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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 (3-D) is held appropriately in front of a light, the projected shadow of the cube on a plane (2-D) will be a square (2-D). Similarily, if a square (2-D) is held appropriately in front of a light (2-D), the projected shadow of the square on a line (1-D) will be a segment (1-D). Moving the light will project other "dimension-less-one shadows," all revealing important properties of the original shape.

This idea of "dimension-less" 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 three-dimensional "shadow" of a four-dimensional 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 4-dimensional solid (i.e. "octacube") with 24 vertices, 96 edges and 96 triangular faces, which enclose 24 3-dimensional "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 4-D 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.