## n sided polygon, n < 3

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Spambot5546
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### n sided polygon, n < 3

Upfront warning: I have a BA in math, which means I know just enough to ask really stupid questions. Questions like the one I'm about to ask.

We know that every polygon with n sides will have the same sum of its internal angles. Every triangle has a sum of angles of 180 degrees. By my math, said sum is

180n - 360

Which shows why no polygon can have fewer than three sides. A two-sided polygon's angles would have a sum of internal angles of zero, and below that all our angle sums are negative. None of that makes any sense.

But then I thought "There was a time when asking what are two numbers whose sum and product is 1" didn't have an answer that made sense either. Now, thanks to Gerolamo Cardano and Caspar Wessel we have the complex number plane and know that there really are two such numbers.

Now, I'm open to the possibility that I revolutionized geometry while taking my afternoon poop, but I think there are two more likely outcomes here. In the first, this turns out to be a known situation and there is some branch of math I'm not aware of that deals with it. In the second, it turns out there is some reason I haven't thought of why this situation cannot be compared to complex numbers. Any takers?
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Nicias
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### Re: n sided polygon, n < 3

On positively curved or non-simply connected surfaces you can have di-gons (or even mono-gons, I suppose). For instance on the surface of a sphere you can have a di-gon with any angle sum less than 2pi. On a flat torus you can also have a digon with any angle sum of less than 2pi (I think, I haven't worked out the details)

Of course, on these surfaces you have to redifine your lines for the edges of you polygons. For a sphere you would use great circles. For a flat torus you would wrap the straight lines at the boundary, for other surfaces you would use geodesics.

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### Re: n sided polygon, n < 3

I'd just point out that you actually can only make di-gons on the sphere if the two vertices are antipodal, otherwise, the only proper lines between the two points are the two sections of the great circle they both lie on.
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### Re: n sided polygon, n < 3

A digon in Euclidean space can easily be defined and doesn't really break anything important (to my knowledge), but it's pretty trivial. It looks like just a regular line segment, but is really two line segments on top of each other with their ends connected. It has zero area, it has two vertices with 0° interior angles, and its perimeter is twice its length. As for monogons...I don't see how that would work in flat space. I was tempted to say that a line may qualify, but it probably can't "meet up" with itself at infinity without being equivalent to a sphere somehow (thinking of the Riemann Sphere).

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### Re: n sided polygon, n < 3

A monogon could work in Euclidean space if sides could have zero length.

Spambot5546
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### Re: n sided polygon, n < 3

A monogon would also, according to the formula, have the sum of its internal angles be -180 degrees. While it's true that negative angles can be seen as just moving clockwise rather than counterclockwise, It seems like an odd way for it to be expressed.

I'm also interested in the idea of a 0-gon, or a -1-gon. I'm wondering how that might work, or if it could.
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### Re: n sided polygon, n < 3

There's nothing wrong with simply saying the formula only works for n at least 2 (and it does work for n=2). Lots of functions have domains that don't include everything (not least because a universal set is contradictory). We don't expect it to have meaning for n=3.5, so why take it seriously when n=1?
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dudiobugtron
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### Re: n sided polygon, n < 3

gmalivuk wrote:why take it seriously when n=1?

Because it's fun!

When n=3.5, the sum of the interior angles is 270.

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### Re: n sided polygon, n < 3

And when n=cheese the sum is 180cheese - 360.

Sure, we can all string the relevant symbols together, but that doesn't necessarily imbue them with meaning.
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### Re: n sided polygon, n < 3

You can't claim you didn't enjoy writing that, though.

---------------
Re: monogons, the proper name for them is apparently 'henagon'.

Spambot5546
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### Re: n sided polygon, n < 3

gmalivuk wrote:There's nothing wrong with simply saying the formula only works for n at least 2 (and it does work for n=2). Lots of functions have domains that don't include everything (not least because a universal set is contradictory). We don't expect it to have meaning for n=3.5, so why take it seriously when n=1?

Sure, domains are a thing. I'm just asking if the establishing the domain of n >= 2 is really necessary. Basically, is there a reason a polygon with less than two sides makes less sense than the square root of negative one?
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### Re: n sided polygon, n < 3

gmalivuk wrote:And when n=cheese the sum is 180cheese - 360.

Sure, we can all string the relevant symbols together, but that doesn't necessarily imbue them with meaning.

On the other hand, most definitions of polygons allow for a digon and some for a monogon. And that formula is said to hold for every polygon in euclidean geometry.

