If point E is determinable, then there must be math. It doesn't start with proving point E. We already have completed two dimensional formations that can be evaluated mathematically to see if they follow the mathematic laws of the 1X1 square and the two adjacent 1X1 squares. We have those formations in the current market; square ABCD and twin squares forming the rectangle ABGH:

ABCD doesn't look like a square and ABGH doesn't look like a rectangle comprised of two such squares. That's because the proposition is the structures are four dimensional solids being viewed on a two dimensional chart. Or, that's how the story goes. Supposedly, the math will tell us the truth where our vision is unable to sort out the truth.

A very limited objective of this essay; is ABCD a square and is ABGH a rectangle comprise of twin adjacent squares. If the diagonals of ABCD is not root 2 (1.4142) of the side and/or if the diagonals of ABGH is not root 5 (2.2361) of the side, then further inquiry is, likely, unwarranted.

The above insert is drawn using software that computes the Pythagorean Price Time Vector ("PTV") value as described in previous essays. Its calculations are dependent on how closely a visual fit of the vectors can be made. We need more accurate measurements. Following are the measurements:

The black bolded solid lines are "real vectors" On either end, they have points of price and time that have occurred. The grey lines have at least one end point that has not occurred in price and time; they are imagined and not a subject of this essay.

*. Analysts tout their great insights by pointing out, "aha, PTV CD divided by PTV AB is within two one hundredths of the root of Phi." Or "AC divided by AB is within two one-hundredths of the number 2 and WD Gann had a thing about doubling and halving time or price." That does it, a couple isolated observations and we all think its magic...and we buy that book or course. Not that they aren't mathematically related, but they hardly create mathematic structure. On the other hand, I believe each of the ten PTVs are related to every one of the other ten. Its just that we have to go very deep into the several steps that comprise the relationships. I've done a lot of that and found several three deep relationships. Three deep where say three mathematic metrics such as root 3, root 5, Phi are required to explain the difference between two PTVs. Pass on that for now. Just realize you may see PTVs that are apparently related by a known math metric.*

__Seeming hit or miss mathematic relationships__*. I've overviewed this before with PTV values that were created by the charting software as opposed to rigorously determined PTV values. The above table removes any doubt:*

__Root 2 and the square ABCD__
The diagonals of the 1X1 square are 1.4142 (root 2) times a side of 1. That's law. If you add the four PTVs that comprise the ABCD square take their average side value and divided it by the average diagonal the result is 1.407536 or .47% less than ideal root 2.

If you divide the sum of the four side of ABCD by the sum of its diagonals the result is .47% greater than ideal root 2.

And the two calculations averaged are within .001% of of ideal root 2.

I've asked before, can that be coincidence? But it doesn't end there. There is a second real configuration that we can evaluate; the double square.

*. The "law" is that the diagonal of adjacent 1X1 squares sharing a common side will have diagonals that are root 5 times the square of one side. As the contemporary thought goes, "but its complicated."*

__Root 5, root 3 and the twin adjacent cubes__
We can do this lots of ways. Above we find the ave side of 1448.72 divided by the ave diagonal/root 5 is 1.315420. Or we can divide the average side by the ave diagonal to yield .5882739 and multiply by root 5 to again arrive at 1.315420.

But 1.315420 is not the root what we expected. We'd already buried the root 5. We should have gotten the number 1, not 1.315420.

No, that's where it is complicated. We have the fourth root of 3. Recall the primer essay on geometric solids; 3 is the center diagonal of the three dimensional 1X1X1 cube. Square 1.315420 and square it again and you have the number 3. If you leave the calculation at root 3, you find that it is within .001% of ideal root 3.

Is that coincidence?

__. With all this in mind, take another look at the market chart. We have a two-dimensional square near to and facing but it does not "look" like a square. Nevertheless, the math demonstrates the hallmark mathematic property of the square; its root 2 diagonals. Not just close...__

**Morphing two dimensions into an allusion of three dimensions***.*

**but to 5 decimals**
Now look at how the visibly appearing further away, twisting and receding rectangle ABGH morphs from vertical side AB to vertical side GH that now appears far away from us. Over that distance, both the two dimensional metric root 5 and the three dimensional root 3 have interceded to create the vector GH; a vastly diminished reflection of its opposing edge AB.

__. Its very unsatisfying that vector AC divided by AB is 2.02132. We want it to be a perfect 2.00000 to demonstrate the doubling and halving of time, the innermost working of the root 2 growth spiral. But it isn't. Probably every one of the vector relations is imperfect even to the point we doubt they are what they are.__

**Evolving mathematic metrics**
But when we aggregate them and look for the expected value, the metric we seek becomes more accurate. To wit, the ave side of ABCD divided by the ave diagonal, we get close to ideal root 2. Then, if we reverse the process with the gross sides divided by the gross diagonals, we get another measurement that's close to root 2 but in the opposite direction. Still further, when we average the two measurements, we are off from root 2 by less than 4 significant digits.

Yet one step further. The 4 significant digit error in the root 2 measurement of the ABCD square seems to be offsetting to the 4 significant digit error of the twin adjacent squares of ABGH.

Which leads to the question previously posed...as the formation matures, do measurements increasingly approach their ideal expectations? As if a larger vibration, set in motion, overshoots its mark but by decreasing margins until, ultimately, it is perfected.

***

The only two geometric images of the three dimensional twin adjacent cubes are conclusively supported by the level of exact mathematic structure that,

*, we expect to manifest. Its not exactly as we expect because the individual PTVs are wishy washy. But their averages are not. And we have the unexpected intrusion of the three-dimensional root 3 into the mix.***by law**
But what do you expect. We are trying to explain a four dimensional phenomenon, namely price/time, using a two-dimensional chart.

We do not know the structure of four dimensional price/time, or at least, I don't. We can't see it or feel it or smell it. We can only imagine it. But that's what we have. We have the clues to imagine it.

Socrates' allegory of the cave and the prisoners having never been in the light see images on the wall; two-dimensional images. One prisoner leaves the cave to discover the images seemingly having only length and width have depth as well.

Painfully, C Howard Hinton in "The Fourth Dimension" (1906) develops how a being am to sense only the planar existence (length and width) can notice anomalies in his environment that leads him to conclude the existence and certain properties of the fourth dimension. Bradley Cowan relates a similar scenario in which a planar being is confronted with a vertical wheel of five colors. Of course, he can't see the tall wheel that is before him; all he can see is a color. As the wheel turns he sees the colors flash in sequence before him. After many turns of the wheel he realizes the sequence of colors repeats. He has a clue. He can't see the wheel but he knows the structure of sequence that presents itself time and again. Exact time durations and exact sequence. Its a clue.

So, its settled, at least in my mind. The first and only two complete geometric structures in the price/time chart of DJIA from 2000 to 2017 articulate the most important properties of planar geometry to a precision that's impossible to claim as coincidence. And they introduce the third dimension as well. But most telling is their measurement is according to a yardstick comprised of both time and price according Pythagoras' 345 triangle and the Price Time Vector.

Now, back to evaluating point E. Once point E and F are projected, the entirety of the cube will be known. Perhaps point E has occurred or is yet in the imminent future. Point F certainly lies ahead.

Jim Ross