The Larson Research Center

After several telephone conversations have convinced me, I’ve decided to redraw the graphic of the previous post.  In the previous graphic I didn’t have room for two springs of the discrete group, those that are mapped to the sine and cosine, but I thought people would not need two of those.  However, it’s much clearer, if both are included, so I’ve made two graphics, one for the sine/cosine interpretation, which uses two discrete groups, and a separate one for the two continuous groups.

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Figure 1. The Discrete Interpretation of Vibration

Figure 2. The Continuous Interpretation of Vibration  

It should be clear that the relative phases are the same in both cases; that is, in the reference systems of both groups, 1, and -1, is the inverse of 0, or -0.  While the concept of -0 may seem strange to those not familiar with the RST, it’s an elementary concept in the new system.

The difference that the new found groups make, one under addition, the other under multiplication, is the two, inverse, reference systems they constitute.  When the discrete reference system is used, the units are displaced from unity, so 0 is interpreted as zero displacement, to the left (negative) or to the right (positive) of unity.  When the continuous reference system is used, however, units are displaced from 0, and there is no left of negative 0, or right of positive 0 (even though these “directions” were invented by mathematicians, but that’s another part of the story), so 0 should be interpreted as 0 parts of a whole (i.e. “positive” and “negative” parts don’t make sense in the continuous reference system.)

In the SHM of the springs, the point of equilibrium is naturally interpreted as a 0 point of the discrete reference system (its identity element), where either positive or negative displacement is possible, because this kind of reference system is necessary to identify the SHM of the two springs, with the two functions of the angle of a single rotation.  However, we can now see that a point of equilibrium only applies to a system with a restoring force, supplied by the spring tension in this case, and by gravity, in the case of a pendulum.

But consider the case of two, counter, rotations. In this instance, no restoring force is involved. One complete rotation, from 0 to 1 and back to 0, takes place in the continuous direction of rotation, with no reversals, and if there are two such rotations, rotating in opposite directions, the inverse phase relationship constitutes SHM just as surely as it does in the discrete reference system, but it does so as a double rotation, not a single rotation.

Therefore, we conclude that SHM comes in two, reciprocal, forms.  In the discrete form, 0 constitutes the origin of a single reference system, and in the inverse of this form, 0 constitutes the two limits of a double reference system. The reciprocal nature of the two reference systems is illustrated in the graphic below.

Figure 3. Inverse Reference Systems of Discrete (top) and Continuous (bottom) Groups.

Larson was the first to identify these two reference systems and to apply them to the development of RST-based physical theory, but he did so without the benefit of the knowledge of their respective mathematical groups, and the tremendous power they afford the developer. With this much understood, we are prepared to move forward like never before. 

Update, Jul 23:  I’ve updated the graphics to show the correct phases and edited out the text explaining the incorrect phases.