Want to make a trip to University Park (PA), the home of Penn State? If not, let me provide a reason to do so.
One way to to think about dimensions is through the use of shadows. For example, if a cube (3D) is held appropriately in front of a light, the projected shadow of the cube on a plane (2D) will be a square (2D). Similarily, if a square (2D) is held appropriately in front of a light (2D), the projected shadow of the square on a line (1D) will be a segment (1D). Moving the light will project other "dimensionlessone shadows," all revealing important properties of the original shape.
This idea of "dimensionless" shadows is basis of a new sculpture installed in the Penn State Department of Mathematics. Designed by Adrian Ocneanu, Mathematics Professor at Penn State, the sculpture creates a threedimensional "shadow" of a fourdimensional solid object.
Constructed of stainless steel, the sculpture measures six feet in every direction, weighs 1200 pounds, and is mounted on a granite base about three feet high "in order to bring its center approximately to eye level." The sculpture is the "shadow" of a regular 4dimensional solid (i.e. "octacube") with 24 vertices, 96 edges and 96 triangular faces, which enclose 24 3dimensional "rooms." Windows cut in faces allow the viewer to see into the rooms or into the structure, the same way that a window in a cubic room opens to the inside of the cube.
For more information about this 4D sculpture, consider these resources:
 Link 1: Daily Science News (October 20, 2005) announcement of unveiling
 Link 2: Penn State Math Department's Description of sculpture and its underlying mathematics
 Link 3: Animation of a hypercube rotating in four dimensions
 Link 4: Animation, Octacube design and 4D projection method © A.Ocneanu
The last two links should not be visited or viewed unless at least an hour has elapsed since your last meal.
