This puzzle idea of Lin and Wu can be applied to many other polylinks if one can determine the proper length to cut the wooden sticks. As a simple example, the polylink shown in Figure 5 can be built from square stock if the ratio of the outer edge to the cross-section edge is 9.8 to 1. This value comes from the software described in the next section, but as wood is flexible and compressible, some experimenting was still required. So from 1-unit square stock, one can cut 24 pieces, each 11.8 units long. Near each end, but from opposite sides, notch halfway through to leave a 1-unit overhang beyond the notch. Figure 15 shows the assembled result. Figure 16 is the analogous puzzle of thirty sticks assembled into six pentagons woven as in Figure 8. I built both of 0.5-inch square wood bars.

In these two experiments, I found that cutting the lengths a few percent shorter than the software suggests seems to compensate for the flexibility of the struts and the fact that the corners of wood struts may be compressed. So although the software provides a good initial value, the woodworker is advised to plan on some experimentation with scrap wood before investing in quality wood.

To design and build regular polylinks and tangles of concentric polylinks, I wrote a program with sliders that can be adjusted to see a wide range of structures on-screen. The computer-generated images in this paper are from screen-shots of its operation. The user can specify any number of polylinks to be assembled concentrically, and for each set, the user selects the underlying polyhedron used as its basis. Then sliders allow the size, translation, and rotation of the components to be adjusted. At present, the edge cross sections are adjustable rectangles, allowing flat (paper) versions as the special case of width zero. As the sliders are adjusted, the dimensions of length, width, thickness, etc. are displayed, which can be used for making models of wood or other materials. When the user is happy with the form shown rotating on the screen, a click of a button generates an STL file for making solid freeform fabrication models, e.g., Figure 5. Clicking another button generates an image file that can be printed for making paper or cardboard models, e.g., Figure 4.

Three 4-cm examples made by selective laser sintering (SLS) from the program’s STL output are shown below. Figures 17 is ten triangles, arranged as in Figure 14, but with a "tall" cross-section. Figure 18 is twenty icosahedrally arranged triangles, with the minimum possible linkage (analogous to the cubic form in Figure 6). Figure 19 is a weave based on the octahedron but with the triangles replaced by hexagons.

The software is freely available online at my website [4]. It has only been tested on PCs, but it is written in java so should be transportable to other computing environments. For rendering 3D images on the screen, it uses Sun’s freely available java3D extension, which must be installed on the user’s computer. The coding is quite straightforward, with no clever algorithms or data structures needed. I believe it can be straightforwardly replicated by any software engineer with expertise in graphics programming.

Users may verify their understanding of its capabilities by replicating the figures of this paper, or one can simply start playing to create new objects. The software generates all regular polylinks including several icosahedral examples that Holden did not describe. Lang enumerated the regular polylinks with a computer search and showed there are 2 tetrahedral, 2 cubic, 3 octahedral, 5 dodecahedral, and 23 icosa-hedral varieties (excluding the extreme cases where polygons either are not linked or are merged into pairs) [8]. I can replicate these numbers with an independent geometric technique of counting subsets of symmetry axes that pierce the interior of a polygon centrally placed in the stellation diagram.

Many extensions of the program are possible. It could be straightforwardly modified to produce circular, triangular, or other shapes of cross sections for the polygon edges. Another possible addition to the software is a numerical search for slider settings that result in a snug fit. I planned for this feature when originally designing the software, but then discovered that it is simple to see on the screen if there are gaps or overlaps in the components, so manual adjustment seems sufficient.

There are an unlimited number of variations on the above ideas. For example, Holden made several models of linked polygonal stars instead of convex polygons. Similarly, one might try rectangles or rhombs instead of regular polygons; I have not implemented any of these with software. (For related rectangle linkages, see [11].) Another idea is to start with polyhedra other than the Platonic solids. Holden constructed examples based on Archimedean solids, e.g., the rhombicuboctahedron and the snub cube. One can symmetrically combine multiple polylinks based on different polyhedra; sets with the same symmetry combine to form a tangle with the same symmetry. Figure 20 shows the combination of a cubical and a tetrahedral form. It combines the tangles of Figure 2 and Figure 10 so each triangle links three squares while each square links two triangles. Figure 21 shows the union of a dodecahedral and an icosahedral form in which both puzzles of Figures 13 and 14 are intertwined.

Another variation Holden tried was to put squares around the twelve 2-fold axes of a cube, i.e., in the planes of a rhombic dodecahedron. Here the squares only exhibit 2-fold symmetry, so their edges and vertices are not all equivalent, and the polylink is not regular. I find the results less attractive—fun to generate with the software but less interesting to look at. An example is shown in Figure 22. Here the twelve squares are grouped together as the sides of four triangular prisms. Extending his idea, it is natural to try putting thirty squares around the 2-fold axes of an icosahedron, i.e., in the planes of a rhombic triacontahedron. These can be grouped in many interesting ways, including five cubes or six pentagonal prisms. In Figure 23, the thirty squares are grouped together as the sides of ten triangular prisms. (When brought to the origin and fused into fifteen squares, they form five octahedra.)

Spiraling the faces in a series of concentric arrangements is another idea that can be explored with the software. Fig 24 shows a simple example based on an inner cube that is surrounded by nine layers, each slightly larger and rotated five degrees more than the previous. At the end of nine steps, the 45 degree rotation of the squares leads them to be arranged as in a cuboctahedron. This is not technically a "tangle" as no polygons are linked, but it is easy to generate other examples that are. Figure 25 shows an assemblage of triangles that smoothly rotate between an outer stella octangula and an inner form of four triangles, positioned as in Figure 4. Figure 26 shows a linked set of ten triangular helixes that each join a pair of opposite faces of the outer icosahedron. Charles Perry has explored this "face screwing" concept to great effect in a number of monumental sculptures, e.g. his 1973, 35-foot tall, "Eclipse", which starts with an inner dodecahedron. See [3], plate D for a figure.

Weaving is another way to generate variations on a polylink. The oldest example that I know of a form closely related to a regular polylink is the sepaktakraw ball, seen in Figure 27. Traditionally woven from rattan, it exhibits the dodecahedral pattern of figure 8. The design goes back centuries for use in traditional Asian "football" games. Figure 28 shows a "spherical basket" I wove of paper strips. The six dark central bands follow the same weave pattern of six pentagons shown in Figure 8. Neighboring bands, which get progressively lighter in color, simply alternate over and under in the natural weave pattern. Note the difference between this weave and the sepaktakraw ball. In Figure 28 the individual strands weave, while in Figure 27 the group of strands weaves as a whole.

Another variation, illustrated in Figure 29, is to allow a parameter for a relative rotation between the polygon and its hole. This adds a dynamic visual quality to the forms.

A different type of variation is to replace the straight edges of the polygons with curved paths. There are infinitely many ways to choose curves, but a particularly natural one is to perform an inversion about the center of symmetry. Central inversion replaces each point at distance r from the origin with a point in the same direction but at distance 1/r. This transformation is well studied mathematically but little used in sculpture [12]. In this context, the chain of n rectangular struts that form the edges of an n-gon is replaced by a chain of n curved volumes bounded by four portions of spheres. Figure 30 shows an example in which a square-cross-section version of the four-triangle construction in Figure 10 is inverted into a structure that resembles four interlocked three-leaf clovers. The twelve corners are now on the inside, but remain 60-degree angles because central inversion is an angle-preserving transformation.