The Larson Research Center

Arguably, the two most challenging mathematical/philosophical problems for the Greeks were manifest in the attempt to square the circle and to accept the existence of irrational numbers. In modern times, we’ve proven that the former is impossible, and the latter is actually quite useful. However, as discussed in the previous post, it is possible that there are other approaches to meeting these formidable challenges, unknown to us, which might even prove more useful than our current method of handling them.

The crucial analysis of the fundamentals that seems to provide us with the clues that this might be so starts with Larson’s idea of scalar motion. As regular visitors of the LRC site know, scalar motion is a definition of motion without reference to moving objects. The equation of motion, v =ds/dt, simply involves a change in space over time, and a changing location of an object is not required to produce the equation’s change of space, just as it is not required to produce its change of time.

In Larson’s system, the initial condition of the universe assumes a natural space clock as well as a natural time clock, the one being the inverse, or reciprocal, of the other. Hence, this assumption defines a universal motion, as the physical datum of the system.  There are several important differences between the new natural type of motion, with no motion of an object involved, and the motion of objects with which we are familiar. One of the most basic differences is that the familiar motion of an object Y, from point X to point Z, increases the distance XY and decreases the distance YZ. On the other hand, the new natural type of motion changes distance itself; that is, both the distance XY and YZ are increased, or decreased, at the same time, making it impossible to define the motion of the object Y, in terms of the changing distance relative to X and Z, with one increasing and the other decreasing. It’s as if the size scale of the system were changing.

This expansion/contraction motion, though easily observed in nature, is quite unlike the motion of an object from one point to another, specified in some specific direction that can be defined in terms of three dimensions. In a 3D system, scalar motion would change the size of a spatial location in all three dimensions simultaneously. This makes scalar motion more difficult to work with in some respects, because the system’s locations (x, y, z), regardless of size, must continuously expand. While at first this is very disconcerting, it turns out that there are ways to cope with it that are straightforward.

Consider a 1D scalar expansion for instance, disregarding the expansion of the points themselves momentarily, the distance between points A and B increases over time. We can choose location A as a reference and measure the expansion in terms of B’s motion away from A, or we can choose B as the reference and measure the expansion in terms of A’s motion away from B, in the opposite direction. Either way, we can conclude that each dimension of scalar motion has two, opposed, directions. In a 1D system there are two scalar directions, in a 2D system there are four scalar directions, and in a 3D system there are eight scalar directions. 

Assigning numbers to the binary directions in each dimension, we get 2⁰ = 1 direction, in the zero-dimensional system (more on this exception below), 2¹ = 2 directions, in the one-dimensional system, 2² = 4 directions, in the two-dimensional system, and 2³ = 8 directions, in the three-dimensional system. Substituting these numbers in the equation of motion, we would get:

ds/dt = d2⁰/d2⁰, for zero-dimensional motion,

ds/dt = d2¹/d2¹, for one-dimensional motion,

ds/dt = d2²/d2², for two-dimensional motion,

ds/dt = d2³/d2³, for three-dimensional motion,

However, as we observe time, it’s clear that it has only one direction, called the “arrow of time,” which is increasing magnitude only; that is, a point in time has no direction, and therefore no extent, in space. On this basis, we can consider time as a zero-dimensional scalar, something that can be counted, but not expanded. Meanwhile, it’s clear that the space that we occupy is three-dimensional; that is, it extends into three dimensions, and, since scalar motion has no specifiable direction, by definition (i.e. it is motion with magnitude only), the expansion of space must be effective in all of the dimensions of the system (i.e. space is a pseudoscalar). Modifying the equation of scalar motion accordingly, we get

ds/dt = d2³/d2⁰,

where space, s, has 2³ = 8 directions, and time, t, has 2⁰ = 1 direction, the scalar “direction” of increasing magnitude only. By defining space and time this way, as the reciprocals of each other, in the equation of motion, the quantity space is differentiated from the quantity distance, which becomes the product of motion and time, as in the ordinary vectorial motion (i.e. motion with direction defined by locations with three dimensions). However, in this case, using the scalar motion equation, distance, d, is a three-dimensional quantity, not a one-dimensional quantity:

 d = Δs³/Δt⁰ * t⁰
    = (n2)³/(n2⁰) * n2⁰ 
    = (8*1³)/(1*1)
    = 8*1³

for each unit of change, n. For example, for two n, we get

d = ((2*2)³/(2*2⁰) * (2*2⁰) = (64*1³/2) * 2 = 64*1³,

or 64 cubic units of volume expansion in two units of time. The expansion series, or “distance” d, as time, t, marches on then is not the familiar linear series of lengths 1¹, 2¹, 3¹, 4¹, …n¹, but the less familiar, non-linear, series of volumes, 8³, 64³, 216³, 512³, …n³.

Geometrically, the first term in this expansion series corresponds to the initial 2x2x2 stack of one-unit cubes, dubbed Larson’s Cube, at the LRC. It is shown in figure 1 below.

Figure 1. Larson’s Cube as the 8 Unit Stack of One-Unit Cubes. 

The red dot in the center corresponds to the 2⁰ = 1, dimensionless, time magnitude, while the stack of eight 3D cubes corresponds to the 2³ = 8 * 1³ space magnitudes, at t₁ - t₀ = 1. Expanding in the next unit of time, at t₂ - t₀ = 2, to two units of space in all directions, it’s easy to see that the stack of one unit cubes, consisting of of 2x2x2 = 8, one-unit, cubes, in figure 1, expands to a 4x4x4 = 64 stack of one-unit cubes. In the third unit of time, the stack expands to a 6x6x6 = 216 units, then to a 8x8x8 = 512 units, and so on, ad infinitum. Meanwhile, the 2⁰ point at the intersection of the cubes, does not expand.

However, this mathematical expansion of the pseudoscalar does not correspond to a physical expansion, because a physical expansion of the pseudoscalar must expand in all directions, defined by three dimensions, not just the three orthogonal directions that constitute its three dimensions. Thus, the physical expansion is manifested as an expanding sphere, not as an expanding cube, and this presents us with the fundamental challenge faced by the Greeks: “How do we calculate the volume of the sphere that corresponds to the volume of the stack of one-unit cubes?” In other words, we need a geometric algebra of quantities that includes the areas of circles and the volume of spheres, as well as the linear extent of right lines, an algebra, which corresponds to a fully functional, non-pathological, numeric algebra, for doing physical calculations in a scalar/pseudoscalar system. In other words, it’s back to the old conundrum of squaring the circle.

Unlike the Greeks, however, we now know that multiplying the sides of a polygon inside the sphere will always result in an approximation, and thus it can’t be represented by a rational number. Since in our universe of discrete motion, as in the Pythagorean universe of discrete numbers, all is number, this is hardly welcome news.

Nevertheless, as we consider the problem, we see that there are two spheres that can be related to the stack of one-unit cubes. One sphere that can be drawn to fit just inside the stack, and the other that can be drawn to just contain the stack. A two-dimensional view of the one-unit instance of these three figures is shown below.

Figure 2. Two-Dimensional View of 2x2x2 Stack of One-Unit Cubes with Inner and Outer Spheres  

In figure 2, the radius, c, of the outer sphere, S₁, is the square root of 2, by the Pythagorean theorem, while the radius, d, of the inner sphere, S₂, is 1, since the radius is r = a = b = 1. By the formula for the area of the surface of a sphere,

A = 4π * r²,

the area of the surface of the sphere S₁ is 8π, while the area of the surface of the sphere S₂ is 4π. Also, by the formula for the volume of a sphere,

V = 4/3π * r³,

the volume of the sphere S₁ is the square root of 2, cubed, times the volume of S₂, which is just 4/3π, since its radius is 1.

Table 1 shows the tabulated circumferences (2*r*π), areas and volumes for spheres S₁ and S₂, and their ratios, for units 1, 2, 3 and 4.

Table 1. Circumferences, Areas and Volumes for Units, 1, 2, 3 and 4

Notice that the S₁/S₂ ratio is just a power of the radius of S₁, or a power of the square root of 2, in each case, denoted “rⁿ” in the last column of the table. The ratio of the surface areas of the spheres is the square of r, or 2, while the ratio of the volumes of the spheres is twice the radius of S₁, which is equivalent to the square root of 2, or r, cubed.

