Note to future self – the holes in rational parametrizations of the circle, chromogeometry, and complex number rotation

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

x=\frac{1-t^2}{1+t^2},y=\frac{2t}{1+t^2}

This uses ‘t’, which you typically vary from 0 up to like 100 or something.

Then there is one with two variables,

x=\frac{m^2-n^2}{m^2+n^2},y=\frac{2mn}{m^2+n^2}

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 :

rational paramaterization of circle showing gapsrational paramaterization of circle

rational paramaterization of circle showing gaps

dwitter.net allows the quick ability to draw pics in a short hand javascript computer language. You can see the ‘gaps’ towards the axes. Especially the x axis, which goes left-to-right.

This picture shows it too, by drawing a line from origin to each rational point

rational points on a circle, lines drawn from origin to each point

rational points on a circle, lines drawn from origin to each point

Why?


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 = X^2+Y^2

Red geometry= X^2-Y^2

Green geometry = 2XY

You may notice here, if you use these in a formula, like x^2+y^2=5 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.

Red(a,b)=a^2-b^2

Green(a,b)=2ab

Blue(a,b)=a^2+b^2

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:

x=\frac{Red(m,n)}{Blue(m,n)},y=\frac{Green(m,n)}{Blue(m,n)}

Or for short

x=\frac{Red}{Blue},y=\frac{Green}{Blue}

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

leg1=Red,leg2=Green,hypoteneuse=Blue

Think about it. For 3,4,5 we have m=2,n=1

Blue=2*2+1*1=5

Red=2*2-1*1=3

Green=2*2*1=4

—————

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

x=\frac{Red}{Blue},y=\frac{Green}{Blue}

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.

a+bi

For example

4+3i

Or if we say we have 4 rational and 3 imaginary, then we can draw as a vector.

comprot5.3

Vector, showing complex number 4+3i

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.

(4+3i)(4+3i)

4(4+3i)+3i(4+3i)

16+12i+12i+9i^2

16-9+24i

7+24i

Note if you draw this out, you get a 7,24,25 triangle.

doublegrid

Spread of this triangle is s_2=\frac{24*24}{25*25}

Compare with spread of original s_1=\frac{3*3}{5*5}

Using the double spread formula, verifies that s_2=4s_1(1-s_1)

Indeed, \frac{24*24}{25*25}=4\frac{3*3}{5*5}\frac{4*4}{5*5}

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

(a+bi)(a+bi)

(aa+bia+bia+bbi^2)

(a^2+2abi-b^2)

(a^2-b^2+2ab i )

In other words, the complex number  a+bi when doubled becomes

Red(a,b)+Green(a,b)i

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.

x=\frac{Red}{Blue},y=\frac{Green}{Blue}


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:

pythgrid

A grid showing the finite nature of how many vectors can be drawn on it

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

x = 99/101, y = 20/101

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.

rational parametrization of the circle, via dwitter

rational parametrization of the circle, via dwitter

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.

rational paramterization of a circle vs m,n grid lines

rational paramterization of a circle vs m,n grid lines

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 x = 99/101, y = 20/101

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.

 

 

 

rational paramterization of the circle, 3 points

rational paramterization of the circle, 3 points

 

rational paramterization of the circle, 5 points

rational paramterization of the circle, 5` points

 

rational parameterization of the circle, 9 points

rational parameterization of the circle, 9 points

 

rational parameterization of the circle, 13 points

rational parameterization of the circle, 13 points

Finis for now.

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