You can chart the square roots relative to each other starting with a quarter-circle, a square, and a diagonal line. Where the circle intersects with the diagonal, the next square root is found. More here: http://www.perspectiveinfinity.com/root grid.html Hi, his is my first post here. I've been researching geometry as a hobby for years and have been making discoveries, but I have no academic connections since I developed this hobby as a working adult. As a result, hardly anyone is aware of my work. Hence, I'm dumping it on you guys. Hopefully someone will find it useful. The root grid started out as a discovery connecting square roots and circles, but has since grown into additional realizations which go against the grain of conventional math. What's infinity/infinity? I hadn't even asked myself that question until I was looking at the answer. According to the root grid, the answer is 1, but the point lies on a specific part of the 'root circle', the very top. Every number multiplied by it's reciprocal = 1, and yet that answer can exist as a unique point on a quarter circle. Circles as you well know have a potentially infinite number of points so they have no trouble in keeping up with infinity. Below, a half-circle describes cosine (blue) while a line extending from 1 to infinity describes tangent (purple). Tangent is the distance as measured from 1. The labels in this image are showing the distance as measured from 0. So here we can see how tangent is damn near a reciprocal of cosine. Any point on the cosine circle reflects a point on the tangent line, with 1 (the root circle) acting as a mirror. What about sine? Just flip the cosine circle 90 degrees. Also of interest is how phi is connected to the square root of 2, and how all the silver mean ratios intersect with the square roots when they're described as intersecting circles: link to image If you're still interested after all this, you might find this geometric calculator of interest as well.