When I was doing the this work in the early 1980's I was looking through a book on Masonry and saw a picture of something called the
contains 45 cans or the triangle of 9
My program does not allow me to take the spaces from between the different rows so I could put the rows on top of each other but I think you get the idea.
Looking along the right hand side of the triangle you can see the total from the first to the last can in the triangle.
Now that we have the idea of the triangular numbers, in the next chapter we will use just the numbers themselves.
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To me, the triangular numbers are more interesting than the squares because of their use in making other figures.
Later we will have a look at some triangular numbers in the Gann material and their importance to the Square of Nine chart and the Hexagon chart.
But first, let's make some of the triangular numbers in a list so you don't have to be using your calculator so much.
The first number in the list will be the number we will be
adding each time and the second number will be the running total or
the triangular numbers.
(You might think of the numbers on the left as being the root of the triangle in the same way as you think of a number as being a root of a square. For example the root of the square of 16 is 4. The root of the triangle of 10 is 4. That is, when we add up through 4 we will have the number 10. I don't know if there is any such thing in mathematics as the root of a triangle. If not I'm coining that phrase right here! We will say that the triangle of 4 is 10 in the same way that Gann said the square of 9 is 81, meaning 9 squared and not the square root of 9, which is 3.)
For right now we will be making the triangular numbers from 1 through 49:
1 1
2 3
3 6
4 10
5 15
6 21
7 28
8 36
9 45
10 55 11 66 12 78 13 91 14 105 15 120 16 136 17 153 18 171 19 190 20 210 21 231 22 253 23 276 24 300 25 325 26 351 27 378 28 406 29 435 30 465 31 496 32 528 33 561
34 595 35 630 36 666 37 703 38 741 39 780 40 820 41 861 42 903 43 946 44 990 45 1035 46 1081 47 1128 48 1176 49 1225
Have a look at the numbers. You will recognize those we did. The triangle of 12 is 78. The triangle of 36 is 666.
There is another Biblical number by the same writer (of 666) which is also found in the pyramids. But we will deal with that at a later time because we do not want to go far afield, but keep to the work at hand.
You might also recognize some numbers from Gann. There are some in the Gann material that are well hidden, but can be found with some careful digging. Before we start looking for numbers in the Gann material, let's look at some of the properties of the triangular numbers. And always remember to keep your eyes open for PATTERNS
Look at the numbers again. Play with them a little and see if you can find a
PATTERN
Hint. Think squares.
Found it?
Let's look at the first two numbers, 1 and 3. They add to 4, the square of 2.
Can you take it from there?
Let's take the 3 and add to the next number, 6. We get 9, the square of 3.
Got it now?
Pick out two successive numbers anywhere in the list. Add them and take the square root.
Ok. You say 666 and 703.
666+703=1369
The square root of 1369 is 37.
PATTERN?
Yes, add any successive triangular numbers and your answer will always be a square.
And what is that square?
It is the square of the root of the second triangular number. We added the triangle of 36 or 666 to the triangle of 37 or 703 and the square root was 37.
We can see why that is.
Remember that Pythagoras used a series of 1, 2, 3 and 4 dots to represent the triangle of 4 or 10. Then if we made a triangle of 1, 2, 3, 4 and 5 dots to represent the triangle of 5 or 15 we could push the two triangles together and have a square, the square of 5.
So, put down 5 rows of 5 dots to represent the square of 5 or 5x5=25.
Now if we rotated this square so that the left top corner was
straight up and the right bottom corner was straight down we would have something looking like a baseball diamond and there would be a dot on top followed by two under that, then 3, etc.
Turn your head to the side and look at the square made up of dots and you will get the idea. The number of dots going from top to bottom or in our case from top left to bottom right are:
1, 2, 3, 4, 5, 4, 3, 2, 1
They form the square of 5 and the square is made up of two triangles. The apex of the two triangles are at top left and bottom right and the bases run along the two rows that contain the 4 dots and the five dots.
Then we could add them. On the top diagonals we would have 1+2+3+4=10.
Counting from the bottom we would have 1+2+3+4+5=15.
