Slices of Hanging Cubes

Hang a cube from one of its vertices. Now, if you slice it horizontally through its center, you get a hexagon.

What if you do this with a 4-dimensional cube, i.e., a tesseract? The slice will yield a 3-dimensional object--- what does it look like?

Answer: you get a octahedron!

Presentation Suggestions:
Use lower dimensional analogies to help students visualize higher dimensional objects.

The Math Behind the Fact:
It is not hard to see (using symmetry arguments) that the object you get must be regular. By analogy with the slice of the 3-cube, the slice of the 4-cube must cut every "face". The number of "faces" of a 4-cube is eight. The only regular 8-sided solid is an octahedron.

Visualizing high dimensional objects can be taxing, but fun!

How to Cite this Page:
Su, Francis E., et al. "Slices of Hanging Cubes." Mudd Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

References: R. Vakil, A Mathematical Mosaic, 1996. Keywords:    geometry Subjects:    geometry Level:    Advanced Fun Fact suggested by:   Ravi Vakil Suggestions? Use this form. 3.75 current rating Click to rate this Fun Fact...     *   Awesome! I totally dig it!     *   Fun enough to tell a friend!     *   Mildly interesting     *   Not really noteworthy and see the most popular Facts!

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