MA-3329 Modern Geometry Name:
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Gauss-Bonnet Day (Gauss rhymes with mouse, and bonnet sounds like
"bow" + "nay".)
The Gauss-Bonnet theorem states that if
we trace out a closed curve on a surface, the total enclosed Gauss curvature
(total angle deficit) is 2Pi minus the total angle defect (angles of deflection)
around the curve (measured in radians, naturally).
Today, we will consider some
examples of the Gauss-Bonnet theorem. We
will start with surfaces having elliptic or hyperbolic 'cone' points as in our
previous labs, but this theorem works for any continuous surface!
When you were asked to draw lines on your balloons, you instinctively
drew the most logical thing you could think of, but the correct term is geodesic. A geodesic is the shortest distance between
two points. Lines of longitude are
geodesics, but lines of latitude are not!
For a sphere, you can find the shortest path between two points by first
finding the plane which contains the center of the sphere and the two
points. That plane will cut the sphere
in half along a great circle (like
the equator) and will give the geodesic between the two points.
When you drew the triangles
on your paper models, you correctly drew geodesics, because you laid your paper
flat to draw the lines, thus finding the shortest distance between two points.
1) On a blank piece of
notebook paper, draw a straight line. Crease the paper so that the line is on
both sides of the crease. The line you
drew was and still is a geodesic, even though there is a crease in the
paper. You may fold paper, roll it up,
or even crumple it up, but the line you drew will still be the shortest way to
get between any two points on your line!!!
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The quadrilateral in this
picture is made of four geodesics.
There are two 72 degree angles and two 120 degree angles. 2) What is the total angle
defect for this quadrilateral? 3) Find 360 degrees minus
angle defect in degrees: 4) What is the angle
deficit in degrees for the vertex inside the quadrilateral? |
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The triangle in this
picture is made of three geodesics.
Each angle is 84 degrees. 5) Find 360 degrees minus the
total angle defect for the triangle. 6) How many vertices are
located inside the triangle? 7) What is the angle
deficit for each vertex? 8) What is the total angle
deficit in degrees inside the triangle? |
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One way to visualize the
total angle defect is to consider what a ribbon would have to do in order to
remain flat along the surface. Each
time the ribbon has to 'turn', folds must be made in the ribbon to form an
angle so the ribbon can remain flat. 9) Calculate the total angle
defect for this ribbon, assuming that the centers of the hexagons are
connected to form a pentagon. (Hint:
Draw one hexagon and find the angle of the ribbon which lies on it.) 10) Find total angle
deficit in degrees for the 5 vertices inside the ribbon. |
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The paper ribbon on this
cube lies flat and encloses four vertices. 11) What is the total angle
defect for a curve which never turns? 12) What is the total angle
deficit in degrees and in radians (Gauss curvature) inside the ribbon
according to the Gauss-Bonnet theorem? |
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One rule we must observe when
laying out the 'ribbons' on the surfaces is that we may not cut through any of
our 'cone' points. You cannot lay a ribbon
flat through a vertex.
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The curve shown on this star-object
has total angle defect of zero and cuts the object into two portions. The entire object has 12 hyperbolic cone
points, one of which is enclosed on the visible side of the curve. Hyperbolic cone points are comprised of 10
planar sections, each with 15) Verify the Gauss-Bonnet
theorem for the curve on this object (in degrees or radians) for both the
visible portion of this object and for the remaining portion. (You should use other paper as needed!) |
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The photo of the
star-object might as well be replaced by this diagram. If the point c is the hyperbolic cone point
described above, and the remaining points are the elliptic points described
above, you may answer the following question (assuming all other points
inside the curve have are planar): 16) What is the total
deflection of the curve? |
The same diagram can be used
to pose any number of questions such as:
In the given diagram, points a, b, and c are each cone points with angle deficit of 
/5, point d has an
angle deficit of 2/5, and points e
and f each have an angle deficit of -/6.
17) What is the total angle of deflection
for the curve?
18)
If the diagram represents a portion of an object which is topologically
equivalent to a sphere, what is
the total curvature of the object on the 'outside'
of the curve?
19)
If the diagram represents a portion of an object which is topologically
equivalent to a torus (or square
donut) what is the total curvature
of the object on the 'outside' of the curve?
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In the following diagram,
angles are measured in radians, the only non-planar points within the curve
are labeled with their Gaussian curvatures, and the curves shown are all
geodesics. 20) What is k? 21) Is there a 'net'
hyperbolic effect inside the curve or a 'net' elliptic effect? 22) If this diagram is part
of an object which is topologically equivalent to a sphere, and there is only
one cone point outside the curve, what is the Gaussian curvature at that cone
point? |
23) If the diagram is part of
an object which is topologically equivalent to a torus, and there are two cone
points outside the curve, each with equal Gaussian curvature, what Gaussian
curvature does each cone point have?
(Note a cone point may be
either hyperbolic or elliptic.)