My maths teacher always keeps telling me about this 'cutting edge maths' that is going on in the world, amazing maths research, etc. A lot of the google searches I've done for 'Cutting Edge Mathematics' hasn't returned much useful information, so I've taken to mathematics stack exchange.
That would be the same as just cutting a square with a line. What the questions asks for, is the shape of the two new internal faces that appears after the cut. For example, if you cut along an edge to the opposite edge, you would get a rectangle of dimensions $1\times\sqrt2$. It is the intersection of the cube and the plane. I speak Danish by ...
The edge orientations that are specified by the delta-complex structure mean that you end up with something that needs a little cutting and gluing to see that it is your friendly ordinary klein bottle.
This is like a person who is just learning to count asking for a roadmap to integral calculus. Focus on completing the core graduate material in analysis, geometry, and algebra -- the cutting-edge stuff will come in due time.
For such nuances you have to know what your instructor thinks the right definition is. It's not an important enough thing for most people to care about, honestly. (No one is doing cutting edge research in Calculus or analysis that would suddenly be invalidated if someone had a different definition of inflection point.)
Think about a line parallel to an edge of the triangle and cutting it into two parts of equal area.
BGR on MSN: 10 Obsolete Media Formats That Used To Feel Cutting-Edge