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.)

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It’s generally understood that, when the time dimension is subtracted from spacetime, we are left with Euclidean 3-space.  This is usually interpreted to mean that, once we look at spacetime without the time component, we’re left with space, or what in SI units is called volume.  The SI unit for volume is which is consistent with our understanding of direction being simply a ratio between lengths.  In this case, the three orthogonal lengths can be used to define an object without ever having to define orthogonality.  This seems like an incomplete description of what a volume actually is.

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.

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So, what is direction, really? We can call it angular position, and we can call the change in angular position angular velocity, and we can call the change in angular velocity angular acceleration.

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.

Direction is currently considered to be a ratio between lengths.  This is different (mathematically and conceptually) from representing direction as a quantity.  Ratios are not quantities.   They are a comparison between two or more quantities, which is fundamentally different from how quantities themselves are represented.

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The way that it’s mathematically expressed, space actually comprises a set of three 2D planes, arranged orthogonally. In this usage, each individual “dimension” is actually a plane, or a set of two perpendicular directions, where each direction is also called a “dimension.”  The x and y dimensions combine to form the xy-plane which is perpendicular (normal) to the z axis; it is the direction of the z axis and not the location of the z axis that is the thing which is perpendicular to the x direction and the y direction.  For these reasons, it would be more meaningful (less confusing or less arbitrary) to specify our mathematical treatment of space as 2D³ rather than 3D. It’s the reason why our mathematical treatment of Euclidean 3-space has octants… 2x2x2=8.

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