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.*

Cos | Sin | Tan |

1/2 | 1/2 | 1/1 |

1/3 | 2/3 | 2/1 |

1/4 | 3/4 | 3/1 |

1/5 | 4/5 | 4/1 |

By using decimals it's difficult to notice the numerical connections.

Translating values from the image above into *fractional roots* results in the following:

Distance: | 2 = | sqrt(4/1) |

Cosine: | 0.5 = | sqrt(1/4) |

Sine: | 0.866 = | sqrt(3/4) |

Tangent: | 1.732 = | sqrt(3/1) |

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.

## Exponentation

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 |

## Newton's Rings

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.

## Polarity

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.