unit circle, or, how close to 1 turn is 333 radians?

[michel paul]

http://sagecell.sagemath.org/?q=xofxon

6 radians doesn't close a circle.

7 starts one over.

How many radians does it take to close a circle?

: )

This is fun to explore. Start with just a few number of radians to get a clear idea for what's happening, then look at the fascinating patterns that emerge given higher numbers of radians. 

A surprisingly simple but interesting fact to consider - each integer number of radians marks a unique location on the circle that will never get revisited on any future cycle. Each integer has a unique location on this circular number line.

And a really cool thing is - all the code you need is right there. You can explore and modify it, and if you break it, no worries. The original is still right here.

--

​ Michel

​

===================================
"What I cannot create, I do not understand."

- Richard Feynman

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"Computer science is the new mathematics."

- Dr. Christos Papadimitriou
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[Peter Farrell]

On Tuesday, April 12, 2016 at 8:26:47 PM UTC-7, michel paul wrote:

http://sagecell.sagemath.org/?q=xofxon

<snip>

A surprisingly simple but interesting fact to consider - each integer number of radians marks a unique location on the circle that will never get revisited on any future cycle. Each integer has a unique location on this circular number line.

<snip>

Brilliant idea! Now instead of irrational numbers making (seemingly) random points on the number line, integers are the numbers making random points on the circle.

[kirby urner]

Along this same line of things not repeating, you know the tetrahelix right? 

Start with a regular tetrahedron, then face-bond another to any side and keep corkscrewing in a spiral, extending out the helix. 

http://bit-player.org/wp-content/uploads/2013/10/tetrahelix-0688-900x349.jpg

One may compute the vertexes using standard analytic geometry and find the points (vertexes) have rational (Q, Q, Q)-domain coordinates, with the curious fact that they get longer and longer (the digits of the p and q both extend, for x, y, z)... meaning there's no repeating going on either. 

http://tinyurl.com/jokeqqs

I'm not proving this result, just recalling it as an observed phenomenon from some computational studies (Syn-L), likely a well-known example of a rational curve shape.

Kirby

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[Bradford Hansen-Smith]

Michel, here is another view on exploring your question "How many radians does it take to close a circle?"

Given the concentric nature of the circle and that movement goes in both directions, infinitely into and out, there is no fixed center to the circle. The center is an idea relative to scale about the idea of a circle. That would mean the radius is of relative length to a fixed circle unit having little to do with the nature of the circle. The diameter is the full straight line measure of a circle fixed to any scale, it is not the radius.

The circle as a concept of unity can not be measured or constructed by any number of units, one will always come up short. A circle is closed. if it is not closed it is not a circle. Any number of radians will never be a circle, anymore than any number of circles sharing the same axis will ever complete unity in spherical form. Adherence to out of date ideas have confused our thinking towards present experience and understanding. The fun is in sorting out what remains relevant and what is not.

Brad

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[michel paul]

On Wed, Apr 13, 2016 at 1:22 PM, Bradford Hansen-Smith <wholem...@gmail.com> wrote:

> ... there is no fixed center to the circle. 

Initially I couldn't understand what you meant by that, but then I remembered an article I recently came across, Is this a circle? What the answer reveals about you. I think it actually can make sense for someone to think of something as 'circular' that doesn't necessarily have a center, like a game of telephone where the message has gone 'full circle'. Yeah, different backgrounds create different associations and expectations, so someone might think of something as 'circular' where the concept of center and radius don't necessarily apply.

However, in the context of making sense of radian measure, I think having a center, a radius, and a specific circumference is pretty important.

> Any number of radians will never be a circle ...

Well, how about 2 pi radians? Or, in other words, tau radians?

Granted, no integer number of radians will close a circle, right. That was the whole point of the exploration. But is there some of number of radians that will close the circle?

Yep. And that number is tau.   : )

> Adherence to out of date ideas have confused our thinking towards present experience and understanding. The fun is in sorting out what remains relevant and what is not. 

I completely agree with that.

[michel paul]

On Wed, Apr 13, 2016 at 11:27 AM, kirby urner <kirby...@gmail.com> wrote:

> you know the tetrahelix right?

Not really,  but sort of, because yeah, I do recognize that kind of structure. I'm glad you're mentioning this. These things are fascinating.

I had a kid in my computational class who was highly functioning autistic, and he would take these 3-d magnetic puzzles I had in my room and create absolutely amazing structures with them. I mean ... they were almost scary. Like from some other world. The kid was a genius. He'd describe to me how if he had more pieces the structure would cycle back and close itself. He was visualizing all this, and he was right on. Really amazing.  

