Definition of Opposite SidesDate: 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/ |
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