Friday, September 2, 2011

Custom-designed twogs

Twogs are flat shapes that interweave reciprocally to form a three-way structural joint. By reciprocal, I mean that every twog has the same shape, and plays the same role in forming the joint. Twogs are thus a special case of the broader topic of reciprocal structures, also known as nexorades (see O. Baverel, C. Douthe, and J.-F. Caron.)

The design of the planform shape of a twog is fairly arbitrary save for three points at each end. Those are the special points where two or three twogs engage each other. "Points" is not quite accurate. Examining a twogs joint very closely, we can see light coming through a tiny window where they seemed to touch. Where three twogs meet we see a tiny triangular window; where two twogs meet we see an even tinier lenticular window, shaped like the cross-section of a biconvex lens.

In designing a twog we must allow for this non-point-like intersection (which is due to the finite thickness of the material;) but more significantly, we must design a joint that cannot fit together without some out-of-plane bending, or coning, of the normally flat twogs. This forced conical curvature provides a biasing spring-force that holds the joint tightly together. (See Holger Strom's patent disclosure of IQ's for more on this.)

When a structural joint formed by three twogs is carrying its biasing spring-force, it acts as a tensegrity structure reminiscent of a wagon wheel. Radial compressive forces press together all three sides of the central 3-way engagement, which we may call the hub. Meanwhile, the ring of 2-way engagements, what we may call the rim, carry a band of tension around the periphery of the wheel like three links of a chain. (Just as with the links of a chain, the local interactions where two twogs—or chain links—meet are compressive, but the overall effect is to carry a tensile load.)

The shape of the planform at these three precisely located engagement "points" is given a small radius of curvature, but elsewhere the design of the planform of the twog is free. When all the twogs have the same shape, the engagement radius (the distance to the hub) is the same for all three rim engagements. If the structure were to lie flat, the angular separation of the rim engagements, as seen from the hub, would be exactly 120°. To achieve the biasing spring-force needed for a tight joint, the angular separation should be somewhat less: 116° is a good starting point for experimentation with any particular material and size.

It is desirable to specify the planform shape of a twog in a way that facilitates easy customizatio: that's because we might want to shape every twog differently in order to make a smoother basket. One way to do this is to suppose that the basket shape is specified by an arbitrary triangle surface mesh. Cut two neighboring triangles out of the mesh together, and open the "hinge" of their common edge out flat so that they lie together in the euclidean plane. The two flattened triangles together form a quadrilateral in the plane.

The engagement points of one twog lie easily within this two-triangle plot. In particular, the central region of each triangle contains three engagement points belonging to one end of the twog.

The two-triangle (quadrilateral) plot has four corner-points. The basic idea is to express each control point in the design of the planform of the twog as a weighted average (or recipe) of these four corner-points. Changing the four corner-points—as we do when we choose a different pair of neighboring triangles—then automatically yields a new planform. With a little care in choosing the recipes of the control points, the three twogs meeting at any joint will fit together properly, even though their shapes are all different.

The basic trick is to ensure that the engagement points inside any given triangle have recipes dependent only on the corner-points of that triangle. In particular, the hub is located at the centroid of the triangle; and the single rim engagement point associated with each triangle side (only two of these concern any one twog) is given by a set recipe of the two corner-points that are endpoints of that side, and the opposing vertex.

If the twog's planform is drawn on a joined pair of equilateral triangles, the control point recipes can read off the sides of the appropriate equilateral triangle like reading the composition of a ternary phase diagram.

Control points in the transition region (that is, between the two ends of the twog) can use recipes based on all four corner-points in order to avoid an abrupt transition at the middle.

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