To reverse engineer space, it must have properties. A space that is nothing at all couldn’t expand, but ours does, so it must be something that has properties. The previous sections suggest that our space is:
- Discrete. Made up of points with a minimum size.
- Directional. Its points have multiple directions of movement.
- Three dimensional. It has three degrees of freedom of movement.
- Finite. Our universe isn’t infinitely large in size.
- Expanding. Expanding everywhere, from an origin not within itself.
- Scalable. Expanding indefinitely, as far as we can tell.
It follows that our space can’t be Cartesian because a Cartesian space must expand from a point within it, and it can’t expand indefinitely. Cartesian space is so deeply ingrained in western thought that some think it is the only way a space can be, but it isn’t. Euclid derived Cartesian coordinates by extending a point to a straight line, then repeating to get a plane and a volume, but instead of extending linearly, one can extend a point by rotations, to derive polar coordinates (Note1).
Given a point as before, it is taken to be discrete, as point of no extent can’t exist. Rotating a discrete point stepwise gives a circle of points that define one dimension with two directions, as each point connects to two neighbors (Figure 2.3). The distance between points is the number of connections between them, so points 1 and 2 in Figure 2.3 are close, but points 1 and 10 are far apart. The circle is also finite, and it can expand indefinitely from a point that isn’t on it (the circle center).
Then, just as Euclid extended a line, a circle can be extended by rotation to give a two-dimensional sphere surface (Figure 2.4). As a sphere increases, its surface expands everywhere at once, not from a point on itself. Cosmologists conclude that our universe is expanding like this, rather than exploding from a point as the term big bang suggests. Hoyle popularized the balloon analogy to explain how our space expands:
“My non-mathematical friends often tell me that they find it difficult to picture this expansion. Short of using a lot of mathematics I cannot do better than use the analogy of a balloon with a large number of dots marked on its surface. If the balloon is blown up the distances between the dots increase in the same way as the distances between the galaxies.” (Hoyle, 1950).
Today, scientists agree that our space doesn’t expand from a center within it but everywhere at once, like a balloon being blown up. To a reverse engineer, this suggests that it is polar not Cartesian. A two dimensional being living on an expanding sphere would see a space that is discrete, directional, finite, and increasing everywhere. It would act just like our space, but with only two dimensions, so is a three-dimensional polar space possible?
Note 1: Cartesian coordinates are represented by (x, y, z) values, but polar coordinates are represented by (r, a, b), where r is the radius from a fixed point, and alpha and beta are angular directions. Both systems need a (0,0,0) point.