The challenge of geometric dissections is to dissect one shape to another in the fewest pieces (and preferably, without turning any over). Here is an example:
In this example, the five pieces of the square can be rearranged to form the hexagram. It is unlikely that anyone will ever find a four piece solution, but this is part of the challenge: no one has yet found a way to prove that it cannot be done in fewer pieces. This is not just true for this dissection but for almost every dissection on this website.
The following dissection requires two pieces to be turned over. This is considered an imperfection, but is acceptable if no other dissection can be found in as few pieces as is true of this particular dissection.
Most of the dissections of this website are of the above form: the dissection of one simple shape to another. But other challenges are possible such as the following dissection of one triangle to two triangles:
Or of one triangle to five triangles:
Here is the dissection of a square to two identical shapes:
Or we can dissect the square to two different shapes:
Or we can dissect the square to a series of shapes:
Another challenge is to find a set of pieces that can form three different shapes. There are even dissections to form four different shapes but it is hard to find such dissections that do not have far too many pieces.
A new challenge I have come up with is what I call “locked” dissections. In these the challenge is to form a dissection where all the pieces are held in place, like a jigsaw. It is not at all easy to find these dissections so there is definitely room for improvement.
Another challenge is not to dissect the shapes themselves but instead to change the shape of a hole. These I call “hole” dissections.
All the above dissections have been flat two dimensional dissections. Things get a lot harder when we move into the third dimension. They are also much harder to illustrate.
Many dissections are not possible in 3D, for example, it is impossible to dissect a regular tetrahedron to a cube, so only a limited number of 3D solid dissections are possible. But it is possible to dissect the surface of a tetrahedron to the surface of a cube. What is surprising is that this can be done with just two pieces:
Another challenge is to dissect shapes on the surface of a sphere. At school you learn that the angles of a triangle add up to 180 degrees, but on a sphere this is not the case. The angles can add to as much as 540 degrees! This makes these dissections much harder: a whole new set of geometric rules must be learnt. In the following dissection the angles of the triangle are each 90 degrees and add up to 270 degrees. The square has angles of 112½ degrees adding up to 450 degrees.