# Circle becomes a pair of parallel lines

• zmodnz
In summary: So there's a non-Euclidean geometry where 2 parallel lines may meet at 'infinity'?3) What is a pseudometric sphere?In summary, the circle becomes a set of parallel lines in a non-Euclidean geometry.
zmodnz
Does anyone know if there is a non-Euclidean geometry(or something like that) where the circle becomes a line or a pair of parallel lines?

...or even where the circle becomes a set of parallel lines?

...i'm doing some work in Guitar Theory and this situation appears...

Thanks

The Roman Empire Guitar Research Institute

It's hard to believe you are serious, but no, all different "geometries" respect the fundamental topological properties of sets. In particular, if a set is connected (as is a circle) in one geometry, it will be connected in any geometry (and "parallel lines" are not connected).

I'm serious!

I study a little Math also...I'm more on the Algebra 'side'...(i finally reached Group Theory proper...thank God!)

BUT isn't there something about something becoming great circles...or something about great circles becoming something...etc...etc...?

Isn't there a situation where a line becomes a great circle?

Thanks

zmodnz said:
I'm serious!

I study a little Math also...I'm more on the Algebra 'side'...(i finally reached Group Theory proper...thank God!)

BUT isn't there something about something becoming great circles...or something about great circles becoming something...etc...etc...?

Isn't there a situation where a line becomes a great circle?

Thanks
If sounds like you're thinking of the geometry on a sphere--I've heard that geometry be referred to as "Riemannian" geometry, but I've also heard "Riemannian" refer to a much more general class of geometries. In the geometry on a sphere, the geodesics ("straight" lines) are great circles of the sphere. Doesn't that intuitively seem like the shortest way to get from one point to another?

zmodnz said:
I'm serious!

I study a little Math also...I'm more on the Algebra 'side'...(i finally reached Group Theory proper...thank God!)

BUT isn't there something about something becoming great circles...or something about great circles becoming something...etc...etc...?

Isn't there a situation where a line becomes a great circle?

Thanks
Yes, of course, in spherical geometry our "lines" are the great circles of the sphere. But that is still a connected set and has nothing to do with "the circle becomes a set of parallel lines" as in your first post.

But isn't there a non-Euclidean geometry that says that 2 parallel lines may meet at 'infinity'?

(Side: and if so, then isn't 2 parallel lines connected?------Connected at infinity?)

Or am i just wishing for too too much?...i don't even have the definition for "connected".

The situation: We have the guitar fretboard...we 'made' it into an infinite rectangular grid...we found the X and Y axes so every point/node on the grid can be located...the axes are not orthoganal...distances are measured in "semitones"...when we plot a circle it comes out to be a pair of parallel lines...when we compare other theorems to Euclid we realize that we have here some non-Euclidean situation going on.

Any guitar mathematicians here?...ha.

Thanks.

The Roman Empire

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You're sort of correct, except keep in mind that its the same infinity in both directions, so we have four lines coming together at a point, and two lines spanning the plane. Kind of (exactly) like straps of a backpack wrapped around your body.

What you're discussing is different, though! You're talking about metric spheres, (or pseudometric spheres, in this case) which are the set of points a given distance (number of semitones) away from a given point. It's not a "sphere" geometrically or topologically, it just has that property of containing all the points a given distance from the "center."

You get two lines because each string is identical to the last, but shifted by a fifth (or fourth, depending on your direction). The "current" string has two points (count off your interval up and down. Each other string has those same two points shifted by five frets (except for that pesky g-b), so you get two points there.

Your space is discrete (each point is "separate" from the others, rather than connected by a continuous array of points) so no subset of it at all is homeomorphic to a Euclidean circle. You do, however, have metric circles.

On last point! The reason you have a pseudometric, not a metric, is becauase metrics require that "no two points lie at the same point." For any point, there are infinitely many other points (one on each string) that are "0 semitones away." In a metric space, only a point and itself may be 0 distance apart. This means that circles of radius 0 in metric spaces are just a point. In your space, the circle of radius 0 is a single line, since the two parallel lines come together.

Yeah Alexfloo! You're correct on these points.

This really helps me a lot.
It really focuses my/the investigations a lot.
It shows me some/the limitations of 'the situation'.

I /we have been formulation this situation upon the guitar fretboard for decades now.
We can write volumes.
But you seem to have quite a hold on this situation!

Some points:

1) O.k. i see how to interpret 'infinity' now(...wonder if all mathematical realms interpret it this way?...)...i think remember seeing something like this interpretation in class...dealing with Reimann sphere stuff...

Side: i just wrote my other points...pressed 'submit reply'...then realized that i wasn't connected...

so i lost my data!

will rewrite them tomorrow when i get some chemical energy back...

So yes...

2) Thanks. I did some research on the net and checked out definitions of 'connected', metric space...etc...And indeed, it is a metric space OR a pseudometric space that i/we have stumbled upon!

my problem is/was that I've been calling 'the object' a line(continuous) now for so so long that i actually came to believe that it really was a line. Ha. Fancy that.

Thanks for awakening me...ha.

But there's no secret formulay for transforming the discreet into the continuous YET!...ha.
3) If we interpret the situation as being collapsed onto only 1 string...we indeed have a metric space...maybe this is an obvious 'trivial case' and I'm making a fool of myself..?

If we interpret it on the usual 6 strings(or on the infinite plane of strings(maintaining the original ordering on the guitar fretboard..etc..)) we have a pseudometric space. Indeed.

Thanks HallsofIvy & Alexfloo...sorry for thanking you guys so much but this realization za'gonna save me a lot of twirling around in circles.4) Here is the distance formula for the metric/pseudometric:

If P1 is (x1, y1) and P2 is (x2, y2) ----> d = abs(x1) + abs(x2) + abs(y1) + abs(y2)

abs(x)...absolute value of integer x.

If anyone has any information on such a distance formula we'd be quite happy to have it...ha. ...etc

ThanksThe Roman Empire Guitar Research Institute of Montreal, Canada

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## 1. What is a "circle becomes a pair of parallel lines" phenomenon?

The "circle becomes a pair of parallel lines" phenomenon refers to a mathematical concept where, in certain conditions, a circle will appear to transform into a pair of parallel lines.

## 2. How does a circle become a pair of parallel lines?

This phenomenon occurs when a circle is viewed at an angle that is perpendicular to its plane. This creates an illusion where the curved shape of the circle appears to flatten and become two straight, parallel lines.

## 3. Is this phenomenon only applicable to circles?

No, this phenomenon can also occur with other curved shapes, such as ellipses and parabolas, when viewed at the correct angle.

## 4. What are the practical applications of this phenomenon?

While this phenomenon may seem like a simple optical illusion, it has practical applications in fields such as computer graphics, where it is used to create 3D effects.

## 5. Can this phenomenon be explained by science?

Yes, this phenomenon can be explained by the principles of perspective and geometry. It is a result of the way our eyes and brain perceive and interpret visual information.

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