Just replace n with the number you want the square root of.
The equation is much easier if you define (n+1)/2 separately which is the intersecting circle radius:
n = 9; //get square root of this number
ic = (n+1)/2; //intersecting circle radius
sqrt = Math.cos(Math.asin((ic-1)/ic))*ic;
Just popping by to update with an equation:
If I didn't write the equation properly or if you can simplify, please do.
I've worked with infinity a lot so I'll chime in. I also take issue with some of the conventional wisdom surrounding ∞ so the following is non-standard.
There are different uses of the term infinity in math. Perhaps the most common is in reference to transcendental numbers. Pi for example is an infinite decimal expression, but as a distance it's not ∞. It exists somewhere between 3 and 4. Pi is a finite distance from 0 (even if we can't quite pin down the exact distance).
When I refer to ∞ I mean the farthest possible distance from 0.
0 is the lower limit, ∞ is the upper limit. (think radial or scalar dimension)
0 * 0 = 0 (can't get any smaller)
∞ * ∞ = ∞ (can't get any larger)
Like 0, ∞ is non-polar. +∞ and -∞ are the same thing. If there is a way for +∞ and -∞ to be distinct, there also must be a way for +0 and -0 to be distinct (which there might well be but I'm not willing to get into that much detail just yet).
As measured from 0 to the tangent, ∞/∞ = 0 ± 1. Details here, because words alone don't do it justice: [ http://www.perspectiveinfinity.com/root_grid.html#infdivinf ]
0 * ∞ offers a similar result but exists 90 degrees from ∞/∞ on the unit circle. All points on the unit circle can be understood as distinct fractional representations of ±1.
In addition to 0 ± 1, ∞/∞ also = 0 ± every other natural number. From an xy perspective when tangent is parallel with the x axis, ∞/∞ results in the entire y-axis. Only by measuring to specific tangents can we limit the result, such as is the case with 0 ± 1.
0 is an origin. When we count from 0, everything is relative to 0. We don't preface every number with 0+ or 0-, even though that's exactly what we're doing. ∞ can also be used as an origin. Now instead of counting from 0 towards ∞ we can count from ∞ towards 0. ∞ + 1 is a finite distance from ∞, but as far as 0 is concerned ∞ + 1 = ∞. Different perspectives, different answers.
I refer to ∞ along with numbers relative to ∞ as "shadow numbers" since ∞ is not considered a "real number".
∞ + ∞ = ∞
∞ - ∞ = ∞
Remember, non-polar! Like zero, infinity is timeless, non-dualistic. Try splitting something timeless into finite parts and all we split is our perception of it.
Nothing is undefined.
Aww too bad the thread went off-topic. Anyway I just stopped by to add that while people intuitively go from a circle to a sphere when thinking of 2D to 3D projections, this geometry is easier to project as a helix when going from 2D to 3D.
A sphere is a surface while a helix is still just a curve, so all that's required is to add a dimension and stretch the circle in the direction of the new dimension (in this case, z). This geometry matches that of a sine wave when viewed from a certain perspective: http://www.perspectiveinfinity.com/sinewave3d.html
r (radius) = any number
y (cosine) = any number between 0 and radius
tangent = √((r/y)²-1) or Math.sqrt(Math.pow(1/cos,2)-1) - I prefer computer speak
x (sine) = tangent x cosine
Say you want to make a circle with a radius of 100 on a computer. What you'd do is get every pixel on the y axis and find it's x value using the aforementioned algorithm, then draw a line to each point.
I made a function for html5 canvas that does this in quarter circles. It's optimized to not draw outside the screen boundary, though despite my efforts the built-in arc function is still more efficient. I suspect it takes some kind of shortcuts, though my method is certainly 'cleaner' as you can see with this comparison: http://www.perspectiveinfinity.com/images/arc-circle_root-circle_comparison.jpg
If your still interested here's the link where I talk about it in more detail: http://www.perspectiveinfinity.com/root_circle.html
And here's the actual function in case you wanted to use or analyze it: http://www.perspectiveinfinity.com/Scripts/rcf.js
What's the official term for when you subtract a number with it's reciprocal? I can't seem to find anything on a google search, probably due to using the wrong keywords.
