Usually a puzzle mat is divided into 32 identical pieces out of soft plastic, which can be sticked together to form a mat. (The picture shows one of those pieces; in the following pictures each piece will be represented by a simple, randomly colored square.)
Of course three dimensional structures are more fascinating than flat mats, thus I spent some time on exploring the possibilities of puzzle mats in this respect. This gallery shows my favourite discoveries. If you are aware of an interesting construction not appearing here, please tell me about it.
Boxes are easily built with a puzzle mat, however they do not roll very well, thus I invented a wheel. After a while I realized that even an almost-sphere is possible.
Stellated versions of all regular polyhedra can be constructed with puzzle mats. However with 32 pieces only the stellated cube, tetrahedron and octahedron are possible.
With enough pieces a flat mat can easily be extended to any size. Here are two other structures of this type.
Good question, isn't it? Here is my answer: