iSoul In the beginning is reality

From 1D to 3D in two ways

Among the instruments on a vehicle there may be a speedometer, an odometer, a clock, and a compass, which provide scalar (1D) readings of the vehicle’s location. But what is the location of the vehicle in a larger framework? The compass shows two dimensions must exist on a map of this framework, but of what are they dimensions?

The identity of the two dimensions depends on whether the dimensions are associated with the odometer reading (the travel length) or the clock reading (the travel time). Let’s represent the travel distance by s, the travel time by t, the speed by v, and the travel direction by angle α clockwise from North.

Consider a simple example in which the vehicle is traveling at a constant speed and not changing direction. Then the ratio of the travel distance to the travel time is a constant, which equals the reading on the speedometer: v = s / t.

The vehicle location may be envisioned in two different kinds of maps: (1) In the first kind of map, which is the familiar one, the travel direction is associated with the travel distance. Then the odometer and compass determine the vehicle location, which may be specified by the polar coordinates (s, α) = s. This ordered pair represents a spatial position vector, s. A velocity vector may be constructed from it as v = s / t.

(2) However, we could just as well associate the travel time with the travel direction. So for the second kind of map, the clock and compass determine the vehicle location, which may be specified by the polar coordinates (t, α) = t. This ordered pair represents a temporal position vector, t. A legerity vector, u, may be constructed from it as u = t / s.

Let’s look at another simple example. Consider a vehicle on a curve that turns for an angle θ at a constant angular velocity of ω with a turning radius of r. The travel distance on the curve is s = = ωt. The travel time is t = /ω = s/ω. In the first case the spatial vector is s = (r cos(ωt), r sin(ωt)). In the second case the temporal vector is t = (r cos(s), r sin(s)), which is found by reparameterizing by the arc length.

Note that in the first kind of map the travel time remains a scalar, which is not associated with any particular position on the spatial map and so is a universal time. Note that in the second kind of map the travel distance remains a scalar, which is not associated with any particular position on the temporal map and so is a universal distance.

The question, “What time is it?” refers to scalar time, which is associated with all points of 3D space. Similarly, one could ask, “what space is it?” referring to the scalar distance, a 1D space, which is associated with all points of 3D time.