I have several ideas that I would like to explore more thoroughly. Here are a few of them: 1. A Higher-Dimensional Geometry I would like to do a proper but simple geometry of four or greater dimensions, of the scope of your average high school geometry. Important decisions would include postulates and primitives. As a reference, I have pored over Henry Parker Manning's higher dimensional geometry, and I think I have some useful idea of what would be involved in doing something similar.

2. Colorspace While reading a computer graphics textbook, I came across the idea that all colors lie in a unique point within a three dimensional cube, on which the three axes are red, green and blue. This is a very promising idea for graphic representation of higher spaces for me. The idea that a single pixel onscreen corresponds to a three dimensional point seems intuitively a strong one; you could do graphs of four dimensions on a line! There's a lot to explore here, I think.

3. Alternative Interpretations of Geometry There is mention in B3D of a geometry of segments, where in each segment is drawn with opposite endpoints on two respective coordinate planes, thus presenting each segment as an object determined by four variables, namely, the coordinates of its respective endpoints. It is possible to reinterpret a four dimensional geometry to refer to such a geometry of segments, by treating the segments as though they were points. There are lots of other possibilites like this, including geometries of ellipses, spheres, squares, sine waves, and just about anything you can think of. The nice thing about them is that they are of high dimensionality, yet are easily represented in our space. I think there could be a whole lot of fun to be had here; what particularly strikes me as promising is the idea of illustrating a single, demonstrative geometric figure using many different representations. I would really like to have a go at this one.

Keith_Adams@brown.edu