TITLE: Surface dissection of regular tetrahedron to cube
QUESTION [5 upvotes]: Does anyone know what is the fewest-piece 
  dissection
  of the surface of
  a regular tetrahedron to the surface of a cube (of the same area)?

It is well-known that the volume of a regular tetrahedron cannot
be dissected to the volume of a cube, because their Dehn invariants differ.
But their surfaces have a dissection, by applying the
Boyai-Gerwien theorem.
Applying that theorem in one way (among several options) leads to an
$31$-piece dissection (if I calculated correctly), illustrated below. But this is surely very far from optimal (and hardly aesthetically pleasing).
Likely the question has been explored, but I have not found any literature.
It could make a pleasing contrast to the impossibility of a volume dissection.

Let the cube edge length be $1$, so its surface area is $6$.
A tetrahedron edge length of
$L= 2^{\frac{1}{2}} 3^{\frac{1}{4}} \approx 1.86$ 
leads to a surface area of $\sqrt{3} L^2 = 6$.
          


          

Surface dissection of regular tetrahedron to cube.


Related: "Covering a Cube with a Square."

REPLY [5 votes]: I found a solution to this many years ago. Amazingly, this can be solved in as few as 2 pieces. See this webpage for the solution. You will also find a description in Greg Frederickson’s book “Dissections: Plane & Fancy”. (This book is a very good introduction to dissections in general and I highly recommend it.)



In any dissection the two shapes must first be dissected to form strips. These strips are overlaid to give the dissection. The cube is easily converted to a strip element in a number of ways. One is in the form of a zigzag of the six squares. The tetrahedron forms a strip element as a row of four triangles. Overlaying these two strips gives a four piece solution. But the trick with the tetrahedron is in realising that the triangles form a tube, i.e. an infinitely long strip that repeats every four squares. (Try it: cut a paper tetrahedron along one edge and also along the opposite edge. You now have a tube.) This allows the number of pieces to be reduced to just two pieces.
It is hard to believe that my solution can be further improved!
On the above webpage you will also find many other surface dissections including a three piece surface dissection of cube to icosahedron and a four piece surface dissection of tetrahedron to cube to octahedron.