The Larson Research Center

Evidently, some are confused as to how one, 1D, extension/compression of a weighted spring, can be understood as a 2D rotation, as asserted in the previous posts below.  No matter how you look at the two, inverse, reference systems, they say, there’s still no way to get the two dimensions required for rotation, with only one, 1D, vibration.

Well, that’s true and that’s why what I’m pointing out has never been recognized before, because, when the 1D motion of the springs constitutes the SHM (given the proper phase relationship), it takes two, orthogonal, 1D spring vibrations, in the discrete reference system, to define a single rotation, in terms of the sine and cosine functions of the angle of rotation.  This is shown in Figure 1 of the previous post.

However, when the same two vibrations are viewed in the continuous reference system, the change from the fully compressed state of the springs to the fully extended state, and the inverse, from the fully extended state to the fully compressed state, becomes a double system of scalar SHM, defined by change in two, reciprocal states, instead of a single system of vector SHM, defined by two sets of reciprocal change of positions.

Two, orthogonal, 1D vibrations, as sines and cosines, define four lengths, as four radii of space in the unit circle, viewed in the discrete reference system, as the positive and negative positions of the orthogonal x, y coordinates. On the other hand, these familiar x, y positions of the 2D coordinate system cannot be defined in the continuous reference system. In the continuous system, the positions from 0 to 1, and the inverse positions from -0 to 1, represent something else.

In the continuous system, there are limits between -0 and 1, and between 0 and 1, which don’t exist in the discrete system.  In the discrete system, 0 is the point of “direction” reversal, and it’s the only such point, but in the continuous system, there are two points of “direction” reversal, 0 and 1 in the positive unit, and -0 and 1 in the negative unit.  There is no “direction” reversal at 1 in the continuous system.  In other words, there is no cross over point at the “origin” of the continuous reference system, from positive to negative, or from negative to positive, as there is in the discrete reference system.  This is a very significant difference between the two, reciprocal, reference systems.

In the springs, this change from the discrete reference system to the continuous reference system can be described as a change in the interpretation of the spring’s compression|tension property. If the compression|tension force in the spring changes “direction” at the equilibrium point, it provides the necessary restoring force that produces the oscillations of the weights attached to the springs, the motion of which has naturally been interpreted in terms of the discrete reference system, thereby coupling the SHM of two, 1D, vibrations to rotation, through the sine and cosine functions of the angle, as the position of the weight changes. 

But in reality, this is accomplished through the interaction of the spring’s compression|tension property, with the momentum of the attached weights. The interaction is just a physical mechanism for producing the oscillations of SHM. Nevertheless, in terms of numbers, the change in one spring is from -1 (fully compressed) to 0 compression|tension, and from 0 tension|compression to 1 (fully tensioned).  This is the same relation of space and time displacements that exists in the RST.

Equivalently, however, this same range of motion, from maximum compression, to no compression, at the relaxed, or “equilibrium” position, and from no tension to the maximum tension (fully stretched) position, can be seen as one range, bound by two limits (0 and 1). In this view of the magnitudes involved, the “direction” reversals actually take place at the limit of the fully extended point, and at the limit of the fully compressed point, not at the mid point of equilibrium, if the factor of a restoring force’s interaction with momentum is not relevant.

We can see this clearly, if we place the spring between our hands and squeeze it, until it’s fully compressed, and then slowly release it so it can fully extend again. If we then pull its ends apart, we can stretch it to its maximum and then slowly release it so it can once again return to its relaxed position. As far as the numbers alone are concerned, the rate at which we compress, or stretch the spring is immaterial.  Starting from the fully compressed state and stretching the spring to its limit, constitutes a single change of state from -0 to 1. The reciprocal of this state is formed by starting from the fully stretched state and compressing the spring to its limit, from +0 to 1.

When we realize this, then it’s obvious that there is a correlation between the reciprocal states of the compressed, and tensioned, spring and the reciprocal phases of rotation; that is, compressing the relaxed spring is equivalent to 90 degrees of rotation, in a single dimension, from 0 to 90 degrees, and stretching it is equivalent to another 90 degrees of rotation, in that same dimension, but in the opposite “direction” from 90 to 180 degrees, so stretching it from the fully compressed state, to the fully tensioned state, is equivalent to 180 degrees of rotation, in a single dimension, regardless of the changing “direction” of the compression|tension force at the mid point. Conversely, compressing it from the fully tensioned state, to the fully compressed state, is equivalent to 180 degrees of rotation, in a single dimension, in the opposite “direction,” regardless of the changing “direction” of the tension|compression force at the midpoint.

