MAM2000 (Dimension)

A rectangular region with horizontal and vertical sides and one corner at the origin is an example of a two-dimensional object. Every point in the region is determined by two coordinates, (x,y). We can determine any such region by giving the coordinates (c,d) of the point farthest from the origin.

A rectangle with one corner at the origin can be specified by the coordinates of its opposite corner. An interactive demo is available for you to experiment with this yourself.

A rectangular region with vertical and horizontal sides is a two-dimensional object. Every point of such a region can be specified by giving two numbers, for example the distance over from the left-hand edge and the distance up from the bottom edge. There are other ways of specifying the location of each point, but each way requires two numbers. Note that specifying any particular such rectangle will take more than two numbers. For example, we can give the two coordinates of the lower left corner and two more coordinates for the upper right corner. The collection of rectangular regions aligned with the coordinate axes in the plane will then be a four-dimensional collection of two-dimensional objects.

Of course, the rectangles need not be aligned with the axes: one could rotate the rectangles, or any shape, or even the entire plane, around a point in the plane. This adds another dimension (the angle of rotation) to the collection of objects that results, but the collection now includes all rectangles in the plane. Rotations of this sort form the basis of the rotations that show up in computer animation, and in other applications in two-dimensional computer graphics.

The plane can be rotated about the origin by specifying the new positions of the unit vectors along the two coordinate axes. An interactive demo is available for you to experiment with this yourself.

A disc centered at the origin is another example of a two-dimensional object, with each point other than the origin specified by its distance from the origin and the angle of the ray on which it lies, the polar coordinate representation of the point. The collection of discs in the plane is a three-dimensional collection of two-dimensional objects, three-dimensional since each disc can be specified by its center and radius.

One of the best treatments of the notion of two-dimensionality is the classic book Flatland written by Edwin Abbott Abbott. You can find an on-line copy of that book, together with other materials, linked to Abbott's page.

Another extremely important example of a two-dimensional object is the sphere. Any point on the surface of a ball can be specified by two numbers, its latitude and longitude with respect to a given choice of prime meridian. The collection of spheres in three-dimensional space will be four-dimensional, since we need three numbers to locate the center and one to give the radius. Making maps of portions of the earth onto a flat plane is the subject of cartography, and there are many excellent sites on this topic. A literary treatment of life on the surface of a sphere is given in "Sphereland" by Dionys Burger, accessible through the Edwin Abbott Abbott page.

Other surfaces besides the sphere can be described by a pair of numbers, for example the torus, an inner-tube shape like the boundary surface of a doughnut. As in the case of the sphere, we can determine a longitude for each of the vertical circles in the torus, and then specify the position of a point on a longitudinal circle by one additional angle. See Jeff Weeks.

Julie Strandberg is a choreographer who has created a dance piece based on s . At her site, you can find a. description of that dance, along with a discussion of the dimensionality of dance movements. See how many dimensions it takes to describe classical ballet moves, or tribal dance patterns involves representing one two-dimensional object on a plane. There are many good references for this branch of ethnography.

Math Awareness Month 2000 Last modified: 26 Mar 2000 17:55:42 Comments to: thomas_banchoff@brown.edu Hosted by: The Math Forum

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