We do anticipate that some of the material presented here will be expanded on and that some errors will have to be corrected. This site is intended as a notebook of sorts, and it will be used for tracking new information that relates to the study of this mathematical form. The scope of the subject matter necessitates launching this site with the material presented as a simple synopsis, or at best, an introductory primer.

The impact of these new mathematical relationships on our understanding of physics will also be explored. What seems to have happened here is that by stumbling across a new relationship in Euclidean 3-space we have opened a door to improvising new mathematical and conceptual constructs in the fundamental nature of spacetime. These improvisations cannot be reached without understanding the underlying geometric relationships that exist, and therefore we have not been able to find much information on many of the subjects that we talk about here. The implications of understanding direction are quite profound, and it will most likely affect areas of inquiry that have not even been considered at this time.

Figure 1.1 The Map of Physics – Image by Dominic Walliman

Direction (or orientation) is a component of space. No one seems to disagree with this arrangement. There doesn’t seem to be any authoritative source on whether direction is a property, quality, quantity, attribute, parameter, constant, or whatever. Some experts claim that it’s a thing that isn’t a thing. Additional perspective about this thing can be realized through the use of this geometric approach. It doesn’t appear to change anything that is already known regarding Euclidean 3-space, except for around the extremely remote edges where we get very near to the *chasm of ignorance*.

It may seem that some of the ideas have been overblown, or overthought, or over-complicated, in some misguided attempt to exaggerate the significance of the concepts. Some experts have claimed that the approach is either too trivial or too tedious to try and understand. Sadly, at the time of this publication there doesn’t seem to be anyone outside our group that takes this material seriously.

All we can ask of the skeptics is that they do the math. The implications of the math itself are what are being discussed here. The math is mostly trigonometry and algebra, and although the underpinnings are fairly straightforward, the insights that are gained can be extremely subtle and at the same time extremely significant.

The math appears to be an expansion of spherical trigonometry. This mathematical expansion expresses a relationship between the radius of a small circle on a sphere to the slope of a tangent to that circle relative to the sphere. The physics appear to be concerned with certain solutions that this expansion and extension of spherical trigonometry can produce for defining spacetime.

The fundamental function underpinning the whole concept is an expression of a unique relationship that’s created by the framework of three orthogonal axes, or perhaps three orthogonal planes. These are the normal axes that we are used to, the ones that exist in Euclidean 3-space, or what we know as the Cartesian coordinate system. The three-dimensional framework contains relationships that don’t occur in two dimensions. Heretofore these relationships have gone unnoticed.

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There exists a deeper understanding of what a 2D plane actually is, though. It’s most commonly defined as two lengths (or axes) perpendicular to one another, sure, but what does that really mean? The concept of orthogonality brings the concept of relative direction (or orientation) into the picture. This structure of relative orientation, when combined with length, is the defining characteristic of a two-dimensional plane, and the defining characteristic of Euclidean 3-space.

There is also another use of dimension where it means a particular quantity that can express a distinct physical phenomenon or characteristic. In this usage, the meaning of dimension is analogous to base quantity in physics, and is the foundation for the analytical technique known as *dimensional analysis*.

The current state of the art allows for direction to be expressed as a ratio. It is not really a quantity by today’s standards. The two basic methods are to consider a ratio between two (or more) perpendicular lengths (Pythagoras) or to consider a ratio between the lengths of the diameter and circumference of a circle (). In either case, these lengths are combined in a specific way in order to use them to represent a distinction that can be made between two separate directions. This value is always a ratio, never a quantity.

The way it’s currently done, the radian is the cheater’s way of representing a turn. It is conceptually a two-dimensional construct, and is only an abstraction of what really occurs with an actual physical turn in spacetime. A turn in Euclidean 3-space is different than simply a ratio between two lengths. The same problem exists with representing volume as a product of three lengths. This *assumes* that the three lengths are in three directions that are “orthogonal” to one another. What does that even mean without a base quantity of direction? Orthogonal, but without a unit of direction? What’s going on here?

