A One-Dimensional Space

We can adapt the definition of time to make an example of a one-dimensional space. Recall that the ground-state model of the electron divided quarks into two groups that were distinguished from each other only by their relationship with the frame of reference. The sensory interpretation of this distinction was that the frame provided a backdrop that was dark for some quarks, and bright for others. But this sensory quality was masked by shifting the description to using a time coordinate. The disguise was completed with an assumption that usually there are equal numbers of black and white sensations about. They supposedly cancel each other and fade from awareness, but the time coordinate was usefully retained. Calculus was not required for a quantitative version of this interpretation.

A one-dimensional spatial model of the electron.
A one-dimensional spatial model of the electron.

To make a similar example of a one-dimensional space recall that scalar multiples of the basis vector $\hat{z} \equiv (0, 0, 1)$ are collectively called the polar axis. And remember that the phase $\delta _{\theta}$ characterizes black and white sensations in the background. Objectify any difference in $\delta _{\theta}$ as a variation in direction on the polar axis. Put a number to this rudimentary notion of spatial structure using $dz$ the polar component of the displacement. To illustrate, we use a rod to represent the polar axis. Then the ground-state model of an electron is modified slightly to make a one-dimensional spatial model as shown in the accompanying movie. The $z$-coordinate of an event is defined from a sum of displacements. Then an assumption about symmetry lets us drop explict reference to sensation, but the $z$-coordinate is kept and used to describe position on the polar axis. Recognizing this sort of association between a visual sensation and a spatial arrangement is something that we do constantly and more-or-less unconsciously. It is the first step in defining a particle-centered Cartesian coordinate system.

A look around a one-dimensional spatial model of the electron.
Please notice that empty space has not been defined. The foregoing ideas are based on a specific particle. And later, atoms are required to define fully three-dimensional spaces. Perfect vacuums and voids are not considered out of deep respect for the work of experimental physicists. By insisting on an empirical basis for theoretical physics we avoid paradoxes like those from Zeno of EleaXlink.png. Moreover as Percy BridgmanXlink.png has pointed out, considering empty space to be a physical object can lead to logical inconsistencies.1 Indeed if a vacuum measuring instrument is put in an ostensibly empty space, then it is not empty.
Right.png Next step: a two dimensional space.
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