When I asked you to multiply by 45, in essence you were also multiplying your answer by 5 and then by 9 and in the case of the 18
11.25 divided by 2 is 5.625=SDV 9
We recognize the last line as Gann's division of the circle by 64. We can also see that it is divided by 2 to the sixth power as 2x2x2x2x2x2=64.
You can continue to divide by 2 but the result is always the same. The answer will have a single digit value of 9!
C C ha h a pt p te e r r 4 4 -P - P AT A TT T ER E R N N R Re e co c og gn ni it ti io o n n
A nice thing about SDV is that it allows for PATTERN
recognition. The triangle, square, pentagon, hexagon and octagon
numbers or any other numbers built on the same PATTERNS will show up as certain PATTERNS in the SDV system.
I showed you earlier that using the simple counting numbers 1 through 9 would have the same PATTERN going from 10 to 18. I also showed you how the squares had their own PATTERN.
Even the numbers that make the squares (I showed the
significance of those numbers in my book "On the Square") have their own PATTERN that never goes beyond 9.
The odd numbers which form the squares when added are:
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 and they have the SDV PATTERN 1, 3, 5, 7, 9, 2, 4, 6, 8, 1, 3, 5, 7, 9
We can see that when the odd numbers reach 17, the ninth term, the PATTERN starts over at 19. Again, as Gann said, you cannot go beyond nine without repeating, especially if the numbers have a PATTERN in the first place. The odd numbers do have their own PATTERN. They are all two units apart.
The making of pentagons, hexagon, octagons and even 33-sided figures can be checked in the same way. I will not go into their construction as I did that in my book "The Triangular Numbers."
Suffice it to say that the results are always the same. But here is a little exercise for you. Do the cubes. That's right, multiply the cubes out from 1 through 9. I think that after the first six you will already have the idea.
Remember, always reduce your answer to SDV. Just to be sure, go ahead and do the first 18 cubes.
C C ha h a pt p te e r r 5 5 -S - SD D V V a a nd n d t th he e S S qu q ua a re r e o o f f N N in i ne e C C ha h ar rt t
The PATTERNS also show up if we lay out a square whether it be built from the center or from the side.
Get out your Square of Nine chart.
Some call, including myself, the square of 33 chart the Square of Nine chart so we will use that loosely here. The square has many constructions and can be described as a cycle of 8 as Gann noted in his private papers, a cycle of 7 and even a cycle of 6. But those
arguments I have described in the book, "The Triangular Numbers," so no need to go into that here.
So, let's look at that part of the square of 33 chart which Gann called the "Square of Nine" and the 19x19 chart that totals 361. See page 112 again in the "old"course (Section 10, Master Charts, Square of Nine, page 3 in the "new" course.)
Starting with the 1 in the middle of the Square of Nine chart we go north until we reach 316 which has an SDV of 1 since 3+1+6=10=1+0=1.
And that's the last time I will reduce a number to it's SDV because by now I think you understand the method.
Now, let's go south starting from 1 until we reach:
352 which has an SDV of 1.
Going east from 1 we go out to:
334 which has an SDV of 1.
Going west from 1 we go out to:
298 which has an SDV of 1.
PATTERN?
Yes, they all have an SDV value of 1.
Now check the distance from the center to each of the points we went to, north, south, east and west.
Starting with 1 as the first term we find that each of those points were the 10th term.
PATTERN now?
Yes, like Gann said. You cannot go beyond 9 without starting over.
Let's prove it to ourselves by checking some of the other
points. The next term north of 1 is 4. Now go to the term north of the term 316 which is 391 and has an SDV value of 4.
The next term east of 1 is 6. Using 6 as the first term go out to the term next to 334 which is 411 and we find it adds to 6.
Work out the terms south and west in the same manner to prove the method to yourself.
Note that the four numbers we found earlier are bounded by a circle. Now go to the corners of the next circle, the square of 19.
Go to 307, 325, 343 and 361. When you reduce those numbers you can see that they all are SDV 1.
With those four points, plus the four points we found earlier, plus the one in the center, we have 9 points, all of which have a single digit value of 1.
If we lay out the square of 19 by 19 as Gann suggested going up 1 column to 19 and repeating until we have 19 columns ending at 361, we would also have 9 points with SDV 1 in the same positions.
Confused?
Think of the square of 19 as an overlay on the square of Nine.
1 on the square of 19 would be at 361 on the square of Nine
chart. 19 would be at 307. 361 would be at 325 (Gann notes that a new cycle starts at this number since two 45 degree angles cross here).
343 would be at 343.
For the other points 190 would be at 316. 172 would be at 352.
