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Monogons and Digons - Polygons with Fewer Than 3 Sides

Date: 01/24/2006 at 13:22:38
From: Jay
Subject: monogon and digon polygons ???

What do monogon and digon polygons look like?  One of my students
obtained the list of polygons from your site and asked me about those
two polygons.  Frankly, I do not know.  I have always thought a
polygon had to contain "at least 3 sides".  I teach 7th grade math and
would like to have an explanation for my students about those two
shapes.  Thanks!



Date: 01/24/2006 at 14:10:46
From: Doctor Peterson
Subject: Re: monogon and digon polygons ???

Hi, Jay.

If you search our site for "digon", you'll find this, which explains 
it a bit:

  One- and Two-sided Polygons
    http://mathforum.org/library/drmath/view/54142.html 

The idea is that, although real polygons in the usual sense can't 
have one or two sides, we reserve those names for cases where we 
extend the definition of polygon and CAN have monogons or digons.  One 
example, mentioned there, is in spherical geometry, where "straight 
lines" are great circles, so a polygon is bounded by parts of great 
circles.  Then the nothern hemisphere is a monogon (taking any one 
point on the equator as the one vertex); and the region bounded by 
two longitude lines (say, from the prime meridian to the meridian 
through New York) is a digon, with vertices at the poles.

Another example is in topology, where we don't pay attention to 
whether "lines" are straight in the first place, but just look at 
the way they connect.  If the "sides" are allowed to bend, it's easy 
to draw a monogon or digon.

It's interesting to realize that the basic definitions kids learn 
are often stretched and twisted in higher math, as we try out "what 
if" questions like these: what if we aren't drawing on a plane?  What 
if we don't require sides to be straight?  In each case, some 
properties remain valid, while others change, and we discover new 
features of well-known ideas like "polygons".  Math is a lot more 
dynamic than most students ever get to see, and this is just a taste!

If you have any further questions, feel free to write back.

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



Date: 01/24/2006 at 14:29:44
From: Jay
Subject: Thank you (monogon and digon polygons ???)

Hi Dr. Peterson,

Thank you very much for the very quick and detailed explanation.  I am
very impressed!

I tell my students to visit your site often and have it bookmarked on
my classroom computers.  This is a perfect example of why I feel so
confident making that recommendation to them.

Respectfully,

Jay
Associated Topics:
College Non-Euclidean Geometry
College Polyhedra
High School Non-Euclidean Geometry
High School Polyhedra

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