### Explaining Huffman’s Impossible Pyramid

#### by David Radcliffe

I read about Huffman’s Pyramid from the consistently excellent blog Futility Closet. Huffman’s Pyramid is a drawing of a figure that cannot exist.

However, the impossibility of this figure is hardly obvious. Here is the reason: if the slanting lines were extended, then they would have to meet at the apex of a pyramid. However, the lines do not meet. Contradiction!

Are you convinced yet? If you are, then your geometric intuition is stronger than mine, and you can probably stop reading now. But if you are skeptical, as I was, then please continue reading.

First, we should ask this question: What exactly do we mean by an impossible figure? Surely a figure can’t be impossible if we can draw it. Sometimes we can even make a physical model of an impossible object, such as this impossible triangle sculpture in Perth.

I think that what we mean by an impossible figure is that the figure appears to have contradictory properties. The sculpture shown above is not actually a triangle, but it appears to be a triangle from a certain angle, and this apparent shape is inconsistent with other things that we can observe about the sculpture.

Let’s consider Huffman’s Pyramid again. We’ll label the points to make it easier to talk about.

The drawing *appears* to represent a polyhedron with two triangular faces and three quadrilateral faces. The triangular faces are ABC and DEF. The quadrilateral faces are ABED, BCFE, and CADF. It also appears that AD and BE intersect at I, AD and CF intersect at G, and BE and CF intersect at H. If we accept this interpretation of the drawing, then the shape that it represents is impossible.

Here’s why. Consider the plane Π containing the face ABED. The plane contains G, since G is on the line AD. The plane also contains H and I, since they are on the line BE. Therefore, Π contains the triangle GHI. Similarly, the plane containing the face BCFE must also contain the triangle GHI. But there is only one plane that contains the triangle GHI, so the two planes must be the same. Therefore, the entire shape lies in a plane, and it cannot be a polyhedron.

This seems to suggest that pentahedra can’t be *too* irregular. It would be interesting to explore this notion further.

**Edit:** Greg Ross suggests that the figure might be possible if there is another hidden edge. Can anyone explain this?

I get it now. If the back face has a hidden edge (AF or CD) then the lines AD and CF are skew, so they do not cross at G.

4th edge:

How do you know that the points G, H, and I exist?

Well, I and H are both the unique points of intersection of two coplanar lines (by assumption, ABED and BCFE are planar faces).

However, G only exists if lines AD and FC are also coplanar. if there is more in the back, then AD and FC need not be coplanar and G doesn’t exist.