Author: Derrin Covayende
Some Terms To Remember While Reading
Channelers original Power:
This is the original amount of the Source that a channeler can draw, be it saidin or saidar.
Letter used as Refrence: A
New Channelers Strength:
This is the new ammount of the Source that a channeler can draw while using an angreal.
Letter used as Reference: B
Quantitative power rating of the angreal:
This value is used for determining the effects of adding two angreal together on a single channeler.
Letter used as Reference: C
Relative power of the angreal:
This is the value of the angle between A and B, also known as the Power Rating of the angreal.
Letter used as Reference: X
Multiple channelers linked together, also known as a Circle.
Letter used as Reference: N/A
The number describing how powerful an angreal is - because this needs to be constent between different channelers, the angle in degrees is used as the angreal's Power Level -> 0-89.9 Letter used as Reference: N/A
Now For the Actual Theory...
(refer to above section for definition of terms) Some basic trig math: The Hypotenuse of a triangle is its largest size. This is why I chose the hypotenuse (B) of a triangle to be the Channelers New Strength. The bottom line (A) is the Channelers Original Strength. Depending on the power of the angreal (x) this will be similar most likely to the New Power of the Channeler (B).
Now, they Hypotenuse of the triangle is defined by the function: cos x. Cosine is Adjacent over Hypotenuse, or in our case, Channelers Strength (A) over Channelers New Strength (B). So: cos (x) = (A/B). This is a ratio of the power growth an angreal can give. So x is the Relative Power of the angreal.
C is not entirely important in this example - C comes into effect when dealing with multiple angreal on the same channeler.
Now some more basic trig math: If we take a second channeler, of power A2, and state that A2 is less than A, then we get a series of similar triangles. With similar triangles, all angles are the same, and all sides are related. All that really matters to us is the property for angles - because we have chosen an angle to represent the angreal's Power Rating.
With these two similar triangles we have the same value for x - which means that the ratio of growth is similar. However, a stronger channeler gets more out of it, because her new strength is more powerful still more than the weaker channelers new strength.
This supports why in the books angreal are often given to the strongest woman in a group - it allows her to take in more power to offer to the Ring.
Now here is where C comes into play, with multiple angreal on the same channeler. First, find the C value of one of the angreal (the triangle composed of A,B, C). Second, find the C value on the Second triangle (Labled here, C2 on the triangle, A, B2, C2 - A is the same, because it is the Power of the Channeler)
Now that we have two separate angreal and their Quantitative Power Rating described we can add these two angreal together on the same channeler.
My theory does not simply add the x values of the angreal together - if you did this you would get values of greater than 90 degrees! Doing so would obviously change which leg is the hypotenuse and disrupt the theory. That is why the third leg (C) is used as a quanitative value.
Now stack C2 on top of C, and draw a new hypotenuse. If you want math of it, simply take the arctan ((C+C2)/A) and you will get the new angle of x3. Then, by multiplying A by cos (x3) you can recive a new value for the Channelers New Strength.
For a single angreal
|Channelers New Strength||B= A cos(x)|
Power Rating of angreal:
x = arccos (A/B)
For a Channelers of different strength, using the same angreal:
|Ratio between the Two Channelers||(A/B) = (A2/B2)|
Power Rating of angreal
x = x because of similar triangles
For a multiple angreal:
|New Power Rating of Multple angreal||x3 = arctan ( (C2+C)/A )|
Channelers new Strength
B = A Cos (x3)
Obviously using trig many other functions can be gathered from these few. Experiment!
Rings and this angreal Theory
Many people have likened Rings to turning humans into living angreal. I do not believe this is the case, and as such, my theory does not exactly explain how Rings work - and because of that, my theory cannot be used to describe the power of Rings (and as such, cant be invalidated by pointing out that a ring doesn't seem to work that way, according to the books!). However, my theory can be adapted to the describing how a Ring works, if you understand how I view a Ring versus an angreal.
