Euclid derived Cartesian dimensions by extending a point linearly in three new ways, so why not rotate a point in three ways to get three polar dimensions? In mathematics, rotating a sphere in an orthogonal direction gives a hypersphere, which is a sphere rotated just as a sphere is a circle rotated (Figure 2.5). The result, in theory, is a surface with three dimensions not the two, which are curved not straight.
The idea that our space could be curved was introduced by Bernard Riemann in 1857. His Riemann Sphere describes a hypersphere whose three-dimensional surface acts just like our space. It still applies today, as relativity lets space curve like a surface and cosmology says it expands like a surface. Hence, our space could be a three-dimensional surface within a four-dimensional bulk:
“When it comes to the visible universe the situation could be subtle. The three-dimensional volume of space might be the surface area of a four dimensional volume” (Barrow, 2007) p180.
Davies makes the case even more clearly:
“… the shape of space resembles a three-dimensional version of the surface of a sphere, which is called a hypersphere.” (P. Davies, 2006) p45.
The argument seems solid, but for it to be really so demands another dimension that our space doesn’t have. How can a sphere that already fills the three dimensions of our space be extended? The problem is insurmountable for physical realism but not if our space is virtual. If network connections can represent a circle and a sphere, they can represent hypersphere if the quantum network connects in four dimensions not three. Our space has three dimensions but that doesn’t mean it isn’t a surface. The screen of a computer has two dimensions but the screen of space could have three. How our space behaves suggests it is a hypersphere surface inside a four-dimensional network.
If planets, stars, and galaxies are curved, why not the universe itself? We used to think the earth was flat, but now know that it is curved, so why can’t space curve? But if our space is a hypersphere surface, why does it seem flat to us? The likely answer is that the surface of a hypersphere that has expanded for over 14 billion years would seem flat to us, just as the surface of our earth did centuries ago. But if our space is a curved surface, what does that imply?