Approximating a sphere with a regular tetrahedron I'm studying computer graphics independently through a textbook and I've come across this (seemingly challenging) problem in chapter 1. It is stated as follows:

A different method of approximating a sphere starts with a regular
  tetrahedron, which is constructed from four triangles. 
  Find its vertices, assuming that it is centered at the origin and has
  one vertex on the y-axis. Derive an algorithm for obtaining increasingly 
  closer approximations to a unit sphere based on subdividing the faces of the tetrahedron.

I struggled with this problem for a while before looking up the answer. I couldn't get enough of an idea to even get started with this kind of task. After a few hours I looked up the answer in the back of the book:

We derive this algorithm in chapter 6. First we can form the tetrahedron
  by finding four equally spaced points on a unit sphere centered at the 
  origin. One approach is to start with one point on the z-axis $(0,0,1)$.
  We can then place the other three points in a plane of constant z. One
  of these three points can be places on the y axis. To satisfy the
  requirement that the points be equidistant. The point must be
  $(0, 2\sqrt{2/3}, -1/3)$. The other two points can be found by symmetry to be at $(-\sqrt{6/3}, -\sqrt{2/3}, -1/3)$ and $(\sqrt{6/3}, -\sqrt{2/3}, -1/3)$.

My question(s) from this are:


*

*What gave him the idea to inscribe the tetrahedron in a unit sphere? I suppose because we are trying to approximate one it makes sense to use it as a guide.

*How does he choose the points and derive the answer? Admittedly my geometry is a little rusty, but I felt completely defeated by this question.

*How can I properly visualize this process so I can repeat it on future questions?
 A: The question isn't so much about the maths, but the process. The tetrahedron is bounded by the sphere, and its four vertices touch the surface of the sphere.
Replace each triangular face, with three more triangular faces, with the new shared vertex on the surface of the sphere. The original three vertices remain, because they touch the sphere.
Do that again, and again until the calculation is too small to be significant.
Using a computer language, this can be done by recursion.
Your questions:

What gave him the idea to inscribe the tetrahedron in a unit sphere? I
  suppose because we are trying to approximate one it makes sense to use
  it as a guide.

This is the problem definition: increase the number of vertices of the polyhedron until it is almost like a sphere.

How does he choose the points and derive the answer? Admittedly my
  geometry is a little rusty but I felt completely defeated by this
  question.

He chooses a new point extended from the origin, through the centre of a face, to touch the required sphere.

How can I properly visualize this process so I can repeat it on future
  questions?

The tetrahedron is the simplest polyhedron. The exercise is to imagine how the centre of each of its faces can be extended out to touch the sphere, replaced by three new faces.
The process continues until the polyhedron so much resembles a sphere that it is not worth continuing.
