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Overview of the Sphere


Definition of the Sphere

The n-dimensional sphere is the set of all points in $ n+1$ space at a given radius from a center point. The radius is denoted by $ r$ and the center by $ \vec{\mathrm{\mathbf{c}}}$.

$ \ensuremath{\Bbb{S}^{n}} = \left\{ \ensuremath{\vec{\mathrm{\mathbf{x}}}}\ \in...
...ert\ \sqrt{\overset{n+1}{\underset{i=1}{\sum}}{ (x_i - c_i)^2 } } = r \right\} $

Unless otherwise stated, the center of the sphere is assumed to be $ \vec{\mathrm{\mathbf{0}}}$ and the radius is assumed to be 1. Under this definition, $ \ensuremath{\Bbb{S}^{0}}$ is the two points $ \left\{ 1, -1 \right\} $ in $ \Bbb{R}^{1}$, $ \Bbb{S}^{1}$ is the unit circle in $ \Bbb{R}^{2}$, and $ \Bbb{S}^{2}$ is the object which is commonly called the sphere or globe, in $ \Bbb{S}^{3}$.

Definition of the N-Disk or N-Ball

An n-Disk is simply the set of all points inside $ \Bbb{S}^{n-1}$, or all points less than or equal to $ r$ units away from the center $ \vec{\mathrm{\mathbf{c}}}$.

$ \mathbf{\mathrm{D}}^{n} = \left\{ \ensuremath{\vec{\mathrm{\mathbf{x}}}}\ \in\...
...ert\ \sqrt{\overset{n}{\underset{i=1}{\sum}}{ (x_i - c_i)^2 } } \le r \right\} $

Under this definition, the 1-disk is the line segment between $ r$ and $ -r$ on $ \Bbb{R}^{1}$, the 2-disk is the interior of the circle of radius $ r$ in $ \Bbb{R}^{2}$, and the 3-disk or 3-ball is the interior of the sphere of radius $ r$ in $ \Bbb{R}^{3}$. As above, the center is assumed to be $ \vec{\mathrm{\mathbf{0}}}$ and the radius to be 1 unless otherwise specified.

Definition of a Manifold

Throughout this paper I will use the term manifold in its standard topological sense. A n-dimensional manifold is usually defined as a set with a collection of patches, or 1-1 functions from $ D \rightarrow M$, where $ D$ is an open subset of $ \ensuremath{\Bbb{R}^{n}}$, such that the following two conditions are satisfied:

(i.) The images of the patches cover $ M$

(ii.) For two patches $ \tau$ and $ \sigma$, the composition $ \tau^{-1}\sigma$ and $ \sigma^{-1}\tau$ are differentiable and defined on open sets in $ \Bbb{R}^{n}$ (O'Neill).

Definition of $ \times$, or Cartesian product

The $ \times$ operator is the Cartesian product. It takes two manifolds of dimension $ n$ and $ m$, and duplicates the one at every point along the other, creating a manifold of dimension $ m+n$.

Definition of Generator

A generator is a manifold $ M$ contained in another manifold $ P$ such that $ M$ crossed with some other manifold $ N$ makes $ P$. In other words, if $ M \times N = P$, then $ M$ and $ N$ are generators of $ P$. For example, since $ \ensuremath{\mathrm{\mathbf{T}}^{2}} = \ensuremath{\mathrm{\mathbf{T}}^{1}} \times \ensuremath{\mathrm{\mathbf{T}}^{1}}$, the generators of $ \mathrm{\mathbf{T}}^{2}$ are the two $ \mathrm{\mathbf{T}}^{1}$.

Definition of Equator

An equator inside $ \Bbb{S}^{n}$ is a $ \Bbb{S}^{n-1}$. This can be seen by setting one of the coordinates, say $ x_{n+1}$, to zero, and taking all remaining points in $ \Bbb{S}^{n}$.

$ \left\{ \ensuremath{\vec{\mathrm{\mathbf{x}}}}\ \in\ \ensuremath{\Bbb{R}^{n+1}...
...t{n}{\underset{i=1}{\sum}}{x_i}^2 } = r \right\}

Construction of $ \Bbb{S}^{n}$ by two N-disks

It's fairly trivial to see that two line segments, each end of one segment joined to an end of the other segment, make a circle (or something that can be deformed into a circle). However, the process generalizes to all dimensions, and provides us with a technique of visualizing $ \Bbb{S}^{n}$.

Two 2-disks, or filled circles, can be joined along their circular boundaries, to make $ \Bbb{S}^{2}$. Each disk makes a hemisphere, and the boundary between the two becomes the equator.

In general, an equator as defined above is $ \Bbb{S}^{n-1}$ in $ \Bbb{S}^{n}$. Now, all the other points in $ \Bbb{S}^{n}$ are the points where $ x_n \ne 0$. So the other points in $ \Bbb{S}^{n}$ are split into two groups: $ x_n > 0$ and $ x_n < 0$. We can see that these two sets of points are N-disks because we can let $ x_n$ range from 1 to 0, and get a corresponding $ \Bbb{S}^{n-1}$ for each value of $ x_n$.

next up previous
Next: Visualizing the N-Sphere Up: The Sphere in Three Previous: The Sphere in Three
Adric 2001-03-23