by Beth Walker Ph.D. Mathematics Education
What do you know about space; not the final frontiers of cosmic explorers but our most immediate environment? We live in three-dimensional space, eat off two-dimensional tabletops, and drive or walk in one-dimension at a time. Like a riddle, we don’t create space, but often try to contain it. We assign space measures, but often don’t understand what those measures mean. In mathematics, the study of one-, two- and three-dimensional space is often referred to as geometry. What are dimensions and are there really more than three?
Mathematical understanding is built from a collection of related facts, which contribute to a conceptual understanding based on observed characteristics. The next step in cognitive development is generalization of conceptual understanding so that knowledge can be applied to new situations. For example, students learn to recognize certain shapes as triangles. Conceptual understanding would be that ALL closed shapes with three sides or three angles are triangles. A generalization is that closed shapes are named by the number of sides (and therefore the number of angles) they contain.
A common vocabulary is helpful in constructing conceptual understanding and generalizations so that ideas can be shared with others. In the above example, names of shapes were crucial in forming a generalization.
So what is a dimension? How is it defined? For the sake of simplicity, discussion will be restricted to one-, two- and three- dimensions, which is enough to describe the world that most of us live in. If many one-dimensional examples are encountered, a conceptual idea can be formed about what makes something one-dimensional. This conceptual understanding may result in a generalization or definition. Ideally, this definition will extend to higher dimensions.
Here are some examples of one-dimensional objects from a theoretical perspective and, ignoring the fact that all physical objects have some thickness however small, some examples from real life:
• a line
• a ray
• a line segment
• a piece of thread
• the edge where two walls come together
What do all of these shapes have in common? If you were a fly hanging out on any of the above examples, how many numbers would be required to describe your exact location? In one-dimension, you can only move left/right, up/down, or backward/forward. While you could designate your location from either the ceiling or the floor where two walls come together, only one number measured from a starting point called the origin, is required to specify your location in space. Any unit of measure can be used, but only one number is required. A common characteristic of one-dimensional space is the need for only one number or measurement in identifying a location in that space. If this definition or generalization is valid, it should be transferrable to two- and three-dimensions.
Examples of two-dimensions are a table top, a piece of graph paper, and a soccer field. If the definition created for one-dimension holds, two numbers are required to specify a location in a two-dimensional space. It is most efficient to make the first dimension at right angles to the second dimension. If we want to describe the exact location of a soccer player at any time in the game, both the distance down the field and from the sidelines would be required.
Similarly in three-dimensions, three numbers or coordinates are required to describe an exact location. Consider a hot air balloon frozen in time as it wafts over the countryside. The third dimension would be at right angles (or orthogonal) to the other two dimensions and in this case would be the distance from the balloon straight down to the ground. Paired with the two numbers identifying the location on the two-dimensional ground directly under the balloon, a precise location of the balloon could be determined.
Locations in our physical world can be specified by one coordinate for one-dimensional space, two coordinates for two-dimensional space, and three coordinates for three-dimensional space. These numbers lie along number lines that are at right angles to each other and their directions are usually indicated by up/down, left/right, and forward/backward.
There are other mathematical ways to describe dimensions, but these definitions help to create a map of the space around us and are transferrable from one- to three-dimensions. I will leave it up to your imagination about how they might apply to four- or more dimensions!
[Dr. Walker likes to examine mathematical ideas and concepts in multiple ways, sharing insights with her students, friends and family. A strong proponent of problem-based learning, she has used videos and scenarios from The Futures Channel in many of her lessons. Her current areas of research include questioning methods in mathematics education and models for shifting everyday behavior to professional behavior. People are welcome to contact her at rbowsend-at-rochester.rr.com (replace -at- with @)].