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 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.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

References:
R. Vakil, A Mathematical Mosaic, 1996.

Fun Fact suggested by:
Ravi Vakil

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