Length in two-dimensional space (in other words, length in a plane) defines distance. If the distance is referenced from a point, then the circle is the set of all points that are this particular distance away from that point, or center. The radius is the length or measurement of the distance from the center to the perimeter. Similarly, an angle in two-dimensional space defines direction. If the direction is referenced from a line, then the direction is the difference in orientation between the two rays.  This difference in orientation is expressed as an angle, which is the measurement of the orientation that separates the two directions.

Moving to three-dimensional space, the sphere defines a relative distance from the center of the sphere, with the radius of the sphere being the expression or measurement of length. Similarly, the aperture defines the relative orientation between the axis and the generatrix of the cone, with the angle being the expression or measurement of direction.  In either case, the solid shape is what describes the physical property (quality, quantity, attribute, etc.)

There are a few subtleties that are encountered, and most of the time these are merely conceptual and are not mathematical. When referring to length, we generally mean a measurement or metric, and when we refer to distance we are referring to the relationship or separation that exists between the center and the points on the perimeter of a circle, or the surface of a sphere. In a general sense, length is spatially oriented while distance is not. This shouldn’t be a stumbling point or anything like that, but it is much more precise to use these words a certain way, especially when we are going to try and speak about direction in a manner that is symmetrical to length. This symmetry is exceptionally important in understanding the thesis.

With direction the subtlety lies in the observation that direction is the difference in orientation between the rays of an angle, or between the axis and generatrix of a cone. This is a relative construct, similar to distance. When we talk about orientation then we have to apply something similar to the metric used for understanding length.

This is very, very, very subtle. The subtlety of it is probably the thing which has kept it undiscovered for such a long time. There’s a symmetry that becomes apparent once these subtleties are acknowledged and studied.

In order to make sense of length, we instill our coordinate system with a metric, or a system of distances. This is usually accomplished in the form of the Cartesian system, which uses perpendicular or orthogonal axes with identical units of length in order to represent a two-dimensional or three-dimensional space. Once this metric has been applied, it limits how we can mathematically express a direction. It is only possible to indirectly represent direction as a ratio of lengths.

This representation of direction occurs by one of two methods: the ratio between the lengths of the sides of a right triangle (Pythagoras) or the ratio between the lengths of the diameter to the circumference of a circle ($\pi$). Understand what is happening here – because we have a coordinate system that is a collection of lengths, only these lengths are available to us in order to express a direction.

Once again we will try and explain the goofy nomenclature that we’re using. First, this discussion is complicated enough without adding to it all of the vagaries of all of the different meanings for the word “dimensions.” In the convention that we’re using, 2D and 3D will always mean the combination of Cartesian axes that define a plane or a volume. Although there are a few issues with this, it’s best to be specific about this particular use of “dimension.” We could also use $E^2$ and $E^3$, or $R^2$ and $R^3$, to mean the same thing mathematically, if not conceptually.

When we talk about spacetime we’re going to be talking about a different type of “dimension,” which can vary (both conceptually and mathematically) according to usage. This variation will be distinguished by the use of two completely different constructs. These two constructs deal with the dimensions of physical properties (qualities, quantities, attributes, etc) rather than mathematical ones. In other words, the physical property (quality, quantity, attribute, etc) of length is used more than once in the conventional understanding of spacetime. In the conventional version we will have the $3D$, $E^3$, or $R^3$ construct of three orthogonal lengths, plus there will be an additional physical dimension of time. This is view is represented by the traditional ${ct,x,y,z}$ coordinate system.

When we talk about spacetime using synchronous geometry, we assign three completely separate physical dimensions (which are time, length, and direction) in order to construct a mathematically and conceptually different version of spacetime. We’ll try not to use the term “dimension” indiscriminately without explaining the peculiar usage each time it’s used. We’ll also try and add the term synchronous when referring to this alternative construction of spacetime.

In any event, the nomenclature is in obvious need of some refinement and this is the way we’re going to proceed, for now. We’ll leave it to others to pick their own poison where the terminology is at stake. As long as we’re consistent in our method then we should be able to avoid any confusion. Or, when there is confusion, which there will be, at least it shouldn’t be coming from the words that we’re using.

Most of these discussions are either in spacetime or Euclidean 3-space, and if the difference matters for a particular aspect it should be worth noting. As far as we can tell, this entire mathematical thesis has no effect at all on existing math that operates in Euclidean 3-space (other than some conceptual insights), but it seems as though it must have a major impact on how math works in spacetime.