A recent paper on 3D printing is likely also relevant to 3D weaving: "Strong 3D Printing by TPMS Injection" by Xin Yan, Cong Rao, Lin Lu, Andrei Sharf, Haisen Zhao, and Baoquan Chen. The authors show how to modify a level-set approximation of a triply-periodic minimal surface (TPMS) to custom fit the shape and interior stress levels of a sculptural object. Alison Grace Martin has already demonstrated that many TPMS can be woven.
Low aspect ratios (quarter-twist-wavelength over strip-width) are possible if the metal is relatively thick and twisted under high tension. In the picture, an aspect ratio of 0.77 achieved with 0.022" thick soft copper (0.096"/0.125").
The thicker the weaving elements, the larger the engagement window, eventually a locking interference is reached.
The stereogram shows a portion of the woven D-surface built up along the axis of a single triangular helix, that is, built above one of the triangles in this diagram,
The triangle is small, in the view along the axis the weaving elements appear to be engaging a single point.
Since each weaving element in the helix engages only two others, this is not really a weave structure that can hold together. By comparison, in a "Star-of-David" arrangement of 6 triangular helices around a hexagon, each weaving element engages four others. This is might be the smallest "tower" that can be bult along the isometric axis if the weaving elements are held by their own friction.
The straight lines in the D-surface form a NbO (niobium monoxide) net, which is usually illustrated from this perspective.
But viewed along a body diagonal of the cube (an isometric axis that foreshortens the the three primary directions equally) it has an kagome-like aspect. In this view it can be seen as an arrangement of triangular helices. The handedness of these helices is arranged as below:
I've made wire model to demonstrate this.
Here is a direct-view stereogram of the finished model:
These twisted 0.25" copper strips have a twist half-wavelength of 0.65", for a quarter-wavelength/strip-width aspect ratio of 1.3. (The hexagonal opening would be smaller in a completed weaving-- here the crossings are held together by elastic bands cut from the ends of 160Q balloons.)
In this view we are looking down the isometric axis giving the x,y,z Cartesian axes the same foreshortening. A chiral pair of twisted strips are aligned with each Cartesian axis, the six strips cross in a consistent over-under-over-under pattern around the hexagonal cycle. I made the strips by dangling a very heavy vise from the strip and counting turns. I found it difficult to get a consistent twist-wavelength over the length of the strip with this technique.
