**Katie Fong**

**Mentor: Greg Schneider**

Last Fall, I watched the movie “Flatland”, which focuses on three-dimensional objects passing through two-space. The movie caused me to think about how shapes in the fourth-dimension might look to us in three-space. Since we have no concept of a fourth-dimension in three-space, so in order to imagine the shapes passing through, we would need to study the cross sections that we see. My plan this summer is to discover the properties of four-dimensional objects as they pass through three-space. I will start off trying to find a way to transform vectors on one plane to another contained in a cube. Afterwards, I will then try to use the same technique to transform vectors in a plane in a cube to a cube in the tesseract. I will also study the types of cross sections that one can see in three-dimensional and four-dimensional objects. Furthermore, I would like to see how knots in the third-dimension transform to the fourth-dimension. I will do all of this research algebraically, and I will use that information to construct and program figures on the computer. I plan on entering math fairs with this research towards the end of the summer and the rest of the year.