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 4dimensional cube,
i.e., a tesseract? The slice will yield
a 3dimensional 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 3cube, the slice of the
4cube must cut every "face". The number of "faces"
of a 4cube is eight. The only regular 8sided 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.
<http://www.math.hmc.edu/funfacts>.