--

[michel paul]

On Wed, Apr 13, 2016 at 11:16 AM, Peter Farrell <peterfa...@gmail.com> wrote:

> instead of irrational numbers making (seemingly) random points on the number line, integers are the numbers making random points on the circle.

Though consecutive integer radians are necessarily separated by equal distances, the ways in which they cluster in groups as the cycles unfold is intriguing.

This provides a good example of how our math curriculum boxes in thinking in unnecessary ways. The way radian measure is typically presented, I don't think most students (or even teachers) actually get what's going on. I know from my own experience that as a student I somehow got the impression that there always had to be a 'pi' in an expression in order for it to be representing radian measure. I somehow thought that 'pi' was the 'unit'. And that is because radian expressions in math texts usually have a 'pi' in them.

My eyes were really opened to this when I was teaching and would give a problem in class discussion: What is the radian measure of an arc whose length is 20 units on a circle with a radius of 5 units? The answer of course is simply 4, but I saw that students thought it had to be much more complicated than that. They thought they had to 'use' some formula they had 'forgotten'. They thought there had to be a 'pi' in there somewhere, because arc length formulas always had that. They didn't believe it could be as simple as 20 units / 5 units. And I got why they thought like that, I could see it clearly, because that is what our stupid curriculum does to people.

- Michel

On Wed, Apr 13, 2016 at 11:16 AM, Peter Farrell <peterfa...@gmail.com> wrote:

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[Bradford Hansen-Smith]

" Is this a circle? What the answer reveals about you" This is about  psychology. having nothing to do with the circle. At best it indicates how poorly we have been educated about geometry. 

Bradford Hansen-Smith
www.wholemovement.com

[michel paul]

Here is a slight modification: http://sagecell.sagemath.org/?q=vmqkea

I added a chord between the radii bounding each radian. 

The initial triangle naively appears to be equilateral, but it can only be isosceles. The chord necessarily being slightly shorter than the radii has interesting consequences. What appears to be a developing inscribed hexagon at radians = 6 doesn't quite make it.

- Michel

[kirby urner]

I find all this animated circle stuff resonates well with my three chords:

(i)   e to the i times tau equals one <-- unit circle in complex pancake

(ii)  data structures

(iii) time dimension <-- moving around the clock face

It's somewhat unfortunate that we sometimes let convention eclipse imagination. 

Put the camera (point of view) behind the unit circle and rotate it (the camera) by 90 degrees such that a unit circle "second hand" sweeps clockwise (as usual) from (0, 1) at the top (noon), all the way around the clock face for tau radians, back to (0,1).

A shortcoming of 1900s math was little attention was paid to the viewpoint, which was "god's eye" and we were meant to forget about it (no little "man behind the curtain" viewing the vista).  When you look at an XY graph, where do you look from?  If you look from behind it, the x-positive goes to the left.

A lack of fluency with moving in space develops when only planar, fixed, no-observer geometry is taught, with X-positive always to the right (who's right?). 

Spatial fluency is natural and has to be dulled a lot to make plane (aka "land lubber") geometry so all-encompassing.  Don't let that Euclidean stuff dumb you down!  Euclid wasn't into flat either.  His investigations led towards actual volumes in the later books. 

Plane geometry is a means to an end, and in teaching what math is For we need to dwell on the ends, more than just means.

Thanks to 21st Century visualization technology, chances are you can rotate your XY plot around any axis you like, thereby reminding the viewer that viewpoint is variable, never fixed (except by convention).

When showing a plane, don't hesitate to pull back and show more of the surrounding vista, which may well be spherical in nature.  The better school districts all teach this.

Kirby

[kirby urner]

On Thu, Apr 14, 2016 at 11:01 AM, kirby urner <kirby...@gmail.com> wrote:

I find all this animated circle stuff resonates well with my three chords:

(i)   e to the i times tau equals one <-- unit circle in complex pancake

(ii)  data structures

(iii) time dimension <-- moving around the clock face

It's somewhat unfortunate that we sometimes let convention eclipse imagination. 

Put the camera (point of view) behind the unit circle and rotate it (the camera) by 90 degrees such that a unit circle "second hand" sweeps clockwise (as usual) from (0, 1) at the top (noon), all the way around the clock face for tau radians, back to (0,1).

Sorry, (1, 0),

I'll not change the basis in this storyboard. 

