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The number of ways that sticks can be arranged symmetrically in space is very limited. It is convenient to examine this question in terms of unnotched straight sticks. The standard six-piece burr can be regarded as a cluster of six rectangular sticks to which parts have been added (or removed) to achieve interlock and other interesting features. The Pin-Hole Puzzle is an even better example. The hollow space in the center is cubic. In any symmetrical arrangement of straight sticks totally enclosing a hollow center, a little thought or experimentation will show that the faces of the enclosed hollow center must be rhombic (or square). There are only three isometrically symmetrical solids with such faces - the cube, the rhombic dodecahedron, and the triacontahedron (Fig. 91).
Fig. 91
The rhombic dodecahedron has 12 identical rhombic faces. It can be visualized as the solid that results when the edges of a cube are sufficiently beveled at 45 degrees (Fig. 92a). It is one of very few symmetrical solids that pack to fill space, two others being the cube and truncated octahedron. Like the cube, it has three fourfold axes of symmetry, four threefold axes, and six twofold axes (Fig. 92b). When viewed along any of its fourfold axes it appears square in profile, while along any of its threefold axes it appears hexagonal (same as the cube).
Fig 92a
Fig. 92b
The rhombic dodecahedron can be totally enclosed by a symmetrical cluster of 12 sticks having equilateral-triangular cross-section, a property not only intriguing but of great practical significance. This arrangement has a pair of mirror-image forms, as shown in Fig. 93.
Fig. 93
Theory of Interlock
Stellations
The Second Stellation
The Four Corners Puzzle
Color Symmetry
The Second Stellation in Four Colors
The Third Stellation in Four Colors
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