iSoul In the beginning is reality.

# Fixed sizes and rates in space and time

A ruler is a measuring device with a fixed size. A ruler with a fixed length marked in standard linear units is a linear ruler. A linear ruler measures linear space, i.e., length or distance, which is the extent of an object or motion that is contiguous to a linear ruler. The units of linear space are metres, millimetres, kilometres, feet, miles, etc.

The common measuring rod marked in units of length is an example of a linear ruler. Other examples are measuring tapes, measuring wheels, odometers, and laser rulers.

A ruler with a fixed size marked in standard angular units is an angular ruler. An angular ruler measures angular space, which is the extent of an object or motion that is contiguous to an angular ruler. The units of angular space are radians, degrees, minutes, seconds, etc.

A common protractor is an example of an angular ruler. Other examples are bevel squares, theodolites, sextants, etc.

Since linear rulers are more common than angular rulers, the unqualified term ruler defaults to a linear ruler unless specified otherwise.

A clock is a measuring device or method with a fixed rate of motion. A clock with a fixed rate of angular motion (or rotation) marked in standard angular units is an angular clock. An angular clock measures angular time, which is the extent of an event or motion that is simultaneous with the motion of an angular clock.

The common circular clock with hands that indicate the angular time is an example of an angular clock. The motion of the sun across the sky functions as an angular clock, or its shadow may be marked with a sundial. Other examples are angular clocks based on oscillations of pendulums, crystals, or electronic circuits.

A clock with a fixed rate of linear motion marked in standard linear units is a linear clock. A linear clock measures linear time, which is the extent of an event or motion that is simultaneous with the motion of a linear clock. The linear time may also be indicated by a (possibly imaginary) device that represents the typical amount of linear time.

The location of a regularly scheduled train can be used to measure linear time. It is not unusual for people to speak prospectively of a journey or a drive in terms of the time taken by a typical traveler, which is an example of an imaginary linear clock with a typical rate of travel. The speed of light functions in astronomy as a linear clock with units in light-years.

The units of angular and linear time, i.e., duration, are the same: seconds (the SI base unit), minutes, hours, days, years, etc. Since angular clocks are far more common than linear clocks, the unqualified term clock defaults to an angular clock unless specified otherwise.

Angular rulers and clocks don’t go anywhere, except in circles, as they are only magnitudes, which are one-dimensional. Linear rulers and clocks do go somewhere, and can do so in three dimensions. As there are three dimensions of linear motion, so there are three dimensions of linear rulers and clocks, that is, of linear space and time.

Which dimensions are observed depends on whether one uses angular or linear measures for space and time. Three dimensional linear space and angular time go together, as do three dimensional linear time and angular space.