Noon = (1, 0) not (0, 1), such that cosine(Noon) = 1 (as usual).  cosine(12:30) == -1.  And so on.

Kirby

[Joseph Austin]

You get a similar phenomenon, but rational, considering the "circle of fifths" in music,

i.e, a succession of tones in frequency ratio 3:2, as compared with the "octaves", or succession of frequencies in ratio 2:1.

[michel paul]

On Thu, Apr 14, 2016 at 11:01 AM, kirby urner <kirby...@gmail.com> wrote:

I find all this animated circle stuff resonates well with my three chords:

(i)   e to the i times tau equals one <-- unit circle in complex pancake

Yep!    : )

For awhile now I have found it significant that one radian corresponds to e^i.

The circular number line we are constructing by sequential integer radians is e^0, e^i, e^(2i), e^(3i), ... . It is a sequence of rotations.

The beauty of Euler's e^(i x) = cos x + i sin x deserves exploration much earlier in our math curriculum than it receives. You can develop it through binomial expansion of the compound interest formula, throwing in some trig. It doesn't have to wait until so late where only a few see it.

THIS is the unit circle!  --->   [e^(i*t) | t <- [0 .. tau]]. 

- Michel

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[michel paul]

​

On Thu, Apr 14, 2016 at 3:26 PM, Joseph Austin <drtec...@gmail.com> wrote:

You get a similar phenomenon, but rational, considering the "circle of fifths" in music,

i.e, a succession of tones in frequency ratio 3:2, as compared with the "octaves", or succession of frequencies in ratio 2:1.

Yes! That's a really good point. Thanks for bringing this up. 

It's the 'Pythagorean Comma'. I love music theory, and yes, the circle of pure fifths never reaches the octave in a pure tuning ... yes, this is something I found fascinating a long time ago, but I never connected it to this example of integer radian measure. There really is a very deep connection here.

When Bach created the well tempered scale, each half step jumps by the 12th root of 2. In Western music we never hear 'pure' fifths. But, it's good enough for rock 'n roll!  : ) 

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[michel paul]

Here is a kind of interesting observation - the area of a 1 radian sector is 1/2 the square of its radius.

In this diagram:

The area of the sector is 1/2 the area of the enclosing square. It is not a very intuitively obvious way of dividing the area, but the area has in fact been divided into two equal halves.

The relation of the radius with a one radian sector is rational, but not its relation with the circle.

Somehow that struck me as worth contemplating. 

- Michel

On Thu, Apr 14, 2016 at 9:36 AM, michel paul <python...@gmail.com> wrote:

[michel paul]

A 1 radian sector has an arc length of r and an area of 1/2 r^2.

By scaling, a full rotation has an arc length of tau*r and an area of 1/2 tau*r^2.

An interesting fact to consider about the unit circle and the relation between pi and tau is this - 

the circumference of the unit circle is tau, and its area is pi.

[kirby urner]

I'm not sure how aligned this blog post is, vis-a-vis these 2D considerations.

I'm looking at some well-known 3D ratios, cross-referencing the volumes table I'm circling.

http://mybizmo.blogspot.com/2016/04/shape-arithmetic.html

Kirby

[michel paul]

Interesting - this process of plotting integer radian arcs around the unit circle turns out to be a version of the old textbook 

wrapping function!

Trying to find an integer number of radians that gets as close as possible to tau without exceeding it also turns into a case for discussing modulo. For a long time I have felt strongly that it would do a lot of good to teach modulo as a 5th arithmetic operator. Kirby has had a lot to say about that as well. 

So, how close to tau is 333 radians? 

In a Python shell:

>>> from math import *

>>> tau = 2*pi

>>> 333 % tau

6.274364026661516

Pretty close, but can we get closer?

Sure:

>>> 9563 % tau

6.275147779849341

>>> 99733 % tau

6.282804445607326

>>> 729371 % tau

6.2831719792710174

>>> 4272943 % tau

6.283184757766655

Of course there is no end, we can keep getting closer and closer indefinitely, but look at the size of the integers you have to check to find the next one that is closer! From 0 to 1000 radians, 333 gets the closest to tau without exceeding it. After that, the next number of radians that gets any closer is 99,733! I found that a little surprising.

This is not something where you can just guess and check using a calculator. You need to be able to code it. However, the code is very manageable, just a one-liner. A list comprehension.

We usually use modulo with integers, but Python accepts reals.

I find that interesting, because when you try the same thing in Sage you get an error. Sage doesn't like using '%' with non-Integers.

- Michel