Replicating Figures in the Plane

Abstract

In [1], C. D. Langford asked for those plane figures which can be dissected into four “replicas”, congruent to one another and similar to the original figure. (An equivalent formulation is that four identical figures are to be assembled into a scale model, twice as long and twice as high.) In addition to triangles and parallelograms, which always have this property (see Figure 1), he also exhibits three trapezoids (Figure 2) and three hexagons (Figure 3) with this property. Finally, Langford gives an example involving a stellated hexagon (Figure 4). He then asks if the list is complete.