10 would be at 298. 352 would be at 334. And at the main center would be 181. Reduce those and they will all be 1's and they will overlay the 1's on the Square of Nine.
I have made two charts below, one is from the Square of Nine and the other from the 19x19 chart. The numbers I show are just the numbers from the 9 different points in each chart to show the idea.
307 316 325
298 1 334
361 352 343
Square of 19 on Square of 9 Chart
19 190 361
10 181 352
1 172 343
Regular Square of l9
You can turn the 19x19 overlay upside down, to the east or to the west and the results will be the same, the 1's will match up with the 1's on the Square of Nine chart.
Of course these are not the only 1's. In the square of 19x19 there are 41 ones plus 40 twos, 40 threes, etc. since 9x40 equals 360.
The other circles on the square of 33 suggests other PATTERNS, but that is a discussion for another time.
C C ha h a pt p te e r r 6 6 -D - D ia i ag go on na a ls l s i in n S S qu q ua ar re e s s
Diagonals in various squares can be studied with the SDV in
mind. You can work the PATTERNS in your mind without ever writing down the numbers with a little practice.
Let's have a look at the square of 19 again.
One of its diagonals runs from northwest to southeast or top left to bottom right. Close your eyes, give it some thought and see if you can figure the SDV numbers on this line.
If you said that all are 1's you are correct!
If you study the chart a few minute you will see why that is true. At 19 the SDV is 1. The next part of the diagonal is at 37 also a 1. We can get to 37 by adding 18 to 19 and we already know that when we add any multiple of 9 to another number we do not change its value. And by adding 18 to 19 we come to 37 since it is one less than the number at the top of the chart which is 38. Add 18 again and we are at 55, dropping one unit down again, etc. Work out the rest by adding 18 each time.
A good way to remember what you are doing is to use an "adder"
which is one less than its side. Or 19-1 is 18.
Dropping down to 11, SDV 2, we find that all the numbers going down on a 45 degree angle from 11 will all be 2's since we are adding 18 to 11 and then 18 to 29, etc.
By establishing these two rows I believe you will quickly see how the other rows can be filled in without any additional figuring.
But for the fun of it, let's figure the other diagonal, the one
that goes from southwest to northeast or from the lower left corner to the upper right and corner.
We will start with 1 and will need an adder to keep us on the diagonal. Going up to 19 and then down to 20 and then up to 21 we will be on the diagonal. So the adder is 20 since 1 plus 20 is 21.
The adder in this case is one more than the side. 19+1 is 20.
Since we can change the 20 to an SDV value of 2 we can go up this diagonal by starting at 1 and adding 2 plus 2, etc. The answer will be 1, 3, 5, 7, 9. When we add 2 to 9 it becomes 11 and therefore 2 and when we add 2 plus 2 plus 2, etc. we get 2, 4, 6, 8, 10 (1), 12 (3), etc.
You will note that after the ninth term on this angle, which is
8, the single digits repeat the same order or PATTERN. Remember, you cannot go passed 9 or the ninth term without starting over.
Going on the diagonal from 58 which is 5+8 or 13 or 4, we find the SDV numbers are 4, 6, 8, 10 (1), 3, 5, 7, 9, 11 (2), 4, 6, 8, etc. Again when we reach the number after the ninth term we have repeated the 4 which starts the same series all over again.
Each number on the extreme bottom row corresponds to the same number at the extreme top in the same row as 20 and 38 are both 2's, etc. The SDV on the extreme left row corresponds to the SDV on the extreme right row.
C C ha h a pt p te e r r 7 7 -T - T he h e S Sq qu ua ar re e o o f f 1 12 2
Let us now leave the square of 19 and look at another square,
one that is mentioned by Gann as being one of the important ones, the square of 12.
Its side is 12. The adder from top left to bottom right must be 12-1 or 11 or 2. Since 12 is 3 we simply add 3+2+2+2, etc. to get the SDV's on that diagonal. They are 3, 5, 7, 9, 2, 4, 6, 8, 1, 3, 5, 7.
12 6 9 3 6 9 3 6 9 3 6 9
11 5 8 2 5 8 2 5 8 2 5 8
10 4 7 1 4 7 1 4 7 1 4 7
9 3 6 9 3 6 9 3 6 9 3 6
8 2 5 8 2 5 8 2 5 8 2 5
7 1 4 7 1 4 7 1 4 7 1 4
6 9 3 6 9 3 6 9 3 6 9 3
5 8 2 5 8 2 5 8 2 5 8 2
4 7 1 4 7 1 4 7 1 4 7 1
3 6 9 3 6 9 3 6 9 3 6 9
2 5 8 2 5 8 2 5 8 2 5 8
1 4 7 1 4 7 1 4 7 1 4 7
On the diagonal from bottom left to top right the adder is 12 plus 1 or 13 or 4. We add 1+4+4+4, etc. to get 1, 5, 9, 13(4), 8, 3, 7, 2, 6, 1, etc.
