To understand how basic this concept of scale is, consider specifying a point in space.  We need a reference system for this, traditionally thought of as a three-dimensional set of axes.  One axis exists for each dimension and locating a point in space requires one measurement made along each axis.  Our current model is that three axes (& three measurements) are all we need to locate a point in space (see Figure 1: from Wikipedia Wikipedia.org:3-dimensional).

Figure 1: Three-dimensional Cartesian coordinate system with the x-axis pointing towards the observer.

One example of specifying a location in our Direct Sensory World is to use the lay-out of a city giving the intersection of two streets and the floor above street-level: say the corner of South and 34^th streets and on the second floor.  As the streets intersect at a corner, we are provided two practical axes along the ground.  The third axis is the floor above the ground.  So we have South Street, 34^th Street, and the second floor as the three measurements.  Such a system is useful for specifying a room or the location of a person, all objects of essentially the same human scale.  However if we expand beyond our human scale, these measurements will be insufficient if we are specifying the position of a molecule that is part of the surface of a pen sitting on a table in this second floor room at the corner of South and 34^th streets (see Figure 2).  Our current 3-D model requires that three dimensions are sufficient to define the location of any object in reality.  However all measurements for locating an object in space inherently include the scale of our 3-D reference system.

The scale of our measurements is an implicit specification along the continuum of scale.  Because we normally are locating objects at the same scale, we use that same scale for all three traditional measurements to locate an object.  This implicit use of the same scale for our measurements obscures the inclusion of a specific scale as part of our means of locating an object in space.  This implicit specification of scale also means all our usual measurements of position already include this fourth measure of location.  For most objects, there is nothing to change of our current measurements – they are already four-dimensional (4-D) measurements.  We just have not made explicit the location of scale with our measurements.  This does mean we require four measurements and four axes, which also means the appropriate geometric model of our ‘normal’ space is four-dimensional.  This dimension is not hidden or beyond what we already can ‘see’ with technology.  It is simply implicit and not accounted for in a 3-D human-centric model of space.  Most of our human-centric experiences do not extend far from the scale of our bodies, so we don’t need to account for this dimension in our normal day-to-day activities.

Figure 2:  Table in room on 2^nd floor of 34^th and South Street.  Molecule of pen on Table.

Let us expand our example of specifying a location in space to explicitly include scale in the location.  We can do this by considering how we might specify the position of a molecule that is part of the surface of a pen sitting on a table in the second floor room at the corner of South and 34^th streets (see Figure 2).  In this figure, showing the position of the molecule is a series of expansions, or magnifications, starting with the figure of the table.  These expansions or magnifications occur along the continuum of scale.  And since scale is acknowledged to be a physical ‘continuum’, it is an axis of reality along which we can build out these successive expansions or magnifications.  Placing these expansions along an axis, our model might look something like Figure 3.

Figure 3:  Molecule of pen on Table in room on 2^nd floor of 34^th and South Street – Scale View.

In this figure we provide a visual means of projecting all four dimensions onto the two-dimensional surface of the paper.  In so doing we open the possibility of measuring a ‘scale-distance’ between objects of different scales – say between the table and the molecule.  In an appropriate model of space, we should be able to measure the distance between any two objects.  Using a 3-D model, how could we measure the distance between the book on the table and the molecule on the surface of the pen?