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Puzzle pieces made up of rhombic dodecahedra joined together different ways are fascinating to play with and offer practically unlimited possibilities for geometrical puzzle constructions and mathematical analysis. The blocks are fairly easy to saw from square stock using a special jig, described in Chapter 23.
There is one way that two rhombic-dodecahedral (R-D) blocks can be joined, five ways that three can be joined, and 28 ways that four can be joined, as shown in Fig. 167.
Fig. 167
An interesting exercise is to catalog the practical geometrical constructions that might be possible using R-D blocks. All of those shown in Fig. 168 have isometric symmetry. They are arranged by increasing size, starting at the top. Those in the left-hand column have four blocks coming together at the center. Those in the middle column have six blocks coming together at the center. Those in the right-hand column have a single block in the center, and hollow versions of these are possible using one less block. Only those constructions using 20 or fewer blocks are shown. It is convenient to identify these constructions by names such as tetrahedron, even though the shape is obviously not an exact tetrahedron but only suggests it using a little imagination.
Fig. 168
At this point, one has a choice of many possible avenues of exploration. What are the fewest pieces that will construct all these shapes, or most of them, or other shapes of your choice? To put it another way, for a given number of pieces - say four or five or six - find the most versatile possible set that will construct figures. Which sets have all dissimilar non-symmetrical pieces? Which figures have unique solutions? The recreational possibilities here are practically unlimited and largely unexplored.
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