# Stewart Coffin'sSymmetric Stick Puzzles

Author: Aaron Siegel Symmetric Stick Puzzles, in which notched or drilled sticks interlock into an assembly governed by principles of polyhedral symmetry, have fascinated puzzle designers for many decades. As in many other areas, it was the pioneering work of Stewart Coffin, beginning with his classic designs Hectix and Locked Nest, that initiated the modern era of such puzzles. Yet although they are based on simple and elegant mathematical principles, symmetric stick puzzles can seem mysterious at first encounter.

This article is a gentle introduction to the geometric principles behind Hectix and Locked Nest (and many other similar puzzles). It will be clearest if you're already at least somewhat familiar with those classic designs; they are discussed in Coffin's books Geometric Puzzle Design and AP-ART.

I learned most of the concepts in this article from George Hart, whose inspirational paper Symmetric Stick Puzzles surveys the area in significantly greater breadth.

## The Distorted Cubic Lattice

To understand the basic geometry of Hectix and Locked Nest, start with an arrangement of twelve dowels in a cubic lattice, as shown (viewed from various angles) in the following pictures. We can envision the assembly as the projection of a single dowel under various rotational symmetries of the cube.  The cube has a total of 24 rotational symmetries, but since the dowels themselves are symmetric, each one is generated equivalently by each of two different rotations. This reduces the number of projections from 24 to 12, corresponding to the familiar twelve edges of the cube.

Next, rotate one of the dowels along the perpendicular axis running diagonally (at a 45° angle) through the center of the cube. The following pictures show one of the dowels (highlighted in pink) rotated by 20°, along with a thin line demonstrating the axis of rotation.  Now project that rotation onto each of the twelve dowels via the corresponding rotational symmetry. This produces a distorted figure with 24-fold symmetry. Here's the result of applying a rotation of 10° to all twelve dowels:  Notice how the three dowels surrounding each vertex seem to "pull away" from each other as the angle of rotation increases.  When the angle reaches exactly θ = arctan(√2) (approximately 54.7°), the three dowels surrounding each vertex become parallel, framing an equilateral triangle. Geometrically, the original cube has transformed into a cuboctahedron, the dual of the rhombic dodecahedron. Here are some additional views of the final (angle θ) assembly.  The angle θ is the "canonical angle" of the rhombic tetrahedron, and it features in many of Coffin's designs. Envision a line segment L joining the center of a cube to one of its vertices; then θ is the angle formed between L and an adjacent cube edge.

## Now Let's Make Some Puzzles!

Next let's replace the dowels with hexagons and enlarge them to a radius r = 1/√6 (approximately 0.408), where the unit length 1 is the side length of the original cube. Then the hexagonal sticks overlap so that each stick cuts exactly halfway into its neighbor. (In these images, the hexagonal sticks have been beveled slightly so that the images are clearer.)  Elegant interlocking puzzles can be created by notching the sticks in appropriate places. This is the twelve-piece assembly introduced by Stewart Coffin's Hectix and later explored by Bill Cutler, Derek Bosch, and others. Suppose instead we shrink the radius by half, to r = 1/24 (approximately 0.204). Then instead of overlapping, the sticks exactly touch.  Now overlay the mirror image of this assembly, using thin dowels instead of hexagonal sticks. The mirror image assembly can be generated using the identical process, but with an angle of -θ instead (that is: rotate the sticks by an angle θ in the opposite direction). The dowels in these pictures have a radius of 0.08, but that is an arbitrary choice.  This is the basic assembly used in Coffin's Locked Nest. Drilling holes in the hex sticks where the dowels go gives the basic building blocks of Locked Nest:  On the left is a single hex stick with appropriate holes drilled; there are five holes, each drilled through a long diameter of the hexagon. The centers of the holes are evenly spaced along the stick, and they progress in a repeating mod-3 pattern at 60° angles to one another.

On the right, one can see an "elbow piece" with dowel attached. Recall that the sticks were originally obtained by rotating a cube's edges by an angle of θ = arctan(√2), and the dowels obtained by a similar rotation of -θ. Therefore the wide angle between the stick and dowel is exactly 2θ (the famous tetrahedral angle), and the acute angle between them is 180-2θ, or approximately 70.5°. This is the angle at which the holes must be drilled in order to obtain the correct geometry for the Locked Nest assembly.

Many variants are possible using the same ideas. Coffin's Cuckoo Nest employs shorter sticks, with three holes drilled at the same 70.5° angle, but in a different pattern (alternating rather than mod-3): There are an enormous variety of other forms achievable using these ideas. An excellent reference is George Hart's paper Symmetric Stick Puzzles, Proc. Bridges 2011, which is accompanied by Mathematica code for visualizing various symmetric stick assemblies.