Take out a sheet of paper. Pop quiz! (just kidding).
Draw any number of dots on your page.
Now connect the dots with lines, subject to the following
rules: lines may not cross each other as they move
from dot to dot, and every dot on your page must be
connected to every other dot through a sequence of
Now count the number dots (D), lines (L), and
regions separated by lines (R). (Don't forget to count
the outside as a region too.)
Compute D-L+R. What do you get?
No matter how you started, the
number you will always get is 2!
In Figure 1, D=9, L=12, R=5, and indeed, D-L+R=2.
If the lines represent fences, and the dots fenceposts,
then the regions separated by the fenceposts are
the pastures. So, if you are a farmer who wants to
fence off 4 pastures together with 55 sections of fence,
you can calculate exactly how many fenceposts you need,
no matter how you arrange the fences!
(L=55, R=5=4+outside, so D=2+L-R=2+55-5=52 fenceposts.)
You may wish to have everyone shout out their answer
at the same time... students will be surprised they all get
the same answer.
The Math Behind the Fact:
The number D-L+R is called the Euler characteristic
of a surface.
It is an invariant of a surface,
meaning that while it looks like it may depend on
the system of fences you draw, it really does not
(as long as every pasture, including
the outside, is topologically a disk with no holes).
Thus the number only
depends on the topology of the surface that you are on!
For planes and spheres, this number is always 2.
How to Cite this Page:
Su, Francis E., et al. "Euler Characteristic."
Math Fun Facts.