We present a construction method for Sierpinski tetrahedron objects that does not consider individual tetrahedrons as basic building blocks but rather regards connectors of two tetrahedrons as the fundamental units. It uses pieces with an identical shape that is a union of two fragments of regular tetrahedrons connected at their vertices, and it has the property that one can only build finite approximations of the Sierpinski tetrahedron if four corner pieces are supplemented. We have 3D-printed two sets of pieces that have different mechanisms to link the pieces. One uses 3D-printed joints and it can be used for constructing Sierpinski tetrahedron objects. The other one uses magnets and it can also be used as a puzzle. We describe how one would construct Sierpinski tetrahedron objects by solving this puzzle, even without any prior knowledge about Sierpinski tetrahedrons.