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Math Shapes

Our familiar space is a space of three-dimensions. A rectangular room with sides aligned with the coordinate planes is an example of a three-dimensional object. Each point of the room can be specified by three numbers, for example the distance over from the left hand wall, the distance back from the front of the room, and the distance up from the floor.

Three dimensional objects can be rotated in space, just as two-dimensional ones can be rotated in the plane. Knowing the new positions of the rotated coordinate axes allows us to rotate any figure in the same way, and such rotations play a crucial role in computer animation, and in other applications in 3D computer graphics.

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

Crystal growth takes place both in the two-dimensional plane and in three-space, and this subject is described in Marjorie Senechal's page.

Other subjects that involves three-space are astronomy and astrophysics. One good example is the geometry of light patterns studied by Arlie Petters in his work on gravitational lensing.

As mentioned in the discussion of paleoecology in dimension one, each observation involving a core sample involves three dimensions, two to give the latitude and longitude of the sample point and a third coordinate to tell the distance down from the surface. Another way to describe a point other than the center of a ball as a three- dimensional object is to give its distance from the center and the latitude and longitude of the point where the ray form the origin through the point meets the boundary surface of the ball. Similarly we can describe each point of a solid doughnut by identifying its latitude and longitude , then specifying how far in the point lies from the surface.

Although much of familiar physics and engineering takes place in our three-dimensional world, there are many concepts that require more dimensions for their effective description, such as velocity and acceleration of a moving particle. We can also talk about higher-dimensional collections of three-dimensional objects, for example the collection of all rooms aligned with the coordinate planes will have six dimensions. One way to see this is to give the three coordinates of one corner of the room and three more coordinates for the corner farthest away. As mentioned before, the space of two- dimensional spheres in three-space is four-dimensional, with three coordinates for the center and one for the radius.

[MATH FORUM] Math Awareness Month 2000
Last modified: 26 Mar 2000 18:23:17
Comments to: thomas_banchoff@brown.edu
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