It's possible to chart the square roots of the natural numbers geometrically by starting with a quarter-circle, a square, and a diagonal line as in the animation below.
The numbers in this animation refer to distance from 0 (not x or y).
A different Sine and Cosine
Typically sine and cosine are described as perpendicular vectors coming from the x and y axis' which meet at the same point on the unit circle's perimeter, like so:
But this works too:
Instead of measuring from x and y, the measurement comes straight from 0 (or the z axis) as a diagonal line which intersects with the blue circle for the cosine value and the red circle for the sine value.
In addition to Sine and Cosine I do a few other things differently as well compared to conventional Trig. Below I list a couple more:
- My Unit Circle starts at the top and goes clockwise instead of starting on the right and going counter-clockwise. The top of the circle = 0° and the right = 90°. This allows both tangent and the number line to run horizontally.
- Distance from 0 is the primary measurement which all other measurements are relative to. If Distance is between 0 and 1 it's value is the same as Cosine. If Distance is greater than 1 it's reciprocal (1/Distance) is the Cosine value.
When I first discovered "The Grid" I wasn't aware of it's many connections with Trigonometry. That realization came later after studying the Grid for a while. So it's not as if I purposely went out of my way to do Trig differently, it just developed this way.
Trigonometry and Square Roots
Trigonometric values may be understood as fractional square roots relative to each other. For example see the table below.
Measurements are in square roots.
By using decimals it's difficult to notice the numerical connections.
Translating values from the image above into fractional roots results in the following:
As you can see, by using fractional roots it's much easier to notice the numerical connections between trigonometric measurements.
Convergences at 0 and 1
The following image shows why the square root of 0 = 0 and the square root of 1 = 1. Pictured is the Cosine Curve from the Root Grid. As you can see, the whole number fractions (left side of image) converge with their square roots at 0 and 1.
As usual, all measurements are from 0 and are relative to the distance between 0 and 1.
Not all 1s are equal
Numbers multiplied with their reciprocals = 1. A reciprocal is simply a flipped fraction. So take the number 2 which you can write as 2/1. Flip the fraction and you have it's reciprocal 1/2 which equals 0.5 as a decimal expression.
In the image below you can see 2 on one side of the Unit Circle, and 0.5 on the other. They're reciprocals of each other so 2 x 0.5 = 1. Same with 1.414 x 0.707 (the square root of 2 and it's reciprocal). What's interesting here is that each "1" occupies a different point of the Unit Circle.
Saying "square root" is the same as saying "to the power of 1/2" or "exponent 1/2". If I change the exponent of the Root Grid from 1/2 to 1, the Unit Circle changes into a diamond (or tilted square).
|Grid when exponent is 1/2||Grid when exponent is 1|
I guess it shouldn't be called a Unit "Circle" anymore. Also, the grid is no longer a "Root Grid" but rather a "Whole Number Grid". Below, a couple more examples of the Grid using different exponents...
|Grid when exponent is 2||Grid when exponent is 1/4|
If I turn everything off but the Scalar grid in my RootGrid Calc while the exponent is 1/2, I get the following image:
The spacing between the concentric circles are the same as Newton's Rings, a phenomenon related to interference patterns.
Below, an image showing the polarity of the Root Grid.
What's interesting is how the polarity is flipped inside the Unit Circle (scalar polarity). This isn't based on a whim, but rather by studying the curvature of Sine, Cosine, and Tangent in 3D as it relates to the Root Grid.
The scalar is a 4th dimension which relates to big and small, macro and micro. It's "axis" is the Unit Circle itself, which here isn't merely a typical flat circle only occupying x and y, but is broken and stretched into the z dimension as well.