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# Category Archives: Length & Duration

Explorations of multidimensional space and time with linear and angular motion.

# Interchange of length and duration

Length and duration are measures of the extent of motion. They are measured by comparing them with a uniform reference motion. Although uniform linear motion is simpler in theory, uniform circular motion is simpler in practice – and essential for unstopped motion. With one addition, the classic circular clock with hands serves as a reference motion. The addition is to mark the circumference in length units along with the duration units of the angles between the hands and the vertical.

Galileo uses horizontal uniform linear motion to mark length and duration below (from his Dialogues Concerning Two New Sciences, Fourth Day):

The horizontal uniform motion of a particle coming from the right at a-b is continued with b-c-d-e as the horizontal component of the particle descending with uniform acceleration b-o-g-l-n. Because the horizontal motion is uniform, it can measure either length or duration of the motion. The vertical component represents the dependent variable, which has the form of a parabola.

To interchange length and duration requires four steps starting with a parametric function in time: (1) replace length components with their radius, which is the stance; (2) switch time and stance; (3) expand time to include angular components; and (4) linearize stance so that it is an independent parameter. The steps (2) and (4) have the effect of inverting the parametric function (2), and then a kind of re-inversion that switches the dependent and independent status of the variables (4).

The result of this interchange process is that the equations of motion for length and duration are interchangeable without functional change. All of the equations of physics in terms of parametric functions of time may be adopted as parametric functions of stance. In that sense it would be best to abstract a functional representation that applies to both length and duration, time and stance.

# Length and duration

Although it has seemed natural to speak of “space and time”, that is a confused designation of length and duration, as well as their metrics, stance and time (see glossary here). Space is the space of motion, so length space is only one side; the other side is duration space. Speaking consistently is challenging but a must in order to bring clarity to a confused subject.

Here is a diagram of the combinations of length and duration, stance and time (click to enlarge):

The extent of motion is measured by length and duration. The space of motion has three dimensions. Length space and duration space therefore have three dimensions as well. The space of motion is  represented by an ordered pair of length space and duration space coordinates. If there is absolute inter-convertibility between length and duration, then the space of motion is a six-dimensional space of either length units or duration units.

The metric of length is distance and the metric of duration is distime (though often called time). Stance is the distance of a point from the origin in length space. Time is the distime of an instant from the origin in duration space. Events are located in length space and chronated in duration-space.

As one can speak of distance in three-dimensional length space, one can speak of distime in three-dimensional duration space. In this sense, time is a three-dimensional concept. There are three dimensions of duration rather than three dimensions of time per se, but time has a three-dimensional aspect.

# Two measures of motion

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.

# Newton’s laws and their duals

The following is based on Classical Mechanics by Kibble and Berkshire, 5th 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).

The most fundamental assumptions of physics are probably those concerned with the concepts of length and duration. We assume that length and duration are continuous, that it is meaningful to say that an event occurred at a specific point in length space and a specific instant of duration space, and that there are universal standards of length and duration (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 chronations 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 chronations and stance, each observer may choose a zero of the stance scale, an origin in 3D duration, and a set of three Cartesian co-ordinate axes. We shall refer to these collectively as a time frame of reference. The chronation 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.

# Length and duration spaces inverted

In geometry, a spherical coordinate system specifies points 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, which is the horizontal angle between the origin to the point of interest.

A spherical coordinate system is implicitly behind length space (space-time) (3+1), with the temporal directional ignored for scalar time. That is, every instant in duration space (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 length 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 duration space (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 length space and duration space requires six dimensions (3+3), three for length and three for duration: ((r, θ, φ); (t, β, α)) or (r, θ, φ; t, β, α). Then length-duration (space-time) (3+1) can be represented by the coordinates [r, θ, φ; t] and duration-length (time-space) (1+3) by the coordinates 〈r; t, β, α〉.

If rectilinear coordinates are used for duration space, say (ξ, η, ζ), then the radial distime t = √(ξ² + η² + ζ²). The corresponding length space concept, stance, is the radial distance, which if rectilinear coordinates are used for length space, say (x, y, z), then the stance r = √(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 its (3+3) function, inversion (i.e., interchange length and duration), and then contraction to (1+3). In symbols with ↑ as expand, ↓ as contract, and ↔ as invert: (3+1) ↑ (3+3) ↓ (1+3), or (r, θ, φ; t) ↑ (r, θ, φ; t, β, α) ↓ (r; t, β, α). In this way length and duration, time and stance are interchanged.

In more detail, an invertible parametric length space function is inverted to a parametric time function: r = [r, θ, φ] = r(t) = [r(t), θ(t), φ(t)] ↑ [(t´, β´, α´), θ´(t´, β´, α´), φ´(t´, β´, α´)] ↔ ((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)]) ↑ [(t´, β´, α´), θ´(t´, β´, α´), φ´(t´, β´, α´)] ↔ ((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 length space and duration space are interchanged, with length space vectors becoming stances and time becoming duration space 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.

