Definition of Opposite Sides

Date: 01/18/2001 at 11:37:11
From: Stephen Stortz
Subject: Definition of "opposite sides"

I need a formal definition of 'opposite sides' of a polygon that will 
address such issues as whether a regular pentagon has opposite sides. 
Also, does a concave polygon have opposite sides?

I tried to use "sides that can be connected by a line segment 
perpendicular to each" but that rules out parallelograms. I try to 
discourage my students from using "they just look opposite" as 
justification, but I have not hit on a formal definition. It's not 
indexed in our math textbook.

Thanks.

Date: 01/18/2001 at 12:39:20
From: Doctor Peterson
Subject: Re: Definition of "opposite sides"

Hi, Stephen.

It might help to know the context in which you have seen the phrase; 
but I would say that, in a polygon with an even number of edges, the 
side opposite a given side is the side that is separated from that 
side by the same number of sides in each direction. Perpendicularity 
or convexity is not required, only an even number of edges. You could 
define it a bit more formally by saying that, for a 2n-gon, the 
opposite side is the nth side counting from the given side in either 
direction.

In a pentagon, there is a vertex opposite each side, but not a side 
opposite a side.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   

Date: 01/18/2001 at 17:04:27
From: Stephen Stortz
Subject: Re: Definition of "opposite sides"

The context is this: One of my students wanted a general term for 
polygons that are created entirely of pairs of parallel opposite 
sides, so we called it a Calligram (in honor of her name). I wrote the 
following definition of a Calligram: "a polygon whose opposite sides 
are parallel." Then I asked the students whether the following would 
be properties of all Calligrams:

     1) They must have an even number of sides.

     2) The measure of the exterior angles is always even.

     3) They are convex.

Properties (2) and (3) are obviously false, but property (1) gave rise 
to the question; "what about a pentagon with two right angles?" If the 
two parallel sides count as 'opposite', then all the opposite sides 
are parallel, and the remaining three sides don't count as opposites 
(I would then have to reword my definition). I cannot get a formal 
definition for "opposite," however. If it means parallel sides, then 
that would give rise to the situation where a trapezoid has only one 
pair of opposite sides. By your definition, you could get some pretty 
mean looking concave polygons with two sides called 'opposite' even 
though they could be contained on the same line. Everyone knows that 
"opposite sides of a parallelogram are congruent" for instance, but 
again, I do not have a formal definition of 'opposite'.

Thanks for your time
Steve Stortz

Date: 01/18/2001 at 22:31:27
From: Doctor Peterson
Subject: Re: Definition of "opposite sides"

Hi, Stephen.

I love this sort of open-ended problem, where you're all discovering 
together. I seem to recall discussing a similar concept once, which 
either the writer or I called a "parallelogon," but I can't find that 
in our archives. I just did a Google search for "opposite sides 
equal," and found these, which might be of interest, though both 
require opposite sides to be congruent as well as parallel for their 
discussions (look for "zonogon" in the first):

  Zonohedrification - George W. Hart
  http://www.georgehart.com/virtual-polyhedra/zonohedrification.html   

  Parahexes - Barry Schnorr
  http://www.imsa.edu/edu/math/journal/volume4/webver/parahex.html   

Another site that defined zonogons called them 2p-gons, which supports 
my contention that you can only talk about opposite sides when there 
is an even number of sides.

One thing you're learning together is why mathematicians have to 
define all their terms before they can state, and especially prove, 
conjectures. If you can all agree on some definition of "opposite," 
that would work fine for your purposes. But I don't see anything wrong 
with my definition, though it's probably one of those things that we 
tend to assume we all understand. Certainly your definition that 
opposite means parallel makes your whole conjecture circular, and to 
count a pentagon as fitting your definition when by your definition 
any sides that aren't parallel simply don't have opposites seems 
really odd.

I 'm a little curious about your third conjecture - do you have a 
counterexample? Using my understanding of your definition, I'd say 
it's true, though I don't have a proof immediately. Ah - here's a 
counterexample, which I just got by imagining three pairs of parallel 
lines moving around to form a shape:

     +-------------+
      \           /
       +         +
      /           \
     +-------------+

This also leads me toward an example of what you mentioned, a polygon 
with opposite sides that are collinear. Nothing wrong with that, since 
such a polygon is pretty twisted in other ways, too. The sides are 
definitely opposite.

Have fun!

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   

Date: 01/19/2001 at 09:10:32
From: Doctor Peterson
Subject: Re: Definition of "opposite sides"

Hi again!

I've been thinking about your question more, because this is a great 
opportunity to explore how math is done, at a basic level. The problem 
you have, of course, is that the concept of "opposite" is so "obvious" 
that mathematicians, as far as I can see, don't bother to state the 
definition they all assume is well known. That gives you a chance to 
work out a "new" definition as if you were on the cutting edge of 
math.

How do we define a mathematical concept? Generally, we start with 
common sense, then make it mathematical by turning that into a precise 
and general definition. In this case, "opposite" is a natural concept 
if we start by looking at a circle. In fact, any definition we choose 
will identify the same point as opposite a given point: halfway around 
the circumference; having a parallel tangent; at the other end of a 
diameter, which divides the interior into equal halves; or whatever 
you want to say.

Now move to a regular polygon. We lose just a little bit of 
generality, because we see that only with an even number of sides can 
we call one SIDE opposite another; but otherwise all the definitions I 
can see yield the same result. We haven't clarified the definition at 
all.

