QR2.2.3 A Cartesian Space

That space is a “something” raises the question What does it do? It seems strange to talk of what space “does” but modern simulations of it do just that:

…we think of empty spacetime as some immaterial substance, consisting of a very large number of minute, structureless pieces, and if we let these … interact with one another according to simple rules … they will spontaneously arrange themselves into a whole that in many ways looks like the observed universe.(Ambjorn et al., 2008) p25.

For the purposes of geometry, Euclid gave our space a structure many years ago as follows. First, he imagined a point with no dimensions. Then he extended that point continuously to create a line, that was again extended at right angles to give a plane, that was again extended to give a cube. This defined a Cartesian space with three orthogonal dimensions, where every point was represented by three real number coordinates (x, y, z).

War-gamers don’t use Euclid’s space because squares only allow four directions to attack an enemy but divide their maps into hexagons instead, to give more interaction directions. In general, a space requires:

1. Dimensions. That define the degrees of freedom needed to create it.

2. Locations. That define when two objects are “in the same place” and so interact.

3. Directions. That define the number of ways a point can interact with its neighbors.

Simulating space as a network isn’t a new idea, e.g. in Wilson’s networks a node is a volume of space and in Penrose’s spin networks a node is a point event with two inputs and an output (Penrose, 1972). All these models, including loop quantum gravity (Smolin, 2001), cellular automata (Wolfram, 2002) and lattice simulations (Case, Rajan, & Shende, 2001) map nodes to a Cartesian space, so they all encounter the problem of scalability.