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### Re: n sided polygon, n < 3

When did "poly" stop meaning "many"?

Polygons in Euclidean geometry must adhere to Euclidean accounts of points and lines. How in that system do you propose to make a line with only one point?

Just like sqrt(-1) really *doesn't* have meaning in the reals, I'm fine saying monogons don't have meaning in Euclidean geometry.
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Spambot5546
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### Re: n sided polygon, n < 3

gmalivuk wrote:I'm fine saying monogons don't have meaning in Euclidean geometry.

You're trying to be dismissive, but you've actually given me exactly the kind of stepping off point I was looking for: non-Euclidean geometry. Not that I have any idea what a polygon with -2 sides might look like in a non-euclidean space (plane?), but at least I have some reference frame now!
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### Re: n sided polygon, n < 3

There's no reason to assume that the 180n-360 formula will work in such a geometry, the sum of the internal angles of a triangle isn't 180° in hyperbolic geometry, but this formula holds in euclidan geometry and that is why there's a constrain on n, monogonos have no meaning in euclidean geometry.

But of course it'll be interesting to discuss a geometry where polygons can have a negativs number of sides, if there's a meaningful way to define such a thing
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### Re: n sided polygon, n < 3

Spambot5546 wrote:
gmalivuk wrote:I'm fine saying monogons don't have meaning in Euclidean geometry.

You're trying to be dismissive, but you've actually given me exactly the kind of stepping off point I was looking for: non-Euclidean geometry. Not that I have any idea what a polygon with -2 sides might look like in a non-euclidean space (plane?), but at least I have some reference frame now!
We've already talked about some non-Euclidean geometries. The surface of a sphere is one, and there the internal angles can sum to a wide range of values. For a triangle, it's anything from 180 to 900 degrees. For a digon, anything from 0 to 720. Hyperbolic geometry is another non-Euclidean one, in which the sum of internal angles can be anywhere from 0 up to the Euclidean value.

In other words, the formula already doesn't work for normal polygons in non-Euclidean geometries, so you're even farther from being able to extract meaning from it in cases where n is less than 2.
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Spambot5546
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### Re: n sided polygon, n < 3

Well, I did warn you that it could turn out to be a stupid question...
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### Re: n sided polygon, n < 3

It wasn't a stupid question, it was just a question with an answer that was perhaps less interesting than you'd hoped.
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### Re: n sided polygon, n < 3

gmalivuk wrote:When did "poly" stop meaning "many"?

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### Re: n sided polygon, n < 3

dudiobugtron wrote:
gmalivuk wrote:When did "poly" stop meaning "many"?

Um... no.
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### Re: n sided polygon, n < 3

Just stopping by to point out that an n-gon with n=a/b (b does not divide a) has meaning if n>2, you just get a self-intersecting polygon - or rather, a star (e.g. 5/2 gives a pentagram). On the other hand, the sum of the angles of the star is actually 180(a-2b) degrees, so...
Last edited by Arcorann on Sat Oct 19, 2013 4:05 am UTC, edited 1 time in total.

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### Re: n sided polygon, n < 3

Yeah, I'd say that's a sort of abuse of notation. There is no sense in which a star lies between a digon andr a triangle. We could as easily talk about a 6/2-gon, even though they are both even, which is also familiar as the Star of David.
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### Re: n sided polygon, n < 3

There's a good generalization if you look at polygons that don't have straight edges.

At any point on a smooth curve, there is a unique circle which approximates the curve up to second order. The curvature at that point is defined to be the inverse of the radius of that circle. (A straight line is approximated by a "circle of infinite radius", so it has zero curvature.)

Then we can define a "polygon" to be a simple closed curve which is smooth except at n points (the vertices). At a smooth point curvature can be positive or negative depending on whether we curve inward or outward. Angles are defined in the obvious way at each vertex. The total curvature of a polygon is just the integral of the curvature with respect to length. Then (n-2)π = (angle sum) - (total curvature). This holds in all cases, n can be 0, 1, or 2.

A simple example is a semicircle of radius r: this is a 2-gon with two right angles. The curvature of the diameter is zero, and the curvature of the circular edge is 1/r. Integrated over the length, which is πr, gives a total curvature of π.

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### Re: n sided polygon, n < 3

gmalivuk wrote:And when n=cheese the sum is 180cheese - 360.

Sure, we can all string the relevant symbols together, but that doesn't necessarily imbue them with meaning.

I am fascinated by your new cheese-based geometry - tell me more!
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### Re: n sided polygon, n < 3

gmalivuk wrote:When did "poly" stop meaning "many"?