And 10+15=25, the square of 5.
Let's make that a little more graphic. Instead of using dots
were will use numbers and put those numbers inside little squares.
And I have numbered the squares 1 and then 1, 2 and then 1, 2, 3
to show how many numbers (dots) are in each row.
And then I have renumbered them to show the total going up to 10 from one corner and up to 15 from the other, representing two triangles, one consisting of 10 dots and the other 15.
And from that we can see that the triangle of 4 or 10 plus the triangle of 5 or 15 is equal to 25 or the square of 5x5.
Look at the numbers again. See another PATTERN?
Since we have 1, 2, 3, 4 from one end and 1, 2, 3, 4 from the other and 5 in the middle, what does that suggest?
If we double the triangle of 4 (2x10) and add the middle number we also have 25. So two times any triangular number plus the addition of the next number (the number itself and not its triangle) equals the square of the next number. Try a few for yourself.
How to make triangles and how to add them to form squares, I learned from the ancient arithmetic man.
But, I also made a lot of my own discoveries. They are probably known by mathematicians. And then again, maybe not!
That's the nice thing about studying PATTERNS, they often lead to other PATTERNS.
Let's put down three triangular numbers in a row. For example the triangles of 4, 5, and 6.
They are 10, 15, 21.
See any PATTERN?
We could add them:
10+15+21 and get 46.
No PATTERN there, at least not one I can see.
We could multiply them.
10x15x21=3150
Can't see anything there either.
Any suggestions?
Let's multiply the two end terms, 10x21 and get 210.
Any PATTERN?
No, but what if we ADDED the middle term, which is 15.
Then we would have 225, the square of the middle term, 15.
As I noted in Book IV-"On the Square," we don't have to go way
up in numbers to check for PATTERNS. We can always use the bottom of the ladder and if it works there it probably works anywhere.
So starting from the bottom of the ladder we can put down the triangular numbers of 1, 2, 3 and get:
1, 3, 6
And 1x6=6 and 6+3 is 9, the square of the middle number. So that probably works every time. Try a few and prove it to yourself.
There is another relationship between the triangular numbers and the squares.
Look at the three numbers below:
10, 16, 10
Let's add them. 10+16+10=36 PATTERN?
We know from our triangular list that 10 is the triangle of 4.
Sixteen is not a triangle, but it is a square. The square of 4.
PATTERN?
36 is a square, but from our triangular list it is also the triangle of 8.
Let's try three more numbers:
15, 25, 15
We know from our triangular list that 15 is the triangle of 5.
25 is not a triangle, but we know that it is the square of 5. Let's add them:
15+25+15=55
55 is not a square as 36 was, but it is the triangle of 10.
PATTERN?
Yes, I can see the gleam in your eyes.
Two times the triangle of any number plus the square of that number equals the triangle of a number that is double the first number.
In our first example we saw that two triangles of 4 (2x10) plus the square of 4 (4x4) equals 36 and 36 is the triangle of 8 which is 2 times 4.
In our second example we saw that two triangles of 5 (2x15) plus the square of 5 (5x5) equals 55 and 55 is the triangle of 10 which is 2 times 5.
PATTERN made!
It even works with 1:
1, 1, 1
The triangle of 1 is 1 and 2x1=2. The square of 1 is 1 and 1 plus 2 is 3. 3 is the triangle of 2 and 2 is 2x1.
(You will find, as I pointed out in Book IV-"On the Square" that the number 1 represents more than just the single digit 1. It can be a single number, a square, a number to any power and a triangle among other things. That's why the natural numbers begin with 1 and not zero, which has puzzled Gann students for a long time. But now you know!)
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We will get back to the triangular numbers and where they appear in the Gann material later, but let's first go on and make some other
"numeric figures."
In Book IV-"On the Square," I said that instead of drawing a
square, we would be making squares numerically. And I showed how that was done by adding the odd numbers and noted that other figures could be made by the same method.
We have seen how that method was used to make triangular
numbers, but that ancient arithmetic man also showed how to make some other number figures.
