3-Sphere

I've been currently reading "Reality in Not What It Seems: The Journey to Quantum Gravity" by Carlo Rovelli (translated from the Italian in which it was originally written). It's a fascinating book but early on, Rovelli makes mention of the 3-Sphere and says:

Einstein’s idea is that space could be a 3-sphere: something with a finite volume (the sum of the volume of the two balls), but without borders. The 3-sphere is the solution which Einstein proposes in his work of 1917 to the problem of the border of the universe.

Well, this was a new concept to me and naturally I needed to investigate. I was fortunate to stumble across a question in Quora titled: How can one visualize a 3-sphere? All three answers to the question were illuminating in different ways. Of course, the 3-sphere is a specific type of n-sphere: there is a 1-sphere, a 2-sphere as well as a 4-sphere, 5 sphere and so on. It helps to start with the 1-sphere and 2-sphere before tackling the 3-sphere.

In the diagram shown below a 1-sphere (circle) existing in 2 dimensions is represented in 1 dimension as two lines joined together at their ends:

In the next diagram, a 2-sphere (sphere) existing in 3 dimensions is represented in 2 dimensions as two circles joined together at their boundaries.

This gives us some idea of how to represent a 3-sphere, or hypersphere, existing in 4 dimensions. It can be envisioned as two spheres joined across their boundaries. This is shown in the diagram below:

The following YouTube video does a very good job of explaining the mathematics underlying the 3-sphere in terms of the Hopf fibration:

During this video, the lecturer (Niles Johnson) mentions Cartesian products and Quaternions. I've heard of both but should try to reacquaint myself with them.

ADDENDUM: added on January 11th 2019

I was reminded of the 3-sphere by a recently read article that begins:

Our universe has antimatter partner on the other side of the Big Bang

In a CPT-symmetric universe, time would run backwards
from the Big Bang and antimatter would dominate

The two three dimensional spheres that make up the fourth dimensional 3-sphere seem perfectly suited to accommodate the matter-antimatter pair of universes. If one accepts that time is an illusion (see this blog post of mine title The Illusion of Time) then these two structures are eternally existent in a timeless 4-D Universe.