No time for a full run down but basically consider this.
Consider rational parameterizations of the circle.
Now there is a simple rational parameterization to get rational points on a circle
This uses ‘t’, which you typically vary from 0 up to like 100 or something.
Then there is one with two variables,
It is related to some old formulas of Euclid in ancient Greece but I digress. I like this one because the ‘t’ version is a special case – set m to 1. Also m and n can be integers and you still get a circle. Here is a drawing using http://dwitter.net :
This picture shows it too, by drawing a line from origin to each rational point
First off, I like to think of the two variable parameterization slightly differently than what was said above. I take ideas from NJ Wildberger’s Chromogeometry which focuses on three nice forms of numbers in a Descartes 2 dimensional coordinate plane. Chessboard.
Blue geometry =
Green geometry =
You may notice here, if you use these in a formula, like then you get a circle. Blue = circles, red = hyperbolas, and green = hyperbolas rotated from the red by a quarter turn. For full details please look up Wildberger’s papers and videos, I am really not explaining it properly here like he does… this is more of notes to my future self, not a precise math paper.
Now you may notice something nice about that Chromogeometry. It can give you a shorthand notation for rational parametrization of a circle.
Now if we break our old x,y and think of a whole new x y plane… and then we also think of an m,n grid of integers… then a rational parameterization of a circle can be described like so:
Or for short
For people with synaesthetic upbringing or biology, this mnemonic may be easier to deal with than all the pluses and squares. x is Red over Blue, y is Green over Blue.
In fact Red, Blue, and Green can be combined to make many types of rational parameterized shapes, like spheres, or toruses, or other things, dig through my site https://github.com/donbright/piliko for some ideas. But I digress.
It is kind of linked to the idea of the Pythagorean Triples or pythagorean triangles – triangles where the lengths are all rational numbers. Like 3,4,5 or 5,12,13
Think about it. For 3,4,5 we have m=2,n=1
Another way to think about this, along with the parameterization of the circle, is that we are basically dealing with two worlds. The m,n world, a grid of integers, and then the circle drawing world, where we use
They might call this a ‘mapping’ between the grid and the circle.
This is where the complex numbers come in.
Consider a complex number, that is a number with two parts ,one imaginary and one rational.
My favorite descriptions about this idea is from the original 1800s and William Rowan Hamilton, http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/ whose works are collected on that page at at Trinity College Dublin.
Or if we say we have 4 rational and 3 imaginary, then we can draw as a vector.
Note this is a pythagorean triangle / pythagorean triple as well, good old 3,4,5.
Now what happens if you multiply two complex numbers together? You get another complex number…. but you also get a rotated vector. In fact in the language of angles, you get a vector with twice the angle against the x-axis.
4,3i complex vector, doubles to 7,24i complex vector.
Note if you draw this out, you get a triangle.
Spread of this triangle is
Compare with spread of original
Using the double spread formula, verifies that
In angle language, this means the 7-24-25 triangle has twice the angle of the 3-4-5 triangle.
Now… what does this mean for our circle? Basically, every complex vector, when doubled, becomes a point on some rational circle.
In fact this is basically how we make a rational circle. Let us continue and see how.
Do you notice something about the complex multiplication? It has our old friends Red and Green inside of it. Let’s see this again with variables
In other words, the complex number when doubled becomes
In other words, a complex number a,b when doubled forms a triangle Red, Green, Blue
This is the same Pythagorean Triples we were discussing earlier.
And if you divide by Blue, it forms the same rational parametrization that we were discussing earlier.
So what does this have to do with the ‘gaps’ in the appearance of the circle? Ah!
It’s the grid. In the land of M and N, we are dealing with a grid. There are only a certain number of vectors that can be drawn on that grid. For example in a 10×10 grid here are a few of those possible vectors:
In fact there is even some Online Encyclopedia of Integer Sequences at http://oeis.org/ that describe these possibilities. For a grid of size nxn, there are only a certain number of distances that are available, a certain number of Quadrances of Blue nature.. etc etc. But I digress.
Now, what does this have to do with our circle? Well, look at the grid, and realize that every m,n on that grid is also thinkable as a complex number vector. And what do we do to the complex number vectors to form our circle? We double them. In angle language, we double the angle.
In other words, the mapping from the m-n grid world to the Rational Parameterization of the Circle world, basically means we take some vector on the m-n grid, double it, and draw it as a point on the circle.
But what about the points way out to the right, and just above the x axis on our grid? If we have a 10 by 10 grid, that will be where m=10, n=1.
Do you see that if we double this vector, we get something quite high?
Red = 10*10-1*1 = 99
Green = 2*10*1 = 20
Blue = 10*10+1*1=101
Do you see this big number for y? That is what lifts our first point way of the axis! There is a gap! For no other vector on the m,n grid will be lower in slope than the one where m=10 ,n=1.
And this, friends, this, might have something to do with the gaps we saw above.
Let’s take another look at the picture.
Now let us draw some of our grid lines. Say, m=from -10 to 10, but make n only go from -1 to 1. Now our grid is gonna be huge, because remember to make the circle we divide by Blue, so the grid massively swamps it in size. So I have drawn diagonals to each grid point, rather than the grid itself. The actual point for m=10,n=1 is way off the screen. Coordinates are actually 10*10*300-1*1 *300 or something like that. Any ways.
Do you see there? The line for m=10, n=1 goes out about halfway between where our first ‘rational point’ from our Red/Blue, Green/Blue rational parameterization of the circle lives.
That’s because that lowest point, the top of the gap for the x-axis, represents the point where the complex number 10,1i got doubled to be Red/Blue,Green/Blue of
And that, my friends, maybe explains why Rational Parameterizations of the Circles typically look like that, with that gap in there. At least when you are feeding a square grid into the formula that makes them.
Thanks for reading. Hello future self.
Update. Using http://dwitter.net, some pictures
The dots on the circle represents our rational parameterization
The grid represents integer grid of m,n, so m by n different values.
The lines on the grid represent vectors – for complex numbers that when squared and divided, per our parametrization, result in the corresponding dots on the circle.
Note how each point on the grid corresponds to one point on the circle – color coded, if you have color vision.
Note also how the number of points on the circle is linked to how many possible angles there are on the grid from origin.
Finis for now.