What is a dimension?
In geometry, a dimension describes measurement and direction of an object. We can measure an object using coordinates, standard units, or nonstandard measurements. When we talk about the direction of a dimension, we refer to length, width, height, and the fourth dimension. The number of the dimension (i.e. the 2 in 2-D) refers to the number of coordinates needed to identify each point on the object. For example, a point on a 3-dimensional object would be named by three coordinates to indicate the location of the point in each direction (height, length, and width, etc.). (http://mathworld.wolfram.com/Dimension.html)
Each dimension can be extended into the next higher dimension by “stacking.” For example, to expand a 1-Dimensional space to achieve a 2-dimensional space, you can “stack” 1-dimensional spaces, then to expand a 2-D space into a 3-D space, you “stack” 2-D spaces, and so on. (http://www.maa.org/meetings/calendar-events/the-fourth-dimension)
What are the different types of dimensions?
Geometry in the fourth dimension still follows the rules of Euclidean Geometry in the third dimension, but is extended to one additional dimension. When we describe objects in the fourth dimension, we often add the prefix ‘hyper-’ to the name of the object (ex: hypercube, hyper sphere, etc.) (http://mathworld.wolfram.com/Four-DimensionalGeometry.html)
An example of a Tesseract:
Why do we study dimensions?
The study of dimensions in math is relevant to our daily lives. For example, we apply knowledge of the 0-Dimension when we use coordinates to find an address, city, or place on a map. When we describe how to get from place to place when we are driving, we use 1-dimensional geometry. We use 2-dimensional geometry when we calculate the area of a living room to determine how much carpet we need to buy. We study the 3rd dimension because every object around us has three dimensions. In regards to geometry in higher dimensions, it is much harder to come up with examples of everyday applications, except for use in physics (http://www.math.brown.edu/~banchoff/STG/ma8/papers/anogelo/hist4dim.html).
We study dimensions in mathematics because the concept of ‘dimension’ helps us better visualize or conceptualize “the complexity of any geometric object.”
Studying the dimension of any given geometrical object, either concrete or abstract, gives an idea of the 'size' of the object or the amount of space it occupies. Once we know how big an object is, we can begin to compare it to other geometric objects. When we study dimensions in math, we gain a better understanding of the qualitative properties of geometric objects. In studying dimensions, we gain insight to the topological properties of geometric objects, but we also gain new perspectives that help us better understand the world around us.
As teachers, we introduce students to the concept of dimension when we teach students about shapes; how to plot points, lines, and segments on the coordinate plane; how to calculate volume and area, etc. Here is a link to some of the specific standards of geometry that allude to dimension:
Here are some cool manipulatives you can use in the classroom to help teach students about dimensions:
Some examples of manipulatives on this website are geoboards (or XY coordinate pegboards),
geosolids, folding geometric shapes, etc.