Earth.
When mapping the Earth onto a flat Euclidean plane, one must consider various geometrical properties of the sphere, including area, latitude and longitude lines, and lengths of these lines. A process called projection allows one to view a surface on a flat two dimensional plane. In the above case, a cylinder is used to project the Earth onto a plane (see image below).
For each point on the sphere, a line is drawn from the center of the sphere outward through the aforementioned point, and this line will eventually intersect with the cylinder (the cylinder does not actually have a "top" or a "bottom" but rather goes on forever). After all the points are mapped, the cylinder is unrolled into a plane, resulting in the fact that going off the right edge of the map goes to the left edge and vice versa. However, when one chooses a point near one of the poles, the line must go a large distance before intersecting the cylinder, and the poles themselves cannot be mapped at all! Despite these problems, this mapping is desirable for bearings, as it maps rhumb lines (or lines bearing in a specific direction, e.g. northwest, east southeast, etc.) and therefore also preserves angles.
Another (perhaps even simpler) useful projection for mapping the Earth (and especially mathematics) is the stereographic projection. It again maps the sphere onto a flat Euclidean plane, but in a different way. The figure below denotes this.
To project stereographically, a point of projection is first chosen. For the purposes of this example, the point of projection is always assumed to be the north pole. For each point on the sphere, a line is drawn from the north pole through this point, and is extended until it intersects the plane passing through the equator of the sphere. In mathematics, this is taken to be the complex plane (see here for a basic discussion of i and the complex plane) and the circle at which the sphere intersects the plane is taken to be the unit circle, i.e. the circle centered at 0 with radius 1.
The figure above shows two arbitrary points. Point A, on the plane outside the unit circle, corresponds to a point on the upper half of the sphere, while Point B, on the plane inside the unit circle, corresponds to a point on the lower half of the sphere. Points on the unit circle obviously remain in the same position. (1 and i are shown as examples)
Additional properties include the south pole of the sphere corresponds to the origin of the plane, and the north pole of the sphere does not correspond to ANY point on the normal plane. This is because the point of projection (the north pole) and the point to be projected (also the north pole) coincide, and the line is therefore a tangent line, which is parallel to the plane and never intersects it. The north pole is sometimes called infinity for this reason.
An image of the Earth using another type of stereographic projection where the plane of projection is tangent to the south pole of the sphere. In all other respects, this projection is similar to the one above; it still matches every point on the sphere with one point on the plane, with the exception of the north pole.
The stereographic projection has many properties that make it valuable to mathematics, the most important of which is that it preserves circles and angles (for a proof, see sources, specifically Needham). And since the points are projected continuously with a one-to-one correspondence, the mapping is a homeomorphism. Therefore, we can make the statement that
The Euclidean two dimensional plane is homeomorphic to the two dimensional sphere with one point removed.
The above reflects that the point of projection itself (the north pole in the above examples) cannot be projected onto the plane. In mathematics, it is possible to remedy this, by including infinity itself as a point on the complex plane, and the new entity that results is known as the extended complex plane. Infinity then corresponds to the north pole under stereographic projection. Note that the actual direction, in which we approach infinity does not matter, as all lines heading away from the origin will travel upwards on the sphere and eventually reach the north pole. In a sense, infinity is an endpoint of any straight line in the extended complex plane.
A generalized view on stereographic projection allows one to project from spheres of any dimension to their corresponding planes. For example, elliptic polychora, or four dimensional finite polytopes (see the polytopes series) are actually tilings of the three dimensional sphere which is the surface enclosing the four dimensional ball (this is the higher dimensional analog of the two dimensional sphere being a surface enclosing a three dimensional ball). Therefore, stereographic projection can be used in a similar way as the above to project such polychora onto flat Euclidean three space.
An example of a convex polychoron projected stereographically (specifically, it is the cantellated 24-cell, see here for more information)
Mappings and projection have revealed that the sphere and plane are fundamentally different, despite being members of the same dimension. Further posts in the manifolds series addresses this topic (see next post).
Sources: http://en.wikipedia.org/wiki/Stereographic_projection, Visual Complex Analysis by Tristan Needham, http://en.wikipedia.org/wiki/Mercator_projection
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