GRADES 9–12
CCSS Geometry: Visualize relationships between twodimensional and threedimensional objects:
Identify the shapes of twodimensional crosssections of threedimensional objects, and identify threedimensional objects generated by rotations of twodimensional objects.
The extension of twodimensional polygons into threedimensional polyhedra continues into higher geometrical dimensions. Just because four, five, or sixdimensional spaces do not exist in real life is no barrier for
mathematicians thinking about them. As a bare minimum, one can use the notion of coordinates to describe these higherdimensional spaces. The position of a point on a line can be specified by one coordinate; on a plane its position can be specified by two
coordinates; in threedimensional space it can be specified by three coordinates. To specify the position of a point in a four, five, or sixdimensional space one simply uses four, five, or six coordinates. Indeed, mathematicians have been known to think about spaces with an
infinite number of dimensions.
The generalized equivalent of a polygon or polyhedron is called a “polytope.” Just as a polyhedron has faces that are polygons, an Ndimensional polytope has “facets” that are polytopes of N–1 dimensions.
The simplest polytope to imagine (and specify) in higher dimensions is the hypercube. A fourdimensional hypercube has its own name: “
tesseract.” To generate a tesseract one starts with a onedimensional line segment and moves it in the direction of the second dimension, making a square. One then moves the twodimensional square in the direction of the third dimension, making a cube. Then one moves the
threedimensional cube in the direction of the fourth dimension, making a tesseract. Obviously one cannot do this in reality, but mathematically the process is simple.
A tesseract is usually represented as a cube within a cube; this is the threedimensional analogy of what one gets by shining a flashlight on a wireframe cube and projecting the image on a screen. The square nearest the light casts a larger shadow than the square opposite it, making the shadow
of the cube look like a square inside a larger square with their corners connected to each other. In the same way, the projection of the fourdimensional tesseract into threedimensional space is shown as a cube inside a larger cube with their corners connected to
each other.
An interesting exercise may be to figure out how many vertices, edges, faces, threedimensional facets, and so on a hypercube of N dimensions has. Following the process of building a tesseract from a point, a line, a square, and a cube is instructive. Every vertex is moved from its starting
position to its ending position so the number of vertices doubles.
Every edge is moved from its starting position to its ending position; in addition, every vertex traces out a new edge as it moves; thus the number of edges in the N+1dimensional polytope is equal to twice the number of edges in the Ndimensional
polytope plus the number of vertices in the latter.
A second class of polytope is the higherdimensional equivalent of the triangle or tetrahedron, called the “simplex.” This also is relatively simple to generate and visualize. To generate an N+1dimensional simplex from an Ndimensional
simplex, one places a point in N+1dimensional space (adds another coordinate to the list of coordinates) equally distant from all the existing points and then draws an edge from each existing point to the new point.
The limitation on the number of regular polyhedra in three dimensions extends into the higher dimensions as well.
In four dimensions there are six convex regular “polychora” (fourdimensional polytopes) and ten more that are not convex. In five or more dimensions, there are only three regular polytopes per dimension: the
simplex, the
hypercube, and the
cross polytope.
Sixty Years Ago in the Space Race:
October 23:
The American Vanguard TV2 was launched to an altitude of 109 miles and landed 331 miles downrange.