For example, if you take the square root of 3 (1.732) and subtract it's reciprocal from it (0.557), you end up with 1.154 (4/3 squared). I'm just wondering what this operation is called.
Btw I found this pattern by experimenting with it: 1/2, 4/3, 9/4, 16/5... etc. (*all squared).
Thanks MathIsFun. I've been using that resource frequently (I had come across it before on a search) but there's still so much to burn into my memory.
I think all these different uses of the word "inverse" have been throwing me off. My first thought when I hear "inverse" is 1/something because that's the first way I learned it. So invtan is the same thing as arctan? Bloody hell...
Ok, so perhaps I should go with "sine/cosine swap" or even the more explicit "1/tan" to differentiate from tan-¹? Sine flip is definitely out (even though I think it sounds catchy).
bobbyn: Thanks for the notation. How is that text generated? Is there a tool on this site?
You guys may or may not find it interesting that the only trig function I use in my program is arccosine, and only because I haven't yet found another way to obtain the angle. I'm not using the unit circle or translating back and forth between radians and real numbers. It's just basic addition, multiplication, and squaring. Knowledge of pi isn't even required, except to use arcos of course.
Hmm, so there isn't a single word or short phrase that could sum up a sine flip? Would inverse tangent make more sense?
And yes my other question implies squaring, so the sequence goes from 0.707 to 1.414. "Square rooted fractions of the natural numbers" is the most condensed explanation I've got.
All those seemingly arbitrary symbols and techno-babble is what turns me off from the more traditional approaches to learning math. I tend to think in shapes, not in words and made up symbols that are used to represent ideas. However, despite this I've been making an effort to learn what they all mean anyway.
bobbym: On my program I have tangent, sine, and cosine displayed along with other values. The "sine flip" button swaps the values for sine and cosine, and inverts (1/n) tangent all at the same time. Pretty simple right? So what I'm wondering if there's perhaps a more common way of describing this operation?
Here's another question: Right now I'm describing "fractional roots" as this sequence: √1/2, √2/3, √3/4, √4/5... all the way to √∞/∞, along with their reciprocals: √2/1, √3/2, √4/3, √5/4... √∞/∞. Such a sequence must have already been in use for a while and have a common name, but what is it?
gAr: Thanks. My geometric calc is done in html5 canvas actually. I used to write most apps in flash until a year or so ago. I'm a graphic designer turned programmer. Art and math go nicely together.
Hi, I'm a bit embarrassed to post this. Despite having a huge interest in math and a fair understanding of the basics, I'm sorely uneducated when it comes to conventional terminology. I've taught myself just about all the math I know using calculators and 3d modeling programs. In fact, I'm less educated in conventional math than most high school graduates because I was homeschooled and didn't have a proper math teacher. As a result I've rediscovered some things on my own but I may call them by different names.
Now I created this geometric calculator (perspectiveinfinity.com/rg.html) to help teach myself (and others) a deeper understanding of math, but some of my terminology may not make sense to conventionally educated people. For example, "sine flip" is a term I coined because I didn't know what else to call it when you swap sine and cosine. Even an internet search didn't turn up an official definition for such a function, which surprises me considering the simplicity of it. The answers are probably there, but I'm asking here instead because frankly, I'm tired of searching and would enjoy a bit more human interaction.
So what I'm asking for is a little help merging my terminology with yours. Symbiosis.
Your prime hexagon. Back in the old days we used that fact to generate primes quicker. We would use numbers of the form 6n + 1 and 6n - 1 for n =1,2,3,4... This was faster than trying every odd number.
That's cool. So the idea has been around for a while then? I can't say I'm surprised given the simplicity of it.
My isolation from academia means I tend to rediscover stuff. It's fun and frustrating at the same time. There is something to be said for reinventing the wheel though. Sometimes possibilities are missed the first time.
Here's a simple geometric method for obtaining a square root using circles.
1: Start with a circle.
2: Duplicate & divide by the number you want to square
3: Make two intersecting circles which touch the inside of the outer circle and the outside of the inner circle.
4: Where they intersect is the square root.