1,001 * 1,001 = 1,002,001 11 * 11 = 121
1,002 * 1,002 = 1,004,004 12 * 12 =144
1,003 * 1,003 = 1,006,009 13 * 13 = 169
1,004 * 1,004 = 1,008,016 14 * 14 = 196 (hmmmmm)
2,001 * 2,001 = 4,004,001 21 * 21 = 441
2,002 * 2,002 = 4,008,004 22 * 22 = 484
2,003 * 2,003 = 4,012,009 23 * 23 = 529 (hmmmmm)
3,001 * 3,001 = 9,006,001 31 * 31 = 961
3,002 * 3,002 = 9,012,004 32 * 32 = 1,024 (oh well!)
OK, so....
1,024 * 1,024 = 1,048,576
2,401 * 2,401 = 5,808,001 5,760 + 48 = 5,808
OK, soooooo....
2 ^ 10 = 1,024 and 7 ^ 4 = 2,401
must be a sheer coincidence, right!
1/2 ^ 1/2 = .7071067813
OK, there is a pattern there somewhere. I'm workin' on it. I'll figure it out sooner or later.....
The arc tangent of 1/3 = arc tangent of 1 - arc tangent of 1/2
The arc tangent of 1/5 = arc tangent of 1 - arc tangent of 2/3
The tangent of half of the arctangent of 1 = the square root of 2 minus 1
So a cool algorithm for determining a square root is approximations using the formulas
(2n)^2 = 4* n^2 and (n + 1)^2 = n^2 + 2n + 1, which aligns perfectly to the bit patterns in a floating point representation.
There is also a pattern involved in determining the sine of a number using the bit pattern found in a floating point number, BUT the ratio is dependent on the form of trigonometry employed. In essence, the angle must first be divided by 90 if measured in Degrees, Pi/2 if measured in Radians, and 50 if measured in Gradients. Once that fraction is obtained the pattern works like this....
for 1/2 we get half of the square root of 2 over 2, which is .7071067813
for 1/4, we get half of the square root of the quantity 2 minus the square root of 2 ( or 1.4142136) which is 0.5857863/2
whereas for 3/4, we get half of the square root of the quantity 2 plus the square root of 2, which is 3.4142136/2
Here is where it gets tricky...the addition/subtraction reverses as it moves logarithmically away from 1/2.
for 1/8, you first add 2 to the square root of 2, then take the square root of that number and subtract it from 2, and you end up with half the square root of that result.
for 3/8, you first subtract the square root of 2 from 2, then take the square root of that number and subtract it from 2 and you end up with half the square root of that result.
for 5/8 you first subtract the square root of 2 from 2, then take the square root of that number and add it to 2, and you end up with half the square root of that result.
for 7/8 you first add the square root of 2 to 2, then take the square root of that number and add it to 2, and you end up with half the square root of that result.
So as long as your bit pattern changes, stating with the default as 1 you get subtraction, and when it stays the same, you get addition.
So for the sine of 30 degrees, which is 1/2, because where are looking at the result of slicing an equilateral triangle right down the middle, when divided by 90, we get 1/3, for which in floating point arithmetic, we get a bit pattern of 01010101010101, where each bit alternates between 0 and 1, you always get subtraction, which gives an infinite regression towards 1, and when you take half of that result, you naturally end up with 1/2. The bit patter for 2/3, is shifted by one bit, so for a 60 degree angle we would expect to get half of the square root of 3. because of the Pythagorean proof where we have (s/2)^2 + x ^ 2 = s^2. Algebraically, we can perform all the square root calculations to see that this is true. Well, our bit pattern is shifted by one bit: 101010101010. If we understand that the final calculation is addition, because of the default being 1, there is no alternation on that bit, but the remaining bits do alternate, which gives us our infinite regression towards 1, but this time we add the 2 instead of subtracting it, and we get 3, which we then take half the square root of 3, and have validated that this algorithm works for that angle as well.