This is how I view an angreal as working. The big white circle is the Source. Normally a Channeler can draw from the small blue circle. When she has reached her limit, the Source is flowing out of that hole at its maximum rate. (Now, if she doesn't use the Source, just draws upon it, she can only hold so much within her. But the rate at which she draws the Source is important too, at least for this example)
The red circle is the new opening from which a channeler can draw on the Source - when using an angreal. As you can see, she can draw much more, much faster. Now strength is not determined alone from how fast and how thick a channeler can draw on the source, but it is one factor of the final strength, and for this example, the part that matters here.
Now a Ring is a different matter. I look upon a Ring of channelers as each individual channeler draws what she can, and offers it up to the leader of the Ring. Looking at the example, the leader is in blue, and all the linked channelers are in red. When the leader of the Ring pulls from the Source, she can draw from any of the openings - achieving the same effect as an angreal, but in a different matter.
Also, I did state that how much you can hold is as important as how much you can draw - well in this example, the other people in the Ring become reservoirs for holding the Source - effectively making the channeler leading the Ring seem stronger.
So how does the Theory relate?
Quite simply in fact. Simply stated because of how I view a Ring versus an angreal, when a channeler draws strength from a Ring they simply add the maximum value they can utilize from the channelers in the Ring. So first calculate the Power of a Channeler if they have any angreal. Then take that new value and put it through some function that determs how much Power a leader of a Ring can draw from members of the ring. Then simply add that to the power of the channeler leading the Ring.
So.... Now What?
So, that's a basic look at my theory - in fact, it's about as in depth as I'm willing to go! (This, in retrospect, is probably a lot deeper than anyone else would go!) Now, please, a good theory only gains power after being attacked and successfully defended. Please e-mail me any point of problem you may have with this theory - I would love to take them and respond to them. I've tried my best to describe what I can, and the next section will be entirely responses to questions I get. If you have a point to address, please email me!
Points Raised in Contention
This one is paraphrased, cause I actually modified the theory after hearing the point - but one part of it is still valid:
This is effectively a linear growth description of an angreal's power. But, for example, the Choedan Kal is hundreds of times stronger than the Wand in the White Tower - wouldn't an exponential growth rate describe angreal's power better?
Yes.... And no. First off, a linear growth rate can reach the same points that a exponential growth rate reachs - it just takes a larger x value. In order to gain that value in my description, simply make the highest power rating (89.9) go to another decimal place (89.99.... 89.999.....89.9999 ect) - This will produce some astronomicly long B legs, the hypotenuse, but that's okay - because these power ratings would only be for the strongest of sa'angreal - the Choedan Kal for example.
Now with exponential growth the main problem lies not with sa'angreal, but with the angreal.
Let us simplify things slightly. Assume the x value on a graph is the stated power rating of an angreal and the y value is the new power of the channeler after using the angreal. (for all numbers x>1)
If we take an angreal of Power Rating 2, and one of Power Rating 3, we get New channeler strengths at 4 and 9, respectivly. This may not seem like much, but lets put this into context of the books.
For our example we shall use Rand's fat little man angreal and Moiraine's ivory carving angreal. These two were described simulary throughout the books - as adding very little to the over all strength of the channeler, just enough to get the job done. (Now, this in paranthesis im not sure of, but didn't Rand compare his angreal to Moiraine's as they were leaving the Stone?)
Unless the angreal were exactly the same Power Rating (possible, but doubtful) even a difference of 1 would give them radicly different power outputs for the Channelers New Strength Even a difference of .5 would be radicly different on the scale we're looking at.
Zoom way out to looking at sa'angreal - if the two Choedan Kal were made even slightly different the difference in their power rating would be astronomical. They'd have to be exactly the same to give a similar power output to the two channelers using it. A .001 difference in their Power Rating and one channeler would be using, well, an exponential amount more than the other.
That's my rebuttle to expontial growth.