In math, three-dimensional space sprawls out to infinity in every direction. With an infinite amount of room, it should be able to hold an infinite number of things inside of it — pearls, peacocks or even planets. The result underscores the delicate endeavor of situating surfaces in space, as well as the intuition-challenging nature of infinities.

It is easily formed from a strip of paper by giving it a half-twist before joining the ends. It is used in the design of necklaces, brooches, scarfs, etc. They have been used as conveyor belts: the entire surface area of the belt gets the same amount of wear, giving the belts longer lifetimes.

Last session we proved that the graphs and are not planar. As an immediate corrolary, we see that is not planar foras all such complete graphs contain as a subgraph; similarly, are not planar, with. Our second observation is the following: suppose we took a graphand made a new graph by adding one vertex of degree 2 in the middle of one of the edges of.

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Recall that a graph is planar if it can be drawn on the plane without edge crossings. A graph is planar if and only if it has neither nor as a minor:. Here refers to the complete graph on 5 vertices, and is the complete bipartite graph on two sets of three vertices.

Try to draw a line on both "sides" without picking up your pencil. It's actually quite simple. That is, when we define a surface normal at a point, it is impossible to extend the definition to the whole surface.

A model can easily be created by taking a strip of material and giving it a half-twist, and then merging the ends of the strip together to form a single strip. If you cut down the middle of the strip, instead of getting two separate strips, it becomes one long strip with two half-twists in it. If you cut this one down the middle, you get two strips wound around each other.

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