Ropes have been made by mankind since at least 18,000 years ago, yet I have had a difficult time finding much mathematical treatment of them, with the following exception: J. Bohr and K. Olsen 2011 EPL 93 60004 doi:10.1209/0295-5075/93/60004 "The ancient art of laying rope". http://iopscience.iop.org/0295-5075/93/6/60004 Ropes are fractal-like, consisting of several levels: fibers, yarns, strands, rope. Fibers are "spun" into yarns, which are "spun" into strands, which are spun into rope. Each level is spun in the direction opposite to the previous level. The numbers of fibers, yarns, strands affect the geometry of the rope. Question #1: Why have any complexity in ropes at all? The tensile strength of a material isn't improved or even affected by subdividing it (to a first approximation), so why subdivide it? A1. Flexibility. Subdivided material can more easily bend without breaking (taking Euler buckling to its opposite limit). This flexibility allows ropes to be stored in coils. A2. Stress relief. Subdivided material automatically rearranges the stress among the various fibers more uniformly. A3. Ease of manufacture from natural materials. Natural fibers have lengths limited by the size of the plant or animal, so making very long ropes requires some amount of splicing. Splicing involves interweaving of fibers in a manner similar to building a rope in the first place, so there is a smooth transition between the concept of a splice and a rope. Question #2: Why have a twist? A. You need to keep the strands together for basic rope operation (e.g., for pulleys, coiling, etc.), so twisted strands stay together better than untwisted strands. Question #3: How come ropes don't untwist under stress? A. That's the cool part of multiple levels of counter-twisted strands: a properly designed rope won't twist in either direction under load. Question #4: What are the best numbers for fibers, yarns, strands ? A. Bohr & Olsen (link above) talk about this to some extent. The final level of 3 strands seems to have been the most popular rope design since antiquity. The numbers for the other levels are not typically special or even specified. ---- Now that we have 3D printers, we could conceivably build ropes of any design that we please. But what designs would we want to build? I would imagine that there are more interesting relationships between integer numbers of fibers/yarns/strands that would have interesting properties. One might investigate the properties of non-round fibers/yarns/strands; e.g., the ancient Egyptians used a kind of "flat rope" ("tape" ?) to hold their boats together. Various kinds of braided patterns come to mind. Carbon nanotubes could form the fiber-level structure, but perhaps there are more interesting designs possible at the molecular level. If one had a 3D printer that could position individual carbon atoms, perhaps structures more interesting than simple nanotubes could be built. These would presumably have features with some combination of nanotube and diamond-like properties. Biological systems, both plants & animals, have developed extremely sophisticated fibers/strands/ropes. For example, muscle tissue itself is a kind of rope that can contract & perform work when properly instructed. There are lots of excellent ideas from these biological systems that could be re-used for more hi-tech ropes. --- Probably the most heavily investigated "high tech" rope is that of the use of carbon nanotubes to construct a "space elevator" -- probably in the form of a flat tape rather than a round rope. At least one elevator cable company is already in production on carbon fiber elevator cables, although these cables are still nowhere close to the strength/weight requirements for an Earth space elevator. But perhaps we're not being clever enough. If we end up building such a structure atom-by-atom anyway, then perhaps there are better designs that can do more interesting things while we're at it: conduct electricity for powering the elevator, micro-mechanical "muscles" that could perform some kind of "peristalsis" that could provide the motive power for moving stuff up/down the elevator, etc. --- One of the hottest topics in architecture is "tensegrity": the integration of strong-but-limp ropes which are strong under tensile loads with stiff-but-not-so-strong members which are strong under compressive loads. Biological systems are particularly adept at tensegrity structures -- e.g., the human hand with its muscles & tendons going all the way up the forearm. In addition to building fractal compression members of the Farr-type, we need to also consider fractal tensile ropes. http://www.london-institute.org/people/farr/fractals.shtml Future 3D printers will have to have the ability to "print" both compressive and tensile members at the same time, and so will require the ability to "print" at least 2 materials. (Such printers may actually require at least 3 materials, because some structures will require the "printing" of scaffolding, which can later be dissolved away, leaving the compression & tension members free to do their thing(s).)