By common experience, we know there are three dimensions of motion. That is, space, which is the space of motion, is three dimensional. To measure the extent of motion requires comparing one motion with another, of which there are two ways: length and duration. The length of a motion is measured by comparing it with symmacronous but not necessarily synchronous motion. The duration of a motion is measured by comparing it with synchronous but not necessarily symmacronous motion.

Length of motion considered by itself forms a length space, which is space with a metric of length. Duration of motion considered by itself forms a duration space, which is space with a metric of duration. Since there are three dimensions of motion, length space and duration space are both three dimensional metric spaces. By convention, both are Euclidean. The length metric is called distance. The duration metric may be called distime.

Each point in length space has a length position (LP) vector that begins with the length origin. Each point in duration space has a duration position (DP) vector that begins with the duration origin. The magnitude of a length position vector is called the stance. Every point in length space that is equidistant from the origin has the same stance. The magnitude of a duration position vector is called the time. Every point in duration space that is an equal distime from the origin has the same time.

Stance and time are vector magnitudes, with their direction ignored. Stance is a radius from the origin of length space. A unit of length is the absolute value difference between two stances, that is, between the radii of two length vectors with unit difference. Time is a radius from the origin of duration space. A unit of duration is the absolute value difference between two times, that is, between the radii of two duration vectors with unit difference.

The rate of motion measured by the length of motion per unit of duration is called speed. The rate of motion measured by the duration of motion per unit of length is called pace. Note that a faster speed is a larger ratio, whereas a faster pace is a smaller ratio. Also, the ratio of a slower speed to a faster speed is less than one but the ratio of a faster pace to a slower pace is less than one.

The vector rate of change in the length vector per unit of duration is called velocity. The vector rate of change in the duration vector per unit of length is called legerity. The vector rate of change in velocity per unit of duration is called acceleration. The vector rate of change in legerity per unit of length is called expedience.

The length position vector of a trajectory evolves as a function of the time. The duration position vector of a trajectory evolves as a function of the stance. These functions are inverses of one another.

The following is based on Classical Mechanics by Kibble and Berkshire, 5^th ed., Imperial College Press, 2004, with the dual version indented and changes italicized.

p.2 The most fundamental assumptions of physics are probably those concerned with the concepts of space and time. We assume that space and time are continuous, that it is meaningful to say that an event occurred at a specific point in space and a specific instant of time, and that there are universal standards of length and time (in the sense that observers in different places and at different times can make meaningful comparisons of their measurements).

In ‘classical’ physics, we assume further that there is a universal time scale (in the sense that two observers who have synchronized their clocks will always agree about the time of any event), that the geometry of space is Euclidean, and that there is no limit in principle to the accuracy with which we can measure all positions and velocities.

In dual ‘classical’ physics, we assume further that there is a universal length scale (in the sense that two observers who have symmacronized their clocks will always agree about the stance of any event), that the geometry of time is Euclidean, and that there is no limit in principle to the accuracy with which we can measure all chronitions and legerities.

p.3-4 To specify positions and time, each observer may choose a zero of the time scale, an origin in space, and a set of three Cartesian co-ordinate axes. We shall refer to these collectively as a frame of reference. The position and time of any event may they be specified with respect to this frame by the three Cartesian co-ordinates x, y, z and the time t. … The frames used by unaccelerated observers are called inertial frames.

p.3-4 To specify chronitions and stance, each observer may choose a zero of the stance scale, an origin in 3D time, and a set of three Cartesian co-ordinate axes. We shall refer to these collectively as a time frame of reference. The chronition and stance of any event may they be specified with respect to this time frame by the three Cartesian co-ordinates ξ, η, ζ and the stance r. … The time frames used by inexpedienced observers are called facile time frames.

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“In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuth angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the polar coordinate system.” (Wikipedia)

A 3D time spherical coordinate system is implicitly behind space-time (3+1), with the two angles ignored for scalar time. That is, every instant in 3D time is projected onto a temporal sphere centered on the origin instant. The scalar time is the radial distime of each instant.

Let the spherical coordinates of space be (r, θ, φ) with r representing the radial distance, and θ and φ representing the zenith and azimuth angles, respectively. Let the spherical coordinates of 3D time be (t, β, α) with t representing the radial distime, and β and α representing the temporal zenith and azimuth angles, respectively.

To represent the fullness of space and time requires six dimensions (3+3), three for space and three for time: ((r, θ, φ); (t, β, α)) or (r, θ, φ; t, β, α). Then space-time (3+1) can be represented by the coordinates [r, θ, φ; t] and time-space (1+3) by the coordinates 〈r; t, β, α〉.

