Editing 3-sphere (section)

===Gluing===
A 3-sphere can be constructed [[topology|topologically]] by [[Quotient space (topology)|"gluing" together]] the boundaries of a pair of 3-[[ball (mathematics)|balls]]. The boundary of a 3-ball is a 2-sphere, and these two 2-spheres are to be identified.  That is, imagine a pair of 3-balls of the same size, then superpose them so that their 2-spherical boundaries match, and let matching pairs of points on the pair of 2-spheres be identically equivalent to each other.  In analogy with the case of the 2-sphere (see below), the gluing surface is called an equatorial sphere.

Note that the interiors of the 3-balls are not glued to each other.  One way to think of the fourth dimension is as a continuous [[real-valued function]] of the 3-dimensional coordinates of the 3-ball, perhaps considered to be "temperature".  We take the "temperature" to be zero along the gluing 2-sphere and let one of the 3-balls be "hot" and let the other 3-ball be "cold". The "hot" 3-ball could be thought of as the "upper hemisphere" and the "cold" 3-ball could be thought of as the "lower hemisphere". The temperature is highest/lowest at the centers of the two 3-balls.

This construction is analogous to a construction of a 2-sphere, performed by gluing the boundaries of a pair of disks. A disk is a 2-ball, and the boundary of a disk is a circle (a 1-sphere). Let a pair of disks be of the same diameter.  Superpose them and glue corresponding points on their boundaries.  Again one may think of the third dimension as temperature.  Likewise, we may inflate the 2-sphere, moving the pair of disks to become the northern and southern hemispheres.