Reaction to Chapter 4 of Beyond the Third Dimension
Review of Chapter's Themes: Computer graphing is the most exciting development in the representation of higher dimensional objects. They have been useful in generating models of two-dimensional shadows of four and higher dimensional objects, such as a simplex or hypercube. From such images, we gain insight into the patterns that exist as we examine more complex dimensions. A table of numbers that describes, for example, the vertices, edges and faces of any shape in various dimensions can be analyzed to derive formulas that predict these features at any given n-th dimension.
Scientists, among others, can use the concepts of shadows and structures to assist in visualizing complicated sets of data. By using multiple dimensions of data, such as color and animations, levels of data can be stacked upon itself and compared simultaneously to reveal interesting relationships.
Comments on Professor Banchoff's Dandelin Page
The explanation of Dandelin's proof that the intersection of a plane and a circular cylinder is an ellipse is enhanced by the use of colors, series of changing images, and see-through pictures (not solid). The explanation proceeds logically from a reminder of the properties of ellipses to a complex three dimensional figure of two spheres, a plane and a cylinder (all of different colors), showing that, by the definition of an ellipse, the intersection of a cylinder and a plane must be an ellipse. The last diagram was a bit confusing to me. I could not fully understand how the description following the picture related to the the picture.