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.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.
References:
R. Vakil, A Mathematical Mosaic, 1996.
Fun Fact suggested by:
Ravi Vakil