If rectilinear coordinates are used for 3D time, say (ξ, η, ζ), then the radial distime t equals √(ξ² + η² + ζ²). The corresponding spatial concept, stance, is the radial distance, which if rectilinear coordinates are used for 3D space, say (x, y, z), then the stance r equals √(x² + y² + z²). For 2D applications such as mapping, polar coordinates would be used instead of spherical, in which case r = √(x² + y²) and t = √(ξ² + η²).

The result is that to convert an invertible (3+1) function to a (1+3) function requires expansion to the (3+3) function, inversion, and then contraction to (1+3). In symbols, (3+1) ⇑ (3+3) ⇓ (1+3), or (r, θ, φ; t) ⇑ (r, θ, φ; t, β, α) ⇓ (r; t, β, α). In this way space and time are interchanged.

In symbols, from a invertible parametric space function inverted to a parametric time function (with ⇑ as expand, ⇓ as contract, and ↔ as invert): r = [r, θ, φ] = r(t) = [r(t), θ(t), φ(t)] ⇑ [r´(t´, β´, α´), θ´(t´, β´, α´), φ´(t´, β´, α´)] ↔ (t´(r´, θ´, φ´), β´(r´, θ´, φ´), α´(r´, θ´, φ´)) ⇓ [t(r), β(r), α(r)] = t(r) = [t, β, α] = t.

Take for example the definition v = dr/dt. We have: v = dr/dt = [dr/dt, dθ/dt, dφ/dt] = [r(t), θ(t), φ(t)]) ⇑ [r´(t´, β´, α´), θ´(t´, β´, α´), φ´(t´, β´, α´)] ↔ (t´(r´, θ´, φ´), β´(r´, θ´, φ´), α´(r´, θ´, φ´)) ⇓ [t(r), β(r), α(r)] = [dt/dr, dβ/dr, dα/dt] = dt/dr = u. The result is that space and time are interchanged, with spatial vectors becoming radial distances and radial distimes becoming temporal vectors.

Functions that are not invertible may be inverted by differentiation, then integration. Take for example, the function s(t) = s₀ + v₀t + ½at². Differentiating twice leads to s(t)´´= a = dv/dt. Expanding, inverting, and contracting results in t(s)´´= du/ds = b. Integrating twice produces t(s) = t₀ + u₀s + ½bs², which has the same form as the original function.

Let us begin with (1) the motion of a body between two events and (2) two ways of measuring the extent of that motion: length and duration (or time). The measurement of length and duration is coordinated so that both measures are of the same motion. Length and duration are measured by a rigid rod and a stopwatch, respectively. A smooth manifold of length is called space (or 3D space), and a smooth manifold of duration is called time (or 3D time).

The length and duration of a motion are commonly measured along the trajectory (or arc) of the motion. The length along the trajectory of motion is the arc length (or proper length or simply length). The duration along this trajectory is the arc time (or proper time or simply time).

Once the length and duration are in hand, the next step is to form their ratio. The ratio with arc time as the independent variable and arc length as the dependent variable is the speed. Which is to say, speed is the time rate of arc length change.

Note that the ratio could just as well be formed in the opposite way, with the arc length as the independent variable and the arc time as the dependent variable. This ratio is called the pace from its use in racing, in which an arc length is first set and then the racer’s arc time is measured. As another way to state this, pace is the space rate of arc time change.

The reference trajectory for measuring the length of a motion is the minimum length trajectory between two event points. The length along this trajectory is the distance between the two event points, which forms the metric of space. Distance is represented as a straight line on a length-scale map.

The reference trajectory for measuring the duration of a motion is the minimum time trajectory between two event instants. The duration along this trajectory is the distime between the two event instants, which forms the metric of time. On a map, two isochrons are separated by a constant distime. Distime is represented as a straight line on an time-scale map.

Motion has direction as well as extent, and direction may also be measured in two ways. Consider the motion of rotation, which can be measured as a proportion of a circle and as a proportion of a cycle. For example, in an analogue clock a minute hand that moves the length of a right angle correspondingly moves a duration of 15 minutes and vice versa. The direction of motion may be measured by either length or duration.

A motion measured with length direction and distance comprises a vector displacement. A motion measured with time direction and distime comprises a vector dischronment. The ratio with time as the independent variable and displacement as the dependent variable is called the velocity. The magnitude of the velocity vector is the speed. The ratio with distance as the independent variable and dischronment as the dependent variable may be called the legerity. The magnitude of the legerity vector is the pace.

There are three dimensions of motion, and correspondingly three dimensions of length and duration. The three dimensions of length comprise space. The three dimensions of duration comprise time. The three dimensions of time come as a surprise, since the distime is often a parameter for ordering events. But the scalar distime should not be confused with the vector dischronment, which has three dimensions of motion measured by duration.