iSoul In the beginning is reality

# Homogeneity and isotropy

A circle or sphere are omnidirectional in two or three dimensions, respectively. This is equivalent to isotropy, uniformity in all directions. A straight line is unidirectional but multiple straight lines may require multiple dimensions. This is equivalent to rectilinear homogeneity.

Pure space or average space is homogeneous and isotropic. Then space may be modeled by one dimension, although since the word dimension usually has to do with degrees of freedom or potential directionality, we say it has three dimensions.

It’s the same with time. Pure time or average time is homogeneous and isotropic, and may be modeled by one dimension, though it has three degrees of freedom and so we say it has three dimensions. If time is isotropic, only one dimension is needed to model it. If time is anisotropic, then three dimensions are needed to model it.

This is like the duality of wave and particle in quantum mechanics. Space and time have one or three dimensions depending on the aspect modeled.

Universal simultaneity requires homogeneity: “the transport of an ideal clock without distortion of time-intervals, requires a homogeneous space” (*).

In surface transportation a distinction can be drawn between congestion-type and current-type hindrances to travel. Radial congestion, such as a simple model of a city with a central business district, is isotropic. Travel across or in a river current could be modeled as rectilinearly homogeneous.

The conclusion is that homogeneity and isotropy come with a pure or average conception of space or time and require only one dimension to model. But the particulars of many situations do not exhibit either homogeneity or isotropy and so require three dimensions to model.