The next figure we will make is a 5-sided figure called a pentagon.
We will put down the numbers to make those figures. The first
numbers will be the numbers with which to make the pentagon and the running total will be the pentagon numbers themselves.
1 1
4 5
7 12
10 22 13 35 16 51 19 70 22 92 25 117 28 145
31 176 34 210 37 247 40 287 43 330 46 376 49 425
Now, without comment, let's make the hexagon numerically. That is, let's make a 6-sided figure. Again the numbers to make the hexagon will be the first numbers. The hexagon numbers which are a running total of the first will be the numbers in the second row.
1 1
5 6
9 15
13 28 17 45 21 66 25 91 29 120 33 153 37 190 41 231 45 276
I had no comment on the 5-sided figure as I wanted to give you a
chance to look for PATTERNS and now I think we have enough material down to give you that chance.
So what about it? PATTERN?
Look at the numbers with which we made the different sided
figures. I will remind you that in Book IV-"On the Square" we made the squares using the natural odd numbers starting with 1 and the numbers were 1, 3, 5, 7, etc.
Got it now?
Let's put down the numbers with which we "made" the different figures in the order, triangles, squares, pentagons, hexagons:
Tri Squ Pen Hex
1 1 1 1
2 3 4 5
3 5 7 9
4 7 10 13
5 9 13 17
Give it another try.
Ok. Let's have a look at your work. You saw that in the first
row where we have the numbers with which we make the triangles, the numbers are 1 unit apart. Add 1 to any unit and you have the next unit. 1 plus 2 is 3, etc.
Looking under the squares you saw that the numbers were two units apart. Add 2 to any unit and you get the next number. 3 plus 2 is 5, etc. (In our work on the square in Book IV-"On the Square" we saw that this was Gann's variable of two.)
Looking under the pentagons you saw that the numbers were 3 units apart. That is, you can add 3 to any number and get the next number. 3 plus 7 is 10, etc.
Under the hexagons you saw that the numbers used to make the hexagon numbers were four units apart. That is, 1, 5, 9, etc.
I mentioned in my advertising material that I would tell you how to make any sided figure very quickly.
So, quick now, how would you make a 33-sided figure?
Yes, knowing what you know now you could make a 7-sided figure, an 8-sided figure and after a lot of other sided figures you would know what numbers to use to make a 33-sided figure, but that would take a lot of time and would not be very quick.
Have another look at the numbers under the triangle, square, pentagon and hexagon.
PATTERN?
Under the triangle the numbers used to make a 3-sided figure are 1 unit part. Under the square, a 4-sided figure, the numbers used are 2 units apart. Under the pentagon or 5-sided figure the numbers are 3 units apart. Under the hexagon or 6-sided figure the numbers are 4 units apart.
Now how do you make a 33-sided figure?
Did you study the PATTERN?
If you did you noticed that the distance between the numbers used are 2 less than the number of the sides.
Now how do you make a 33-sided figure?
If you said subtract 2 from 33 and get 31 and use numbers that are 31 units apart you are correct and those numbers would be:
1. 1 32. 33
63. 96, etc.
Look under the headings of the triangle, square, etc. again.
A good way to check if you are on the right track is to add the first two numbers and they will add to the side you are making.
Under the triangle the first two numbers are 1 and 2 and that equals 3 and we are making a 3-sided figure. Under the square the first two numbers are 1 and 3 which equals 4 and we are making a square, a 4-sided figure.
In the 33-sided example look at the first two numbers used to make the 33-sided figure. They are 1 and 32 and they add to 33!
Now you know a fast way to make any sided figure you want to.
Quick now, how would you make a 12-sided figure?
That's right, the first two numbers would be 1 and 11 and you would be using numbers which are 10 units apart since 10 is 2 less than 12.
Give it a good try!!!!
I have shown you how to make the other figures for two reasons.
The first is to show you the importance of the number 1 and its many characteristics. We know that the single digit is 1 and we also know that 1 to any power, square, cube, whatever is still 1.
