Because many (most) readers are even less savvy with geometry than I, this post is a primer for essays on the geometry of the current market. It will cover relevant metrics of two and three-dimensional structure. I will be brief so you need to refer to high school level and freshmen geometry texts (believe me, I had to retrain myself).
Metrics of the square (two dimension) and cube (three dimension). Derived by the 1X1 square ABCD, diagonal AC is the square root of the square of two sides each squared or 1.4142. AC is the square root of 2. Derived from the 1X1X1 cube, the center diagonal AE is the similar square root of one side and one diagonal or 1.7321. AE is the square root of 3.
Double squares and the root five growth expansion. Two equal cubes sharing a common planar side have important interrelated metrics and demonstrate the relation of growth spirals of Phi (1.6180) and root five (2.2360). The following is my rendition of Bradley Cowan's illustration found on page 51 of "Four Dimensional Stock Market Structures and Cycles" ("Four D" for short):
Phi and root five and mathematically and geometrically related. In fact, Mr. Cowan refers to root five as more important than Phi in the market. Moreover, analysts are frequently misled by the more readily identifiable footprints of Phi (square root of Phi, Phi squared, etc) when the more important growth spiral in the market is the root five spiral.
Adjacent cubes. One last metric for now; the center diagonal of adjacent cubes or root six. I expect this metric to occur in the 2009 to present market, but have not derived it as yet. In the ideal world, here is one way you'd derive it:
And a tool; the measurement value of a spacetime vector. As trademarked by Bradley Cowan as the "Price Time Vector" or "PTV, the PTV is an application of Pythagoras' formula for the hypotenuse of the 3-4-5 triangle. Simply:
From two data points on a chart (typically one is a high and, the other, a comparable low) calculate the change in time and square it, calculate the change in price and square it and take the square root of the two after having added them.
Simple, right...except for "squaring." That thing that WD Gann told us we needed to do to our chart paper and Brad Cowan says we need to read what WD Gann said about squaring. From my studies, Pythagoras' formulation works only as long as both time and price have the ability to influence the value.
To accomodate squaring I have two rules. First I use natural units. I use hours, days, weeks, months. Of course the price side is simple to use the decimal to "balance" time and price. Second, that "balance" word itself. Over a long period of market time, time and price need to balance... Neither should become the primary determinant in the Pythagorean calculation. As well, I try to keep the ultimate output of the formula (the PTV value) to 3 digits by, whatever the price and time metric, reducing them by 10 to make the PTV value, ideally, 3 digits.
The final comment on squaring; I cheat. Brad Cowan used a weekly calculation per dollar up to the late 1980's and thereafter, trading hours to the dollar. I go one step further and reduce trading hours by 10 and dollars by 10 to make the value a 3 or 4-digit number. The ultimate output of the Pythagorean formula is the same in either case differing by only one decimal.
Calculate the PTV value of the vector from the 2000 DJIA high versus the 2002 DJIA market bottom:
That's how its computed and, again, the squaring is trading days X 6.5 hours per day / 10 and price change in dollars /10.
So what if we find all the above numbers in the DJIA from 2000 to present? Gosh, we find Phi everyday. There are resistance and support levels that we see hit all the time. A big "so what?"
Its the structure. If follows the structure of a Platonic solid. Bradley Cowan indicates in Appendix G (hmm, does G stand for Gann) of Four D that the important growth spiral in DJIA is root five and, therefore, the Platonic solid would be the adjacent cubes.
We find Phi all the time but can we predict every instance in which we are going to find it; exactly which day, exactly which price level in reaction to a previous price level? Nope. I defy anyone to assert the inerrant and predictable Phi structure.
I'd say Brad's left something tucked away with only the remotest of hints. I recall the first time I'd read anything about Bradley Cowan; it was in an interview in which he made the comment that every angle in the market is 60*. Ultimately I came to understand we as observers see the 60* angeles differently because the market twists relative to our field of vision. If you look face-on at a 10" X 10" piece of paper it looks square. If we lay it on a table and look at it with our eyes level to the table, we see only the edge of the paper. If its square to our vision, its a 10" long line without any height. What we perceive about the third dimension, what we see, lies to us. If our vision is subjective in the third dimension based on the location of the observer, how can it be otherwise in the fourth dimension?
That 60* hint; its the equilateral triangle in the second dimension and the tetrahedron in the third dimension. The triangle and the tetrahedron is the most efficient of the Platonic solids in many regards. In two dimensions, the triangle forms a solid structure with only 3 sides. In two dimension, the four-sided square is not a solid structure until you add a diagonal to firm it up (which creates 2 triangles by the way). Ditto those observations for the tetrahedron versus the cube. Nature's smallest firm structure; the triangle and tetrahedron.
And what are the fewest number of tetrahedra that can be arranged to comprise a cube? There is one and only one decomposition of a cube into five tetrahedra, the smallest number. And there are twelve methods of decomposing the cube into six tetrahedra. At least, that's what I read HERE. Twelve and one.
Another "so what." Perhaps the cubic structure of spacetime is most evident in the cube, but the cube is comprised of either 5 or 6 tetrahedra. One edge of a tetrahedra appearing on the face of the cube, perhaps two edges of each tetrahedra, forced into the cube provide the market's etching of its history on each cube.
What I believe we will see in the next essay is the PTV of the market creates minimal elements of the sides and edges on sequential faces of the two adjacent cubes. The math confirms this as I hope I can demonstrate.
I expect the tetrahedron to be the basest structure of spacetime, five or six of which coalesce to form the cube. But for now, we need to go about proving the structure in the market itself. That will begin in the next essay related to which I've already identified most of the metrics presented above in a progression that demonstrates the higher level of structure, the cube.
In my current thinking, this is only a very early experiment in spacetime. Ultimately, Mr. Gann was able to use the unique vibration of each person, place, thing (according to Luo Clement, aka WD Gann) to describe and predict its unique future according to the structure of time.