Now make the polygon slightly irregular, and we're forced to make some 
decisions. There probably won't be any parallel side, so that's out. 
Halfway around the perimeter, or cutting the polygon into equal 
halves, may give you an intuitively "opposite" side, but may also give 
a vertex, leaving the choice uncertain - and both ways would be very 
hard to calculate. Actually, once the sides have different lengths, 
such a definition applies only to points not sides; and that's the key 
to our choice. Since we're talking about sides, our definition ought 
to relate to sides. So we go back to the most basic possible 
definition, one that relies only on counting sides - count half the 
sides, and you're at the opposite side. This is the "topological," 
rather than "metric" definition - one that doesn't depend on 
measuring any distances, but only on how the sides are connected. For 
some special purposes a different definition (especially for 'opposite 
point') might be useful, but since we're accustomed to thinking of 
polygons topologically, this feels so natural to most mathematicians 
that we don't bother mentioning it.

Now here's where things get tricky. Once we've settled on a 
definition, we have to follow it where it leads - just as, having 
taken the common-sense idea of a line into an abstract world where a 
line has no color, no thickness, and no ends, we have to accept that 
it takes an infinite set of points to make one. What happens if we 
look at a REALLY irregular polygon? Then our definition of opposite 
will start to feel less right. For example, take a hectogon (100 
sides) and cut it in half to make a semi-hectogon, with 51 sides, one 
long and 50 very small:

                           a
     *-------------------------------------------*
     *                                           *b
      *                                         *
      *                                         *
       *                                       *
        *                                     *
         *                                   *
          **                               **
            **                           ** c
            c ***                     ***
                 ******         ******
                       *********
                         b a

The opposite of the long side is, appropriately, the short side at the 
bottom (a->a); but the opposite of the small side at the upper right 
is the side just to the left of the bottom (b->b), which seems a lot 
less 'opposite'. Does that mean our definition is bad? No, just that 
we've defined opposite in terms of counting, and when we deal with 
different size sides, that won't match a metric definition. We should 
expect a pathological shape to be less intuitive than a "natural" 
shape.

Interestingly, your Calligram is by definition one for which two 
definitions of 'opposite' agree - not an uncommon way to define a 
special kind of object. You sense that 'opposite' ought to mean at 
least approximately parallel (as in a circle), so you ask about shapes 
that work that way. The problem you've had is an unwillingness to be 
inflexible and hold to a topological definition of oppositeness while 
you compare it to a Euclidean version; you let the latter leak into 
the former while you work, until your definition of a Calligram 
becomes so circular that, in the extreme, we could claim that any 
polygon with NO parallel sides is a Calligram, because no two sides 
are parallel ('opposite'), and therefore all of the opposite sides 
(that is, no sides) are parallel.

As I said, this shows how important definition is in math.

Let me know if you or your students come up with any new insights.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   

Date: 01/19/2001 at 13:45:29
From: Stephen Stortz
Subject: Re: Definition of "opposite sides"

Dr. Peterson,

We did have a good discussion again today with a few insights. I saw 
that the definition you provided - counting sides - is really 
topological, but it is also very pleasing because it can be 
generalized to points and angles. So a side (or point, or angle) is 
opposite another side (or point, or angle) if there is an equal 
number of sides between the two counting in both directions. So we 
preserve the idea that in a pentagon there is a vertex or an angle 
opposite each side, but not a side opposite.

One approach that a colleague suggested to try to preserve the common-
sense feeling of opposite was to draw an axis through the "middle" of 
the polygon and define two sets of opposition - X-opposite and 
Y-opposite. Of course, rotating the figure destroys the relationships, 
so I think we set that notion aside.  

The introduction of the idea of a zonogon (all pairs of opposite sides 
parallel and congruent) looked as if it might make the term Calligram 
superfluous until we found the hexagon created by lopping off the 
corners of a regular triangle. Parallel opposite sides then did not 
imply equal opposite sides, and the Calligram (or whatever someone 
already named it) is preserved as a unique set.

Still, it surprises me, after reading the definitions for things like 
'open', 'closed', 'bounded', etc. that 'opposite' wouldn't show up in 
a glossary. Especially when you think of all the geometric theorems 
that use the term - "the side opposite the largest angle in a 
triangle..." etc. If a published formal definition pops up, please let 
me know.

One other question - is there a shorthand symbol for 'supplementary'? 
I let my kids use the lightning S like the symbol used in the name of 
rock bands, AC/DC for example. I would rather not write 'sup.' any 
more then I would want to write 'eq.' for '='.

Thanks again for the response. I am glad to have provoked some thought 
- that's what math is all about.

-Steve Stortz

Date: 01/19/2001 at 23:19:12
From: Doctor Peterson
Subject: Re: Definition of "opposite sides"

Hi, Stephen.

I agree that it seems a little odd that we don't bother to define 
'opposite' precisely; but the comparison to 'open' or 'bounded' is not 
really fair, since in those cases a simple word is being given a 
complicated definition, while 'opposite' is a simple word being given 
the simplest possible definition. Also, the definition probably 
doesn't enter into any proofs in such a way as to require clarity; you 
only need the definition to see what someone is talking about. That's 
important, of course, but there is a lot of ordinary language that 
mathematicians use that doesn't need definition just because we use 
them enough to know we are using them the same way. Often it's only in 
talking to kids that we realize we don't know how to define a word.

I'm not familiar with any symbol for supplementary - unless, of 
course, you use 'A+B = 180'.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/