Now, in making the figures you have seen that it has other characteristics. It is a triangular number, a square, a pentagon
number, a hexagon number and the first number of any sided figure, 33 or otherwise.
The second reason for looking at the other figures is to show you the importance of the triangular numbers.
We have seen that any two successive triangles equal a square.
I am now going to put down the other sided figures and see if you can make a PATTERN. I will simply put down the figures
themselves, not the numbers with which we make them. But in the first row I will put the "terms."
Ter m
Tri Squ Pen Hex
1 1 1 1 1
2 3 4 5 6
3 6 9 12 15
4 10 16 22 28
5 15 25 35 45
6 21 36 51 66
7 28 49 70 91
8 36 64 92 120
9 45 81 117 153
PATTERN?
We have used the terms to make the triangular numbers and when we add any two successive triangles we get a square. Can you take it form there?
You are correct if you said that when you keep adding the
triangular number after making the square you can make the other figures.
3 (the triangle of 2) plus 6 (the triangle of 3) equals 9 the square of 3. Add the triangular number 3 (the triangle of 2) to 9 and we get the pentagon number 12. Add 3 to that and we get the hexagon number 15.
Add 28 to 36 gives 64. Add 28 to 64 gives 92. Add 28 to 92 gives 120.
So triangular numbers can be used to make any sided figure. We could use this method without ever knowing the numbers it takes to make up the various sided figures.
The names triangles, squares, pentagons and hexagons are rather common, but the larger numbers really have no names so when we are expressing them we can simply say that they are a certain-sided figure to a certain term.
We usually say the square of 4 instead of a square to the fourth term. We could say the pentagon of 4 and the hexagon of 4 or simply a pentagon to the fourth term, etc.
With the other sided figures we could say 33-sided to the fourth term, etc.
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I know this is the part you have been waiting to get to. In my advertising material I said that unlike other Gann writers I zero in on the numbers in the Gann material, citing chapter and verse. No theories here.
I spent the first part of this book detailing the triangular
numbers because of their connection to the squares, etc. and because of the part they play on the Square of Nine chart, etc.
Gann speaks of the triangles. I believe most writers think he is referring to the large triangle within the circle, the one that divides the circle into three equal parts of 120 degrees each. I'm
sure Gann had that in mind, too.
But, I think his reference to triangles goes much deeper than that. Although he does not refer to the triangular numbers, there seems to be a lot of evidence for them in his material.
Under the heading "Time and Resistance Points According to
Squares of Numbers" on page 125 of the commodity course (in my copy, maybe a different page in yours,) he says that stocks work out to the square of some number or the triangle point of some number.
Some might believe he is referring to the large triangle in the circle. But he doesn't seem to be talking about circles at this point. It's squares and triangles.
I believe the best evidence for this is found on page 132 of the
"old" commodity course (Section 9, page 11, in the "new" course).
He talks about the 9 mathematical points for price culminations.
And then he adds:
"There are nine digits which equal 45, another reason why the 45-degree angle is so important."
You already know from your study now that the triangle of 9 (adding 1 through 9) equals 45!
Also on page 126 when he talks about the numbers for resistance he notes that 45 is the "master of all numbers because it contains all the digits from 1 to 9."
On page 112 of the "old" course (Section 10, Master Charts,
Square of 9, page 3, in the "new " course) under his discussion of the Square of Nine chart, Gann says, "note the angles of 45 degrees cross at 325, indicating a change in cycles here."
I have not figured out his 45 degree angles coming down, nor his change of cycles, but from your triangular list you know that 325 is the triangle of 25, a triangle of a square in the same way that 666 is the triangle of 36, also a square.
We don't have to go very far to find some more examples.
On the next page, page 113 in the "old course" (Section 10, Master Charts, Hexagon Chart, page 4, in the "new "course). Talking about the hexagon cycles, he says the sixth cycle is completed at 91. We now know this to be the triangle of 13.
And then on down a few paragraphs he mentions 66 several times and we now know this is the triangle of 11.
He even notes that we have an angle of 66 degrees. We noted
above that he talked about an angle of 45 degrees coming down from 325, the triangle of 25.