Phi, the Divine Ratio

Discussion concerning the first major re-evaluation of Dewey B. Larson's Reciprocal System of theory, updated to include counterspace (Etheric spaces), projective geometry, and the non-local aspects of time/space.
bperet
Posts: 1484
Joined: Thu Jul 22, 2004 1:43 am
Location: 7.5.3.84.70.24.606
Contact:

Phi, the Divine Ratio

One of the problems of using a mixed geometry is that "units" aren't the same length, since you have a distance in one aspect and an angle in the other. Using imaginary numbers, that angle can be converted to an imaginary distance, so we can get a ratio of the conversion factors involved. It is a similar situation to the ancient Alchemical "squaring the circle" problem, but in our case, we'll be "circling the square".

Start with a unit area square on the complex plane--an observable unit area, since we are dealing with observed measurements between space and time. In the observable, spatial system, there are NO negative distances. We cannot have a ruler that is -5 inches long. BUT, in the imaginary realm, we CAN have both positive and negative distances, as indicated by CW and CCW rotation or the inductive/capacitive relations in electronics. It makes sense since time does not have direction in space, so space does not have direction in time... one dimension will show up as a bi-vector across the imaginary axis.

In order to plot such a square, we can use a unit distance along the real, spatial axis of 1 unit, extending from 0 to 1. Since time is bi-directional, we must split the imaginary length into two '1/2' lengths, one up towards +i, the other down towards -i, making a total distance of 1 unit, giving us a unit square. (This 1/2 distribution of a unit shows up a lot in the RS... neutrinos, 1/2-1/2-(1) and the specific rotation in Basic Properties of Matter.

To find our polar region, we simply circumscribe the square. Since the real axis runs from 0 to 1 (no negative), the circumscribed circle has a radius of sqrt(12+0.52) = sqrt(5)/2.

To determine the "imaginary" length to use in the ratio, we must find the distance from that circumscribed circle that runs along the imaginary side of our unit square. We can easily get this by using the i axis; the radius of the circle is already known and that covers the distance from the circumscribed circle to the center, half way down the side of the unit square. To get the remainder of the distance, we just need to add the extra 1/2 bit of length that is sticking down the -i axis, giving:

length = sqrt(5)/2 + 1/2 = (1 + sqrt(5)) / 2 = φ (Phi)

Our ratio of real-rectangular to imaginary-polar is then: 1 : φ ... the "golden" or "divine" ratio, 1.618034.
phi.gif (3.47 KiB) Viewed 1148 times
Attachments
RS2 geometry of Phi, the Divine Ratio
RS2 Phi Golden Ratio.png (43.98 KiB) Viewed 1497 times
Every dogma has its day...