Projective Plane 


With ordinary surfaces we have the idea of a point, and the idea that some points are close together. Using those two ideas we can picture the idea of starting at one point and moving from point to point, always taking small steps, never leaping to a point that's not close, and eventually getting somewhere else. We might even return to our starting position.
We can generalise this idea by imagining the collection of infinitely long lines that all pass through a single point in space. We can't tell which way a line is facing. Some lines are close together, so we can imagine starting with one line and moving from line to line, always taking small steps, never leaping to a line that's not close, and eventually getting somewhere else. We might even return to our starting line, and when we do so the ends may have swapped.
This second object is rather odd. If we take three lines in a sort of small triangle, A, B and C (labelled clockwise) and move them around until their ends have swapped, we can get them all close to where they started, but now they'll be labelled anticlockwise. This is the same idea as starting from one place on a Moebius Strip and going around the loop once and ending up on the "Wrong Side", whatever that means.
This collection of lines passing through a single point, along with the obvious idea of "closeness," is the twodimensional Projective Plane. The Cross Cap is one representation of a 2D Projective Plane. It can be obtained by gluing the edge of a Moebius Strip to the edge of a disk, or by taking a disk and attaching each point on the edge to the point opposite. Neither process is easy to visualise in ordinary three dimensional space because some parts of the surface have to go through other parts.
At some stage I hope to put some nice pictures and pretty animations here to help visualise the surfaces. Please, if you're interested in this then please let us know and we'll do it sooner rather than later.
This sort of thing is one tiny branch of Mathematics, even though there isn't a number in sight.
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Quotation from Tim BernersLee 