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© 1999-2010 by Francis Edward Su

From the Fun Fact files, here is a Fun Fact at the Advanced level:

# Koch Tetrahedron

 Figure 1

In Koch Snowflake we saw an interesting fractal snowflake-like 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.

Su, Francis E., et al. "Koch Tetrahedron." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

References:

Keywords:    geometry
Subjects:    geometry
Fun Fact suggested by:   Ravi Vakil
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