This is an amazing fact that we should be able to exploit in order to replace the 20, 2¹, 2², 2³, numerical units that are so hard to reconcile in a non-pathological, multi-dimensional, algebra.

Recall that, currently, for one-dimensional units, we resort to complex numbers (z = a+bi), the algebra of which is not ordered; For two-dimensional units, we resort to quaternions, the algebra of which is not ordered or commutative, and, for three-dimensional units, we resort to octonions, the algebra of which is not ordered, commutative, or associative!

All of these traditional units depend on one or more imaginary numbers to define their dimensionality, arbitrarily defined as the square root of -1, in the different dimensions of the respective algebras. Of course, in reality, there is no unit that can be physically identified that, when multiplied by itself, is equal to -1, in any dimension.

However, we should remember that the purpose of using the plus and minus signs is only to differentiate between a given dimension’s two “directions.” There’s nothing meaningful about them otherwise. As already noted above, in scalar motion, the choice of a fixed reference (point A or B), with which to measure scalar change, is completely arbitrary.

The same thing is true with numbers. Each number has its inverse and the designation as to which is the number and which is the inverse number is completely arbitrary. Nevertheless, with the number 1, we say that it is its own inverse, and we use this convention to build group theory, where 1 is the identity element.

However, if we could change our number system, from one based on multi-dimensional numbers, using imaginary numbers to define their dimensions, and plus and minus labels to define the two directions of each of their dimensions, to one based on the properties of spheres (i.e. 1D circumferences, 2D surfaces and 3D volumes), the inverse of 1 would no longer have to be itself, but would now be 2, the inverse of 2 would be 4, etc, by the formula for inverse geometry, r’² = r * r’’.

In this way, negative numbers are eliminated conceptually, although the change is actually only one of perspective. It’s like saying that the inverse of -1 is 2 units above it; the inverse of -2 is 4 units above it; the inverse of -3 is 6 units above it, etc. In this case, however, the unit referred to is the square root of 2, r, which is not imaginary, but is the relation between unit dimensions, defining the radius of a sphere.

Just like in the traditional mathematics, the new unit, r, defines the identity element of a group. Figure 3 shows the number 1 of the group, P, the group identity element, P’ (equal to the square root of 2), and the inverse of number 1, p’’ (equal to P’ squared, or 2).

Figure 3. The number 1 of the group (P), the identity element (P’), and the inverse of number 1 (P’’).

In figure 3, P is the radius (1) of the inner sphere, the generator of one 1d (circumference) quantity, one 2d (surface) quantity and one 3d (volume) quantity. Radius P’ generates the 1d, 2d and 3d quantities of the identity element (square root of 2), while P” is the radius (2), or the inverse of radius P (by P’² = P * P’’), the generator of its 1d, 2d and 3d quantities.

This is no different than the number line, where -1 is one unit removed from 0 and two units removed from +1. The difference is huge, though, because we can represent all three of the dimensional numbers with one radius, and do away for the need of imaginary numbers (C = circumference; A = area; V = volume for the given dimension’s P, P’ and P” quantities):

1) 1D: C_Sp = 2π (i.e. -1); C_Sp’ = 2π*r (i.e. 0); C_Sp” = 4π (i.e. +1)

2) 2D: A_Sp = 4π (i.e. -1²); A_Sp’ = 4π*r²  (i.e. 0); A_Sp” = 16π (i.e. +1²)

3) 3D: V_Sp = (4/3)π (i.e. -1³); V_Sp’ = (4/3) π*r³ (i.e. 0); V_Sp” = (32/3)π (i.e. +1³)

The fact that each successive dimension has it’s own “zero” quantity, or identity element, might take some getting used to, but it would be well worth it, if it enables us to get rid of imaginary numbers and the pathology of higher-dimensional algebras.

In that case, we would have an algebra of 0D scalars, an algebra of 1D pseudoscalars, an algebra of 2D pseudoscalars, and an algebra of 3D pseudoscalars, each one with all three algebraic properties of order, commutativity and associativity.

We’ll see.