Dissections are Fun, No Matter How You Slice It

Some folks have referred to me as “a real cut‑up.” Of course, those folks are generally septuagenarians who still use words coined in the mid‑1800’s. But I take it as a compliment, perhaps because I’m a fan of dissection problems.

Speaking of dissection, here’s one of my favorite quotes about jokes, which comes from E. B. White:

Humor can be dissected as a frog can — but the thing dies in the process, and the innards are discouraging to any but the pure scientific mind.

Like jokes and amphibians, geometric figures can be dissected, too. The following are three of my favorite dissection problems.

My all-time favorite is the Haberdasher Problem, originally proposed by H. E. Dudeney in 1902. It’s my favorite because of its simplicity — it can be described with just a few words, and it asks the would‑be solver to turn the simplest of all polygons, an equilateral triangle, into a square:

With three cuts, dissect an equilateral triangle so that the pieces can be rearranged to form a square.

The second isn’t really a problem to be solved. Instead, it’s a dissection that offers a surprising result. (At least, I was surprised the first time I saw it.)

Remove a tetrahedron of edge length 1 unit from each vertex of a larger tetrahedron with edge length 2 units. In total, remove four tetrahedra, one from each vertex. The image below shows one of the tetrahedra being removed. What is the shape of the solid that remains?

 

Finally, here’s an original dissection problem.

Form a pentagon by placing an isosceles triangle with height 1 unit on top of a square with side length 2 units. Then find a point P along the perimeter of the pentagon such that when P is connected to two vertices, the three pieces into which the pentagon is divided can be rearranged to form a square. Find the area of the largest piece. 

As usual, these problems are presented without answers or solutions. More fun can be had if you solve them yourself.