How to Calculate Square of 9 Chart Angles

Ever wonder how to calculate an angle on WD Gann's square of 9 chart? Here are some handy formulas

let "P" be your number or price, time… whatever


1. Degrees = MOD(180*P½ - 225),360) (angle in degrees)
   (we are taking the square root of our number and multiply by 180
    this formula is valid for a chart with zero in the center)

equation 1 assumes the center of the square is zero.  But most 
square of 9 charts have 1 in the center. We can still use equation
1. with a minor modification.   We subtract 1 from P first.

note*  MOD is short for Modulo which means divide by
the number, in this case 360, throw out the integer portion
and the remainder is the result. example MOD(1632,360) = 0.533

2. Degrees = MOD((180*(P-1)½ - 225),360) ( angle in degrees)
	     (valid for a chart with one in the center)

Sometimes we are only interesting in the difference (in degrees)
between to numbers.  In this case we subtract the square roots of 
the numbers and multiply them by 180 to get the total number of
degrees. If we want, this can also be reduced to a nubmer between
0 and 360 using modulo math

3. Degrees difference = MOD(180*(P1½-P2½),360)

Other times we are interested in adding a certain number of
degrees to our number. Two equates to 360 degrees on the square of 9
Therefore one equates to 180 degrees and ½ to 90 degrees.  

Lets say we want to find the number that is 90 degrees above our P.
Take the square root, add an increment (inc) 0.5 (because that
equates to 90 degrees on the square of 9) and square it again to 
arrive at the result P'

4. P' = (P½ + inc)2

The formulas shown above do not give us the exact number on an actual square of 9. It is an approximation (or perhaps more accurately vice versa). I have always used the above formuala with great success. That said, however, there are those who insist on finding the actual number that would be in a particular cell on an actual square of nine. With (more effort) this can be done as Daniel Ferrara explains in the post below:

How to Find the Actual Number on the Square of 9

In my experience with working with this (square of 9) method, price & time must balance on a hard aspect. The hard aspects are 45, 90, 135, 180, 225, 270, 315 and 360 or 0 degrees. The most important being the squares or 90-deg harmonics (0, 90, 180, 270). In terms of selecting a past date and price to start from, I have found that the lowest low over the past 365-days and the highest high over the past 365-days have the greatest influence on these balance points. This technique can be used to generate the horizontal support & resistance levels for intraday trading.

This is extremely useful when you anticipate that a particular day will be a trend change as the result of cycles or time counts, etc.

On another note, I do not agree with the formula you have below: MOD 360[180*abs(price distance or Time distance)-225]. For what you are assuming, it should read =MOD 360 ((price distance or Time change)^0.5*180-225).

This is based on Carl Futia's formula. However, this formula assumes that the Squares of Even numbers fall on the 135-deg angle and that the Squares of Odd numbers fall on the 315-deg angle, which is not true on Gann's actual Square of Nine chart. If you start with a "1" in the center, the Squares of Odd numbers will fall on the 315-deg angle, but the Even Squares (16, 36, 64, 100, 144....) will gradually float towards 135-degrees. For example, on the actual Square of Nine, 16 is on the 112.50-deg angle, 36 is on the 120-deg angle, 64 is on the 123.75-deg angle, 100 is on the 126-deg angle and 144 is on the 127.50-deg angle and so on.

Starting with "0" in the center, the Squares of Even numbers will line up on the 135-deg angle and the Squares of Odd numbers will Float. Could this amount of inaccuracy or "Lost Motion" be important? After all, it is impossible to draw or actually build a Square of Nine Chart based on the MOD 360 formula above. If you want to work with calculations that are based on W.D. Gann's printed Square of Nine chart, the following formulas will be of great use to your research:

Ring# = Round(((SQRT(Price)-0.22 / 2),0) {This rounds to the nearest whole number, i.e. it eliminates the decimals}

Example the number 390 is in Ring #10 if you crunch the above formula.

315-deg Angle. This is the most accurate angle of the entire chart and is used to calculate all other values. The Squares of Odd numbers are all on this angle.

315-deg Angle = (Ring# * 2 +1)2. Example, 390 was in ring# 10 so the 315-deg number is (10 * 2+1) ^2 or simply (21)^2 = 441

The Zero Angle on this Ring = ((Ring# * 2 + 1)2) - (7* ring#). So you would get 441 - 70 = 371 This number is needed to calculate the angle that the 1st value of 390 is on.

Angle = Sum ((Price- Zero Angle) / (Ring/45)). So we have ((390 - 371) / (10/45) = 85.50-deg

You may have to occasionally adjust the Angle calculation because sometimes you will get a negative value when you have a number that is approaching the 0-deg angle of the next ring. For example we know that 371 is a zero-deg number. If you try to find the angle of the number 370.5, which is a number in the previous ring approaching the next ring, you get Sum ((370.5 - 371) / (10/45)) = -2.25-deg. If you get a negative number, just add 360 to correct it. So this would actually be 357.75-deg.

A simple formula to correct this is If Angle<0 then +360 else Angle = Angle.

To generate other values on the Square, use this formula: (Ring# * 2+1)2) - (7* Ring#) + ((Ring# / 45) * Angle)

Angle is this formual is your input value. For example, we know that 390 is on the 85.50-deg angle. If we want to know the value of the number that is 45-deg to this number, we would be interested in the angle of 130.50-deg (85.5 + 45). Inputing this in the above formula gives us: (10 * 2+1)2 - (7 * 10) + ((10 / 45) * 130.5). Simplified a little, we have 371 + (28.99971) = 399.99 is 45-deg to 390. Keep in mind that if you add or subtract an amount that will change the original angle (85.5-deg) to an amount greater than 360 or less than 0, that you JUMP rings. For example, if you subtract 90-deg from 85.5 to potentially find a square aspect, you get -4.5-deg. Add 360 gives 355.50-deg in the previous ring. We were using Ring# 10 in the formula, but for this calculation, we would have to use Ring# 9. Similarly, if you added 315-deg to 85.5-deg, you get 400.50, which is 40.5-deg in the next ring. So you would have to use ring# 11 for this calculation

Daniel Ferrera

Whether you choose to use the Futia or the Ferrara formulas, you will find that the square of 9 is an amazingly powerful tool in the hands of any competent technical analyst. It is well worth spending some time to explore square of nine methods.