# Why are parallelograms defined as quadrilaterals? What term would encompass polygons with greater than two parallel pairs?

It seems the definition of a parallelogram is locked to quadrilaterals for some reason. Is there a reason for this? Why couldn't a parallelogram (given the way the word seems rather than as a mathematical/geometric construct) contain greater than two pairs of parallel sides? In a hexagon for example, all six sides are parallel to their opposing side. Is there a term for this kind of object?

It seems to me there must be some value in describing a polygon with even numbers of sides in which the opposing sides are parallel to each other. While a hexagon, octagon, decagon, etc. all match this rule, you could have polygons with unequal sides as well.

Edit 1: Object described by Mark Fischler

Zonogon:

• Indeed, I see no reason why the word "parallelogram," which has origins in Middle French where it refers to "bounded parallel lines," should have come to mean specifically 4-sided plane figures. In solid geometry, again the term "parallelpiped" is reserved for six-sided figures, now with 3 pairs of parallel opposite faces. – Mark Fischler May 9 '19 at 21:37
• @MarkFischler Yes basically! Also, your comment briefly hurt my brain at the switch from 2D to 3D terminology of 'sides' (2D: side = edge; 3D: side = face). – duct_tape_coder May 9 '19 at 21:43
• @ Mark Fischler What word did Euklid use for "parallelogramm"? – user May 9 '19 at 21:44
• I have added an answer to what Euclid called them in my answer below; the comments don't seem to speak pasted Greek. – Mark Fischler May 9 '19 at 22:03

Interesting question. Parallelograms are quadrilaterals for historical reasons. They could have been defined to include your examples, but weren't. Now the meaning is so common that it can't be changed.

I don't think there is a name for your class of polygons. The reason is in this:

It seems to me there must be some value in describing a polygon with even numbers of sides in which the opposing sides are parallel to each other.

If there were some value - if these polygons came up often in geometry - then someone would have named them. If you have interesting things to say about them and publish your thoughts you'll invent a name in your paper. If it's widely read the name will stick.

I thought parallelogon would be a good possibility, but that name is taken: https://en.wikipedia.org/wiki/Parallelogon .

The convex polygons whose sides come in equal parallel pairs are zonogons: https://en.wikipedia.org/wiki/Zonogon . Your polygons have zonogons as nontrivial Minkowski summands.

• Thank you for your answer. The zonogon concept is fascinating. The question was a bit more r/showerthoughts than math.SE (I'm no mathematician) and I'm quickly out of my depth but you've cut to the quick of my question. I think Mark's suggested object (I had to draw it to understand it so I've pasted it to my question) creates even more questions I can't answer. – duct_tape_coder May 9 '19 at 22:09
• I strongly recommend you reconsider the sentiment that "If there were some value ... then someone would have named them." Experimentation and exploration of math concepts should never be curtailed by that line of thinking. – Zimul8r May 10 '19 at 3:55

I'm going to propose, out of the blue, terms like "hexaparallelogram", "octaparallelogram", and so forth.

I'm wondering whether, for more than $$4$$ sides, you would like your definition of hexaparallelogram to be restricted to having 3 pairs of parallel and pairwise equal sides (as in your picture - evidently these have a name, zonogon), or would you include a hexagon with vertices at $$\{(0,0), (12,0), (16,6), (4,12), (0,12), (-6,3)\}$$ which has three pairs of parallel sides but no two sides of equal length?

Euclid, in proposition 34, introduces the term (παραλληλόγραμμα χωρία) which we can translate to "parallelogrammic area." So much for the etymology sites that trace the word only to Middle French. Euclid himself restricted the word to just four-sided figures. Proclus credits Euclid with having introduced the term "parallelogram," as opposed to bringing down that term from earlier works. So that tells us who to blame.

• +1 interesting history. Your naming convention would require advance knowledge of the number of edges. What about "ultraparallelogram" or "megaparallelogram"? – Ethan Bolker May 9 '19 at 22:07
• Thank your for your answer as well. I've drawn (as best I understood it) and posted your object in the question. The zonogon was sort of what I was thinking originally (n-sides) but your object with sides of different lengths is even more fascinating. Thank you for the etymology as well. – duct_tape_coder May 9 '19 at 22:13
• What is the etymology of zonogon? – Anush May 10 '19 at 4:15