In this case, the fully compressed and the fully tensioned states of the spring, become the limits of the system, which represent one full unit, between them, which is divisible into a continuous spectrum of magnitudes of a given size, where these are the magnitudes of the continuous group, of a given order, that lie within the bounds of a single unit.

What we have done is combine the two, discrete, units of the two opposed radii, described by the sine, or cosine, of the angle of rotation, into a single diameter of the circle, to form one continuous unit, from two discrete units. The question now is, how can we couple this single diameter to a 2D rotation?  In the discrete system, since such a coupling requires four discrete units, the positive and negative units of the sine and cosine, which are equivalent to two, orthogonal diameters, the question becomes, “Don’t we need two diameters in the continuous system as well?”  

The surprising answer to this question is no, we don’t.  The reason why we don’t, is because, in changing from the discrete reference system, to the continuous reference system, the vectorial motion of the spring, from one point to another is no longer the defining property.  It’s not the changing position of a weight, attached to the spring that is represented by the change of compression|tension, but the ratio of the spring’s compression|tension itself, a scalar value.

At the point of equilibrium, the compression|tension ratio is 2|2 = 1|1 = 0, because, from this point, the two units, out to the fully compressed limit and back, and the two units out to the fully tensioned limit and back, are two potential, reciprocal, units. When the spring is fully compressed, the potential ratio is altered, from 2|2 = 0 to 1|2 = -1, and when it is fully tensioned, the potential ratio is altered from 2|2 to 2|1 = 1. These magnitude changes represent the discrete elements of the group under addition, the discrete group.

But now we know that the interpretation of the reciprocal operator, from the difference interpretation, to the quotient interpretation, transforms the elements of the discrete group under addition, to the elements of the continuous group under multiplication.  When we make this change in interpretation, then 1/2 = .5, 2/2 = 1/1 = 1, and 2/1 = 2, and we are now in a new reference system, the reference system contained within the limits of two, reciprocal, units.  In this system, .5 is the negative limit, defining the unit between -0 and 1, while its inverse, 2, is the positive limit, defining the reciprocal unit between 0 and 1, a fact that the chirality of the system disguises, but that the inverse product operation of the continuous group exposes, as .5 x 2 = 1, where 1 is the identity element.

Consequently, under this interpretation, the spring’s compression|tension property is the reciprocal of its tension|compression property, and, while it can take either form, it cannot take both forms simultaneously.  Hence, in order to have both a negative unit and a positive unit, we must have two springs, where, at any given moment in time, the state of one is the inverse of the state of the other, with regard to these reciprocal properties.  This is just another way of saying that, in a coupled system of scalar SHM, one spring must be compressing, while the other spring is tensioning, or, in other words, there must be a 180 degree phase difference, not a 90 degree phase difference, between the two springs. Obviously, a single spring cannot be compressing and tensioning at the same time!

Hence, since, in the continuous reference system, the relative phases of the two reciprocal springs in scalar SHM is not the 90 degrees of the discrete reference system, but the 180 degrees of the continuous system, as shown in the SUDR|TUDR graphic, in the previous post, and shown in a more compact form below, the scalar SHM of the system is derived from a double set of parallel springs, not two independent, or orthogonal, springs.

Figure 1. The Reciprocal States of the Continuous Reference System

The key here is not to think in terms of the lateral motion required to compress and tension the two springs, but rather to think in terms of the change of state in the springs themselves, from a compressed (local) state, to a tensioned (non-local) state. Such a change is a continuous change analogous to the scalar change from point to sphere (expansion), and from sphere to point (contraction), which we use to map the reciprocal oscillations of the S|T units to the reciprocal rotations of two, meshed, gears.

The relevant point here is that one cycle of scalar magnitude, from local to non-local and back, is the equivalent of 360 degrees of rotation, and, when two of these are combined, as shown in the graphic of figure 1 above, the two constitute the equivalent of  720 degrees of rotation in one cycle of oscillation, just as the counter-rotations, of two, meshed, gears constitutes 720 degrees of rotation per cycle.