In case it isn’t obvious, there is a stunning symmetry with length here. We can call it linear distance, where we can call the change in linear distance speed, and the change in speed acceleration.

Even more stunning is the fact that when we add direction to length we get position, when we add direction to distance we get displacement, and when we add direction to speed we get velocity.

Still, even in light of all these facts, there seems to be a winning argument (for reasons no one can explain other than the old “we’ve always done it that way”) that direction isn’t really a base quantity, like time or length. It’s supposed to be a thing that isn’t really a thing, whereas the other two are things that really are things. This approach is rather arbitrary, especially since direction can now be quantified as a scalar value or an amount, exactly like time and length are both quantified.

Figure 2.1 – NIST Subway Diagram

So, what is direction, really?

Currently, direction is an orphan when it comes to understanding physical quantities and how they relate to one another. The basic unit that is used to define an angle, the radian, is the disconnected unit at the lower right in the NIST diagram. Why is that? And why is the unit of the solid angle, the steradian, a component of luminous flux? What’s up with that?

It’s our new understanding that a quantity related to the radian is actually the eighth base quantity, and it belongs on the left-hand side of the diagram, probably just below length. Then, those three base quantities (sometimes referred to as dimensions in physics) of length, direction, and time can be understood as comprising spacetime.

This idea that there are fundamental physical properties that can be related or equated to one another gives rise to an entire method of *dimensional analysis* that is used for examining physical phenomena and theories and inferences about their underpinnings. The foundation for such a view is the understanding that everything can be expressed as a quantity of something. This ability to equate the property to a quantity is the key here, and this website is dedicated to learning how and why direction can be expressed as an actual quantity.

The SI base units and their physical quantities are the metre for measurement of length, the kilogram for mass, the second for time, the ampere for electric current, the kelvin for temperature, the candela for luminous intensity, and the mole for amount of substance. The turn (or revolution) is the manifestation of a generalized property that is expressed as orientation or direction. It is possible that the physical quantity of should be incorporated for the amount of direction, or orientation. There’s a *legitimate argument* to be made that is a better value to use for deriving the base unit, and we’re starting to lean in that direction.

Just as we can use a direction and a radius for identifying a position, there’s also another interesting use for combining a direction and a radius. We can use a direction and a radius to specify a volume, and this raises some other issues that have to be explained somehow.

The SI derived unit for volume is . But that’s not the only way to specify a volume. If we know the radius of a sphere we can calculate the volume inside the sphere using a simple formula. But what happens if we take a partial area of the surface of the sphere and use this to calculate volume? The most straightforward method is to use the steradian (symbol ) as the partial area of the surface of the sphere. The steradian (see the SI chart in the above post) is referred to as the derived quantity for solid angles, but in this case its just as appropriate to look at it as the area of on the curved surface of a sphere.

The procedure is to take the volume of the space that lies within the steradian and the sphere center. To make this into a volume we’ll take the product of the angular quantity of the and the length quantity of the meter. Well call this unit of volume the steradian meter, or . There are steradians in a complete sphere, so there must also be units of volume that will be equal to the volume of the complete sphere.

The first thing to notice is that the can use either direction or length as the unit quantity. A quantity of where is different than the quantity of where . Mathematicians will probably know of a convention that can be used in order to capture this difference. If we put the number in the appropriate location to show which quantity is the unit quantity and which quantity the number is associated with, then the first case would be written as and the second case would be written as . A little arithmetic reveals that:

and also, for example:

If space is simply an empty volume, these three examples of expressing a volume should have identical meanings. They should all be identical, both mathematically and conceptually. The problem is that there is a difference. The expression has no spherical excess, has a spherical excess of , and has a spherical excess of .

I guess it would be possible to pretend that this difference has no relevance at all. Once again, I think were up against the same mathematical quandary. The difference between using unit length or unit direction to express a volume has no impact at all on Newtonian spacetime where ct is a constant. But it does raise a whole universe of issues once we try to use these two different methods to express a volume in relativistic spacetime.

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 (). 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 and , or and , 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 , , or construct of three orthogonal lengths, plus there will be an additional physical dimension of time. This is view is represented by the traditional 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.