Reaction to Chapter 2 of Beyond the Third Dimension

Andrew Miller

Math 8

The overarching theme of Chapter 2 dealt with the extrapolation of algebraic and geometric patterns that exist in lower dimensions to the fourth and higher dimensions. By doubling the size of an object, its one dimensional properties double, two dimensional properties quadruple, and three dimensional properties increase by eight times. In this simple example, it is easy to see that by doubling the length of a four dimensional object, its four dimensional properties would increase by a factor of 16 (two to the fourth power). Chapter 2 made clear that trends that consistently held true from the first to the second and from the second to the third dimension extended to the fourth and higher dimensions.

A fascinating point alluded to in the chapter was the Platonic idea that an ideal square or other geometric figures exist only in the "mind of God" (p. 16). People’s reaction to the study of a fourth spatial dimension is often one of critical skepticism. They claim that it is a pointless endeavor because there is "no such thing as the four dimension." However, similarly, there is "no such thing" as the first or second dimension as things that humans can inspect in their perfect forms. Yet we study them because they provide useful insights into three dimensional space, with which we are accustomed. From my limited knowledge of relativity, I understand that even a perfect third dimension does not exist because space is curved along the dimension of time. The relation of dimensionality to godliness reiterates a theme developed in Flatland. The complexity and abstractness of a four dimension causes us to associate it with godliness. This may be an example of how we attempt to explain phenomena that we do not understand with supernatural explanations, not unlike primitive peoples explained natural events, such as earthquakes and the moon, with godly attributions.

A third point that was important to the chapter was that exponential dimensional expansions are not always by integer values. The example of the Sierpinski gasket and the Koch snowflake represent figures that do not have integer growth exponents.

Exercise

An examination of the architecture and construction of Egyptian pyramids would be fascinating and reveal a famous application of the study of geometry.

Questions

1. When visualizing various dimensions, I often think about the basic properties of the units for that dimension. For example, the unit for the second dimension is area and that for the third dimension is volume. What is the unit of measurement for the fourth dimension? Is it something that can be intuitively conceptualized?

2. I have seen computer images of fractals (p. 32) that are intricately colored. They look like they have elements of both randomness and order. How are these images made and what makes them artistically unique?