# QR2.2.5 A Polar Space

Euclidean space is so deeply ingrained in western thought that one might think it is the only way a space can be but one can derive polar coordinates based on rotations rather than straight lines. Instead of beginning with a point that makes a line then a plane, etc., one can begin with a point that creates a circle and continue from there. In network terms, a circle defines one-dimension, as every node has two neighbors giving left and right directions and the distance between two points can be defined as the number link connections (Figure 2.3). A network defines distance and direction by its architecture, i.e. how the nodes connect, e.g. a node directly linked to another is “near” while one many links away is “far”. Unlike a line dimension, a circular dimension is finite rather than potentially infinite and does not have its zero point on itself.

Now just as Euclid did for a line, the circle in Figure 2.3 can be rotated at right angles to give a two-dimensional sphere (Figure 2.4). A Flatlander confined to this surface would see a space that is:

1. Finite. Has finite number of points.

2. Unbounded. Moving in any direction never ends.

3. Has no center. No point on the sphere surface is the center.

4. Approximately flat. If the sphere is large enough.

5. Simply connected. Any loop on it can shrink to a point.

In other words, this surface performs just like our space but with only two dimensions.

Now just as rotating a 1D circle gives a sphere with a 2D surface, so rotating a sphere gives a hypersphere with a 3D surface (Figure 2.5). A hypersphere is what you get when you rotate a sphere just as a sphere is what you get when you rotate a circle. It is well defined mathematically but while a sphere surface has two dimensions, a hypersphere surface has three.

Centuries earlier, the mathematician Riemann speculated that our space was a hypersphere surface because the facts fit: such a surface is unbounded, simply connected and three-dimensional just as our space is. The logic is even more convincing today, when Einstein says that space curves like a surface and cosmology says it expands everywhere at once like an expanding balloon surface. Logically, our 3D space could easily be a surface in 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 hyper-sphere.” (Davies, 2006) p45

Quantum realism concludes that our 3D space is a hypersphere surface within a four-dimensional quantum network that can transmit quantum waves. Why then does our space appear flat not curved? The simple answer is that the surface of a hyper-sphere bubble that has been expanding for over 14 billion years would seem flat to us.

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