Reaction to Chapter 4 of Beyond the Third Dimension

Andy Miller

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.

Exercises

• Collect data of temperature, time and location of a large geographic region where this information is available for a six month period (from the news or weather records, for example). The region should extend from the northern hemisphere to the southern hemisphere. On a map of the region, indicate the temperature change over time as one half of the region cools with winter and the other half warms with winter. Use the three dimensions of temperature, time and location on a two-dimensional map, using whatever indicators necessary (e.g. color, thickness of lines, numbering, etc.) to represent the shift in heating.
• Produce a simple architechtural design of a theoretical dorm that minimizes the amount of surface area in shadow, and, hence, keeps more rooms warmer by solar energy. Assume that the position of the sun in not directly overhead (as is the case at Brown, especially during the winter). Would a circular, crescent or triangular design be more efficient that another? This exercise would probably best be done qualitatively, with simple models of buildings and light source(s) that simulate the sun.

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.