Polynomials anyone?

Anyway, this topic reminds me of this

On the actual question, I'm inclined to agree, there's no good reason to think that there is any interesting geometry where the concept of n-gons for arbitrary n makes sense while still satisfying an interior angle formula like Eucliden space. That's not to say such a geometry doesn't exist, just that I see no compelling reason that it ought to.

One thought is that all polygons of a type having the same interior angle sum will either force there to be not very many polygons or the space to be Euclidean-ish. Depending on your axioms it's likely to imply things like no curvature, rotational invariance, etc
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### Re: n sided polygon, n < 3

Arcorann wrote:Just stopping by to point out that an n-gon with n=a/b (b does not divide a) has meaning if n>2, you just get a self-intersecting polygon - or rather, a star (e.g. 5/2 gives a pentagram). On the other hand, the sum of the angles of the star is actually 180(a-2b) degrees, so...

The Wikipedia article on internal and external angles usefully says:-

The concept of 'interior angle' can be extended in a consistent way to crossed polygons such as star polygons by using the concept of 'directed angles'. In general, the interior angle sum in degrees of any closed polygon, including crossed (self-intersecting) ones, is then given by 180(n-2k)° where n is the number of vertices and k = 0, 1, 2, 3 ... represents the number of total revolutions of 360° one undergoes walking around the perimeter of the polygon. In other words, 360k° represents the sum of all the exterior angles. For example, for ordinary convex and concave polygons k = 1, since the exterior angle sum = 360° and one undergoes only one full revolution walking around the perimeter.

gmalivuk wrote:Yeah, I'd say that's a sort of abuse of notation. There is no sense in which a star lies between a digon andr a triangle. We could as easily talk about a 6/2-gon, even though they are both even, which is also familiar as the Star of David.

If we describe polygons using sequences of vertices, such as (5, 0), (15, 0), (20, 20), (10, 25), (0, 20) (a pentagon), and (10, 25), (5, 0), (20, 20), (0, 20), (15, 0) (a pentagram), there is no single sequence for a Star of David. But we can still map pentagons to rational numbers by dividing n by k, as long as k≠0.

I like the idea of irrational numbers being mapped to by stars that you never finish drawing, with infinitely many vertices, such as when all internal angles are the same, irrational fraction of 180° and all lines are the same length.

Using directed angles, we can also have negative external angles, polygons where k is negative, and 'internal' angles that are on the outside of the polygons. For example, compare (0, 0), (1, 0), (1, 1) with (0, 0), (1, 1), (1, 0). The formulae for total external and internal angles still work for these cases.

TwistedBraid wrote:There's a good generalization if you look at polygons that don't have straight edges.

I like this. It means we can have monogons, and zerogons, too!
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### Re: n sided polygon, n < 3

mike-l wrote:
gmalivuk wrote:When did "poly" stop meaning "many"?

Polynomials anyone?

You mean the thing which is a sum of many monomials?

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### Re: n sided polygon, n < 3

letterX wrote:
mike-l wrote:
gmalivuk wrote:When did "poly" stop meaning "many"?

Polynomials anyone?

You mean the thing which is a sum of many monomials?

Where "many" can mean as few as 1, 2, or 3? I don't see how it's any different from "polygon".

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### Re: n sided polygon, n < 3

Derek wrote:
letterX wrote:
mike-l wrote:
gmalivuk wrote:When did "poly" stop meaning "many"?

Polynomials anyone?

You mean the thing which is a sum of many monomials?

Where "many" can mean as few as 1, 2, or 3? I don't see how it's any different from "polygon".

Oh for the love of god.
http://en.wiktionary.org/wiki/poly-
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### Re: n sided polygon, n < 3

Derek wrote:
letterX wrote:
mike-l wrote:
gmalivuk wrote:When did "poly" stop meaning "many"?

Polynomials anyone?

You mean the thing which is a sum of many monomials?

Where "many" can mean as few as 1, 2, or 3? I don't see how it's any different from "polygon".
Monomials, while included in algebraic rings and such formed from general polynomials, are in most other cases distinct entities. Someone who speaks one language is not a polyglot, nor is someone knwledgeable about one field a polymath, nor is an institute that focuses on one thing a polytechnic, and so on and so on.
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### Re: n sided polygon, n < 3

gmalivuk wrote:Someone who speaks one language is not a polyglot, nor is someone knwledgeable about one field a polymath, nor is an institute that focuses on one thing a polytechnic, and so on and so on.