In describing his square of 12 on page 111 in the "old" course (Section 10, Master Charts, page 2, in the "new" course), Gann said when you move over three sections you find the square of its own place.
Some have intepreted this as astrological or astronomical since Gann seems to be saying that the square represents 144 months or 12 years and moving over 3 places is 36 months or one-quarter of 144 or one-quarter of a circle (that of Jupiter) or 90 degrees. Since
Jupiter goes 2.5 degrees a month (heliocentric) it reaches 90 degrees in 36 months. Ninety degrees astronomically or geometrically is a square.
That seems logical, but I have often looked in other places for other meaning. You will note that across the top of the square the SDV is 3, 6, 9. Nine is the square of 3. This is another possible explanation of what Gann meant.
Observing the first 3 SDV numbers in the first three rows in
bottom left we can observe the PATTERN going up, 1, 2, 3 and then 4, 5, 6, then 7, 8, 9. Moving up to the next nine numbers in the three rows we can read, 4, 5, 6, then 7, 8, 9, then 1, 2, 3. Up to the next 9 numbers we read 7, 8, 9, then 1, 2, 3, then 4, 5, 6. Then the next 9 numbers repeat the order on the bottom.
If we think a few minutes, this PATTERN will become clear to us.
But while we are giving that one some thought, let's move on to some other squares.
C C ha h a pt p te e r r 8 8 -T - T he h e S Sq qu ua ar re e o o f f S Se e ve v e n n
First, let's look at the square of 7, a popular Gann number.
Since our side is 7 our adder will be one less than 7 or 6 for the top left to bottom right diagonal or 7+6+6+6, etc. We get 7, 4, 1, 7, 4, 1, etc. Going from bottom left to top right our adder will be 7 plus 1 or 8 so we get 1+8+8+8, etc. or 1, 9, 8, 7, etc.
Now let's go ahead and fill it in. Now, let's look at the square again. The square of 7, right?
7 5 3 1 8 6 4
6 4 2 9 7 5 3
5 3 1 8 6 4 2
4 2 9 7 5 3 1
3 1 8 6 4 2 9
2 9 7 5 3 1 8
1 8 6 4 2 9 7
Wrong! Fooled you that time.
What you see now is a section of the square of 52, another square prominently mentioned by Gann.
For our purposes here we will make up a partial square of 52.
We will also use only one diagonal, the one going from top left to bottom right. Since the side is 52 our adder will be 52-1 or 51 which is 5+1 or 6.
52 x x x x x x
51 4 x x x x x
50 x 1 x x x x
49 x x 7 x x x
48 x x x 4 x x
47 x x x x 1 x
46 x x x x x 7
Since 52 is 5+2 or 7 our first SDV is 7 and the next is 7 plus 6 or 13 or 4. We can check this as 52 plus 51 is 103 or 1+0+3=4.
The SDV on the diagonal is 7, 4, 1, 7, 4, 1, 7, 4, 1, 7.
Draw this on down to 1.
Dropping down to 25 and drawing a diagonal down from there we change 25 to 7 and add 51 or 6 and get 4 in the next column. We can check this as 25 plus 51 is 76 and 7+6 is 13 and 1+3 is 4. So our numbers here run 7, 4, 1, 7, 4, 1, etc.
PATTERN?
Now let's drop down to 7 in the square of 52. Going down the diagonal from 7 we add 51 which still has the SDV of 6 and we get 13 or 4. We can check this as 7 plus 51 is 58 or 5+8 is 13 or 1+3 is 4.
As you can see we have the same PATTERN running down from 52, 25 and 7. It is the same PATTERN on the square of 7. For the same reason that we had a repeated PATTERN in the square of 12.
The reason is that the square of 12 is just an extension of the square of 3 and the square of 52 is an extension of the square of 7.
3 is 3 and 12 is simply 1+2 or 3.
The square of 7 on the diagonal starts at 7. The square of 52 starts at 7 since 5 plus 2 is 7. Ditto with 25 which is 2+5 or 7.
In all cases the difference is a multiple of 9. The difference in 12 and 3 is 9. And 7 plus (2x9) is 25 and 7 plus (5x9) is 52. All proving once again that you cannot go beyond 9 without starting over again. Make up a few squares on your own and prove it to yourself.