I am not a mathematician by training, having taken my last math course in high school almost 50 years ago. Still, I have always enjoyed “playing” with math. I was a rugweaver for many years and worked with tessellations and the work of M C Escher. When I first experimented with beadwork, I immediately saw it as 3D tessellation. Much of my current beadwork consists of cutting metal tubes and joining them with thread to form triangles, and combining these into chains of polyhedra. It’s amazing what you can do with seat-of-the-pants math and the Pythagorean theorem. I have learned an amazing amount from such websites as Wikipedia, and the beaded molecules blog, and I am so grateful for the wealth of information they provide.
The edges of an octahedron form as 8 triangles, But they also create 3 quadrilaterals at right angles to one another. I have been playing with the second concept, varying the lengths of the sides of what I think of as the “waist” to achieve certain effects. Here all tubes are the same length except for the front and back waist tubes. If those are about 1.25 times the length of the other beads, they produce a right angle between sides of the octahedron. That allows me to stack them on the bottom of the piece. In a similar way, I have adjusted the inner triangles of the octahedra going around the back of the neck to achieve the curve I need. Finally I added a series of right angle tetrahedra along the bottom to add interest.
This piece consists of 6 tetrahelices, made of oxidized sterling silver and gold-filled tubes . In a perfect tetrahelix, the angle between the faces of succeeding tetrahedra is an irrational number, so the chain never repeats exactly. Fortunately for me, beadwork is never perfect. I can use the imprecision and flexibility inherent in joining tubes with thread to create what seems like a repeating sequence that forms a pinwheel. Ideally the pinwheel would have 6 identical arms, but unfortunately the needs of jewelry require an opening and a clasp.