Humm, some misconceptions here... Circles and squares are topologically identical and both are finite. The problem is that Euclidean space/algebra does not include the infinity concept. The oriented projective plane, on the other hand, does. Your clock with a tangent vertical line was a very clever idea: if you wish to measure distances expressed by values on this vertical line, you can write than with real numbers (but you won't be able to express points at infinity) or you can simply use the values in your clock. For example, 0 can be expressed as 9h, 1 can be expressed as 10.5h or 10:30 (since 45 degrees = arctan(1)), big numbers are almost 12h and the point at infinity at 12h :) . You can extend this concept to values beyond 12h, which will be equivalent to values coming from minus infinity. You can think of it as all points before infinity from 6h exclusive to 12h exclusive. There are two points at infinity, 6h and 12h. Now, comes the surprise, are you ready? From 12h to 6h (both exclusive) we have all set of points beyond infinity. Such extension is necessary to preserve some Euclidean properties, like orientation of 3 points (that's why this plane is oriented). If you extend your clock to a 3D one, you will have an sphere and a tangent plane. Now, we can describe two dimensional points. Keep in mind that our visualization model has 1 more dimension than our actual number of dimensions in the plane. From there we can go on in any number of dimensions you want... A while ago I did a visualization system for the oriented projective plane (using OpenGL). In the following pages, you can find some pictures and videos that helps you comprehend what I mean :-D www.ic.unicamp.br/~rezende/T2inCGALandT2Viewer.htm[^] www.ic.unicamp.br/~rezende/T2Viewer.htm[^] Enjoy! Fabio