With regards to Gannism;

Gannism is esoteric, not easily understood, possibly highly discretionary, and thus always viewed with suspicion.

As to the origins of the time theory element to Gannism the following caught my attention.

In 1913, a PhD was published by Bachelier in France, and demonstrated via rather complex mathematical equations that, price fluctuations grow in range and will be proportional to the square root of time

Stock prices in the United States over the last 100 years have 66% of the time fluctuated within a range of 5.9% on either side of their average.

The range in a course of a year has not been 72% or a multiple of 12 [year] rather, it has averaged around 20%

This is 3.5 times the monthly range.

The square root of 12 = 3.46

quote:

--------------------------------------------------------------------------------

But exactly just what did Mr Gann write about angles? For anyone with an original Gann course, if they will turn to the first page of the section titled The Basis of My Forecasting Method, looking at line 10 from the bottom; they will read the following quote. "There are three kinds of angles-the vertical, the horizontal, and the diagonal, which we use for measuring time and price movements." Today every usage called "Gann angles" uses "diagonal angles" only. Yet the Master says we must use all three angles-the vertical angle, the horizontal angle, the diagonal Angle. Definitely not the last and least important, the diagonal angle alone.

--------------------------------------------------------------------------------

The above caught my eye;

Intrestingly, this is another application of higher mathematics, referring to chaos

This is chaos in the mathematical definition, not your standard day-to-day useage.

In it's simplest form chaos can be written;

4x{1 - x}

Computing the value of *x* of that expression for some initial value of *x* then substituting this answer back into the original expression starts a feedback loop.

Repeating this simple iterative process repetitively produces surprisingly complex, unpredictable mathematical behaviour.

The mathematical behaviour expresses the same kind of disorder produced by non-linear equations

The simplest non-linear equation;

Xn+1 = KXn - KXn(1 - Xn)

This equation determines the future value of the variable x at the time step n + 1 from the past value of x at time step n

This is known as the logistic equation

All well and good, but, what the hell is this to do with the Gannies?

Logistic equations are used in Medicine to predict population expansion, via Birth rates, Death rates, due in part to availability of food, water, arable land, disease etc.

It can also be used in ecology, for populations of insects, crops, etc.

Gann was interested in commodities.

Wait, there's more.

The logistic equation is a quadratic equation with a linear first term, and, a non-linear second term

It is the non-linear, or feedback component that is important.

For a given value of K once a starting point Xo is specified, the evolution of the system is fully determined. One step, inexorably leads to the next.

The whole process can be pictured on a graph.

It forms a parabola, that opens downwards.

There is a short-cut provided via the graphical representation, that avoids endless computations.

Re-read the quote at the start of this post;

The addition of a 45 degree line up from the horizontal axis [representing the line Xn+1 = Xn]

The best course is to steer is from Xo vertically to the parabola to reach X1 then horizontally to the 45 degree line, and vertically back to the parabola.

These paths or Orbits give the first indication of which routes lead to the erratic behaviour of chaos

Whereas some orbits converge, on one particular value, others jump back and forth among a few possible values, and many roam, never settling anywhere.

When K is between 1 & 3, just about every route no matter where it starts, is eventually attracted to a specific value called a fixed point which occurs where the parabola intersects the 45 degree line at x = [k - 1]/k This corresponds to to a steady state or equilibrium

Therefore, taking the previous mathematical work performed by Bachelier, combined with a logistic equation, and you can reproduce seemingly Gann.

The question is, historically, who, and at what date, was the initial work completed in logistic equations?

How did Astrology get involved?

Mathematicians have always historically been associated with planetary movements orbits etc.