Cleverbeans wrote:Oh for the love of god.
http://en.wiktionary.org/wiki/poly-

You're both right. However mike-l is also right:

Wikipedia wrote:Polynomials of small degree have been given specific names.

A polynomial of degree zero is a constant polynomial or simply a constant.

Polynomials of degree one, two or three are respectively linear polynomials, quadratic polynomials and cubic polynomials.

For higher degrees the specific names are not commonly used, although quartic polynomial (for degree four) and quintic polynomial (for degree five) are sometimes used. The names for the degrees may be applied to the polynomial or to its terms. For example, in x^2 + 2x + 1 the term 2x is a linear term in a quadratic polynomial.

(From here.)

Polynomials are not defined to be equations with >=3 terms only, so it's not clear a-prior that polygons would necessarily be limited to >=3 sides.

Maths is partially about generalizing, and sometimes the 'plural' terminology is used for singular situations. It's just how it is, and, to my mind at least, it's a perfectly reasonable shorthand for mathematicians to use.

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### Re: n sided polygon, n < 3

Firstly, monomials are definitely still polynomials, and the very existence of the term degree 0 polynomials guarantees that there are at least some single term polynomials (since a degree 0 polynomial is a priori only 1 term).

Secondly, just because you named something something doesn't restrict what it can be. (oblig http://imgur.com/ZVR36) Just because you named something 'many sides' doesn't mean the concept can't apply to things that don't have many sides. In fact, mathematics is largely about looking at things in ways that don't originally make sense.

So yes, obviously the prefix poly means 'many', but just because something has the prefix poly doesn't mean we are forever bound to only consider things that satisfy the 'many' condition.
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### Re: n sided polygon, n < 3

Yep.

There are some contexts where we wouldn't want the prefix "poly-" to include "only one". Some such contexts are in everyday language, such as the words "polyglot" or "polymath" or "polyamory".

There are some contexts where we do want the prefix "poly-" to include "only one". A word like "polynomial" is such an example. That way, we can say things like "The sum of two polynomials is a polynomial", and not have to make exceptions for things like (5x^2 - 3x + 1) + (8x^2 + 3x - 1).

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### Re: n sided polygon, n < 3

skullturf wrote:That way, we can say things like "The sum of two polynomials is a polynomial", and not have to make exceptions for things like (5x^2 - 3x + 1) + (8x^2 + 3x - 1).

(5x2 - 3x + 1) + (8x2 + 3x - 1) = 13x2 + 0x + 0.

I've tended to sometimes think of single and zero term polynomials as polynomials where some or all of the coefficients are zero.

We can translate single term polynomials of degree greater than zero, such as 13x2, to give polynomials with more than one term. For example, let u=x+1 => x=u-1 => 13x2 = 13u2 - 26u + 13, which is a polynomial.
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### Re: n sided polygon, n < 3

FancyHat wrote:
skullturf wrote:That way, we can say things like "The sum of two polynomials is a polynomial", and not have to make exceptions for things like (5x^2 - 3x + 1) + (8x^2 + 3x - 1).

(5x2 - 3x + 1) + (8x2 + 3x - 1) = 13x2 + 0x + 0.

I've tended to sometimes think of single and zero term polynomials as polynomials where some or all of the coefficients are zero.

We can translate single term polynomials of degree greater than zero, such as 13x2, to give polynomials with more than one term. For example, let u=x+1 => x=u-1 => 13x2 = 13u2 - 26u + 13, which is a polynomial.

You can do that*, but if it is to somehow make it jive with the name polynomial, I don't really see the point. Names shouldn't constrain concepts, they should label them. Depending on the topology at hand, any function can be continuous; we aren't obliged to try and find a way to make that fit with the idea of a "continuous curve".

*Depending on context, I'm not sure that you could say you were dealing with the "same" polynomial after substitution; if you aren't talking about polynomial functions and are being ultrapedantic. Also, if you are talking about polynomial functions, then you run into trouble in certain fields: x3 + x2 + x = x (mod 2), in which case you need to say that some not single term polynomials are monomials, which is another problem of the same form.
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### Re: n sided polygon, n < 3

I mean, if you're going to be pedantic and call a bunch of 0 terms the other monomials in a polynomial, then I can just talk about polygons as having some existing sides and some non existing sides in the same manner (and, for the record, we do do this, see for example homology)

But the entire line of questioning is stupid. The answer to 'does this idea make sense for n < 3' is not 'no because the name someone happened to pick for it requires n >=3'. Good reasons have been posted in this thread (even in the post that started the line of questioning!), but this was not one of them.
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