One of the most important concepts in algebraic geometry and singularity theory is the blowup of a plane in the origin.
The plane is mapped to a surface in 3-space in an interesting way: Every point of the plane outside the origin is mapped to a unique point of the surface above. But the origin itself is mapped to a whole line (the thick green one in the model) where each point on the line on the surface corresponds to a direction through the origin.
In the model this is illustrated in the following way: The blue curve which passes twice through the origin in the plane is mapped to a blue curve in the surface, but the curve on the surface has no self-intersection because the two directions of the curve in the origin are mapped to different points of the green line above.
See our youtube video on
http://calendar.algebraicsurface.net/calendar.php?mode=youTube&day=09 for more information.