Figure 1

In Koch Snowflake we saw an interesting fractal
snowflakelike object that is obtained
when gluing smaller triangles iteratively to the sides
of a big triangle.
So, what happens if you do something similar to a
tetrahedron?
That is, suppose you take a regular tetrahedron
(all side lengths the same),
and glue to each of its triangular faces some smaller
regular tetrahedra, as in Figure 1?
(Each smaller tetrahedron is scaled down by a factor
of 1/2 from the larger one, and placed on each face
in an inverted fashion, so that it divides the face
into 4 equilateral triangles and covers the center one.)
Then iterate this process: at each stage,
take the new object, and glue
still smaller regular tetrahedra (scaled by 1/2 in
the length of each side)
on each of its triangular faces.
You might think that you get a very jagged object
after all the stages are completed, but surprisingly,
in the limit, you get a perfect cube!
Presentation Suggestions:
Draw a picture to help students to see what is going on.
Challenge them to think about the object they get before
telling them.
The Math Behind the Fact:
The cube you obtain has side length T/Sqrt[2], where T
is the length of one of the edges of the regular tetrahedron
you started with.
How to Cite this Page:
Su, Francis E., et al. "Koch Tetrahedron."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
