This does belong in AGN, in my opinion.


Don - once again, I'll mention the unity circle concept to you.

No need for arcs, etc. It can be much, much simpler.

Take a piece of rope of any length X.
Anchor one end. Draw a circle by rotating the other end around the anchor, keeping the rope taut.

Now, in math they may or may not have had:
R = 1/2 of X
Now if you consider two radii drawn to form an angle of 90 degrees at the center of the circle, you have a right triangle where the the base and height is R. The square root of R squared of R squared would be what? R = 1, then it's the square root of 2, R = 2, it's the square root of 8 (2 squared (4) + 2 squared (4) = 8).
However, R = 1/2 of X (the diameter). So .5X squared is .25X, double is .5x so you actually have the square root of .5X for the hypotenuse.
That's the square root of 2 or 1/2 you might need for some claims.
In addition, you can get to the square root of three... Add in another radii splitting that 90 degree angle to 45. Or 30, if you want even more numbers. The height of the chord your using for the third side of your triangle can be the square root of 3 or 5 or... well, you get the idea - consult a table of logarithms for just about any value you want, depending on the central angle.

That's all math. There's a much, much easier way, like I said.

Remember that rope? the one you anchored and rotated about the anchor to draw a circle?
Well, you could have just looped it into a circle first off and declared it to be the diameter, then cut a second rope inside it to fit the diameter.
Presto, use of PI without knowing PI.

Also, having described the circle as above, you could then anchor a midpoint on your rope (1/2 R in math) and describe an ellipse as above (walking the rope about). Do this for each of the cardinal directions and you get an interesting figure of 4 lobes inscribed upon your circle.
Hmm, you could also use 1/3 R or 2/3 R to create different ellipses taht would also develop (via drawing of lines) into triangles w/ heights of sq rt of three again.
That is, where the curves intersect your circle can match your some of your claims... without the use of *any* math, OR knowledge of PI. Well, okay, you have to be able to fold a rope into half or thirds...

I've previously referenced some good material on the unity circle concepts. I'd recommend you read through it? Most of these claims based on math knowledge only require empirical knowledge of geometry... not extensive knowledge of exponential math (implying algebra) or knowledge of PI (implying advanced geometrical math).

Occams razor.

Ever read up on how the Egyptians leveled a surface? I found it instructive in its beautiful simplicity