# Dual calendar systems

The unit for all calendars is the day, the diurnal cycle of daylight and night. A lunar calendar is based on the monthly (synodic) cycle of the Moon’s phases. A solar calendar is based on the annual cycle of the Sun’s height above the horizon. A lunar-solar (lunisolar) calendar is based on the lunar month modified in order to match the solar (or sidereal) year. The solar-lunar calendar is based on the year but includes months similar to the lunar cycle.

“The lunisolar calendar, in which months are lunar but years are solar—that is, are brought into line with the course of the Sun—was used in the early civilizations of the whole Middle East, except Egypt, and in Greece. The formula was probably invented in Mesopotamia in the 3rd millennium bce.” (Encyclopedia Britannica)

The lunar and lunar-solar (lunisolar) calendars are the oldest calendar systems, and are still used in some traditional societies and religions. The Hebrew (Jewish) and Islamic calendars are examples of the lunar-solar calendar systems. Solar and solar-lunar calendar systems came from Egypt, Greece, and Rome. The solar-lunar month departs from the lunar month but combines to equal a year.

The question is why the Moon forms the primary cycle in some calendars, whereas the Sun forms the primary cycle in other calendars. The reason may well be that some societies think in terms of 3D time, whereas other societies think in terms of 3D space. The difference is that in 3D space the Earth revolves around the Sun and the Moon revolves around the Earth, whereas in 3D time the Earth revolves around the Moon and the Sun revolves around the Earth. In the former case the solar cycle is primary, whereas in the latter case the lunar cycle is primary.

When European societies considered the Earth to be the center of all celestial motion, their calendars were already established. So the correspondence between calendar systems and the dominant perspectives (spatial or temporal) applies to the original development of calendars.

# Length and duration

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.

# Space and time as opposites

A theme of this blog is that space and time are dual concepts, which means they are two ways of understanding the same thing. But in what ways are space and time opposite concepts?

Space is oriented toward its origin, the place that motion begins. Time is oriented toward its destination, the time that motion ends. Both length and duration are measured from an “origin,” a reference point, which is a zero point for each, but zero speed leaves a body in space at the beginning, whereas zero pace puts the body in time at its destination.

Length in the denominator of speed is a measure of the progress from the origin to the current location in space, whereas time in the denominator of pace is a measure of the lag from the destination to the current location in time. A body at zero speed will remain at its origin and never reach its destination, whereas a body at zero pace will arrive at its destination in literally no time. A body with a small speed will take a long time to reach its destination, whereas a body with a small pace will reach its destination quickly.

Large quantities in space correspond to small quantities in time. Large quantities in time correspond to small quantities in space. A high speed is fast, and a small speed is slow. A small pace is fast, and a large pace is slow. Mass and vass are inverses, as are energy and lethargy.

The origin in space corresponds to the destination in time. Time in space flows from the past toward the future. Space (stance) in time flows from the future toward the past.

# Arcloge

An arcloge (arc’-loje) is a continuous, independent measurement of length. That is, it measures an ever-increasing length, which is the stance, similar to how a clock shows the time. The term is a combination of arc (as in arc length) and loge (as in horologe, a clock).

What does an arcloge look like? Start with a sector, which is a geometric figure fixed to the center of a circle that sweeps out an angle and a curved edge:

# Basic definitions

Independent variable is a quantity that is not dependent on another quantity, which is either (a) a quantity chosen before an experiment or race, or (b) an ever-increasing quantity. Dependent variable is a quantity that whose value is a function of another variable.

Length is measured by a rigid rod. 3D length space is length measured in three directions of motion.

Duration is measured by a stopwatch, timer, or clock. 3D duration is duration measured in three directions of motion.

Length space is (1) 3D differentiable manifold of length; (2) the order of events on a stance line; (3) the stance, the reading on an arcloge.

Duration space is (1) a 3D differentiable manifold of duration; (2) the order of events on a time line; (3) the time, the reading on a clock.

Length-duration is the 6D manifold formed from 3D length space and 3D duration space. Worldline is the path in length-duration traced out by an object in motion.

Displacement is the vector between two points (events) on a worldline. Distance is the magnitude of a displacement.

Dischronment is the vector between two points (events) on a worldline. Distime is the magnitude of a dischronment.

Reference frame (or frame) is an abstract coordinate system and set of reference points in 3D length space that uniquely fix the coordinate system and standardize measurements. Rest frame of a body is the reference frame in which the body is moving at zero speed, which is the time conversion pace.

Reference timeframe (or timeframe) is an abstract coordinate system and set of reference points in 3D duration space that uniquely fix the coordinate system and standardize measurements. Freeflow frame of a body is the reference timeframe in which the body is moving at zero pace, which is the stance conversion speed.

Proper length is the length of a body measured by a rigid rod moving with it. Proper time is the time of a body measured by a clock moving with it.

Lorentz transformation is a set of equations that relate length space and duration space coordinates of reference frames moving at a constant velocity relative to each other.

Dual Lorentz transformation is a set of equations that relate length space and duration space coordinates of reference timeframes moving at a constant legerity relative to each other.