Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
"Yet I exist in the hope that these memoirs, in some manner, I know not how, may find their way into the minds of humanity in Some Dimension, and may stir up a race of rebels who shall refuse to be confined to limited Dimensionality."
-A Square in Edwin Abbot's Flatland
When we measure something, such as the length of a wooden beam, we are focusing on one particular characteristic of that object and assigning a number to it. Many objects, however, in both our everyday experience and the realm of mathematics, cannot be adequately described by a single number. For instance, if you were to build a house, you would need beams and boards that are cut precisely in three different directions, length, width, and breadth. In other words, a 2×6 that is three feet long will not do if you need one that is eight feet long. All three measurements are independent and important. The more aspects that we can measure about a single object, the more precisely we can describe and work with it.
This way of thinking leads us quite naturally to the idea of "dimension." The word itself comes from the Latin dimensus, which means "to measure separately." So, quite literally, dimensions are aspects of a particular object that we measure separately from one another.
In this unit, we will explore the idea of dimension in a few ways. At first we will define it simply as quantities that can be manipulated independently of one another. We will describe the fairly common concepts of one, two, and three dimensions—most of us can easily grasp these—and then we’ll explore the trickier 4th dimension and discuss how to conceive of higher dimensions. Then we will introduce two concepts, scalability and self-similarity, and explain how these give rise to a different idea of dimension, the "fractal" dimension.
Dimension is a tangible part of our everyday experience; we are accustomed to "navigating the grid" in most cities and towns by moving in two directions, north-south and east-west. Dimension is often referenced in popular culture, too. Think of the "one-dimensional" character in a movie—the person who is concerned with only one thing, to varying degrees, such as the hero of an action movie, or the villain of a crime thriller. Artists such as Marcel Duchamp and Pablo Picasso attempted to present the concept of "higher dimensions" in their works by portraying objects from different angles simultaneously. In many works of science fiction, people use extra dimensions to travel around the galaxy via cosmic wormholes and other fanciful conjectures.
In modern mathematics the concept of dimension, utilized in a number of practical applications, encompasses much more than just the three spatial degrees of freedom—length, width, and height—to which we are accustomed. For example, marketers and matchmakers design computer programs capable of constructing "30-dimensional" profiles of individuals based on their multiple interests and inclinations, hoping to pair these people with products or romantic partners.
Many scientists believe that the very fabric of our universe—of reality—can be understood only by going beyond the traditional three dimensions and studying the mathematics of higher dimensions. Whether it is the five dimensions associated with the theory of general relativity or the 13+ dimensions involved in string theory, we live in a reality that allows for many degrees of freedom.
We can find exciting phenomena in fractional dimensions as well. This entirely new and different way to view the concept of dimension has been applied to the simulation of realistic plants in computer programs and to the authentication of works of art, such as those of Jackson Pollock.
In this unit, we will learn how to leverage our intuitive understanding of the world of three dimensions to enable us to think meaningfully about worlds of many degrees of freedom. Mathematics often is applied to the study of things and worlds that exist only in our minds—that is, the realm of the logically possible. One of the basic tools mathematicians use to get a handle on these mental worlds is the notion of dimension. We’ll develop a mathematical understanding of dimension and gain some familiarity with associated tools, such as slices and projections, which mathematicians use to conceive of and understand our world and other multi-dimensional frontiers.
Next: 5.2 Degrees of Freedom