The ancient art of laying rope

The ancient art of laying rope Jakob Bohr∗and Kasper Olsen Department of Physics, Technical University of Denmark Building 307 Fysikvej, DK-2800 Lyngby, Denmark jakob.bohr@fysik.dtu.dk, kasper.olsen@fysik.dtu.dk Abstract We describe a geometrical property of helical structures and show how it accounts for the early art of ropemaking. Helices have a maximum number of rotations that can be added to them – and it is shown that this is a geometrical feature, not a material property. This geometrical insight explains why nearly identically appearing ropes can be made from very different materials and it is also the reason behind the unyielding nature of ropes. The maximally rotated strands behave as zero-twist structures. Under strain they neither ro- tate one or the other way. The necessity for the rope to be stretched while being laid, known from Egyptian tomb scenes, follows straightforwardly, as does the function of the top, an old tool for laying ropes. The crafting of rope, string and cordage has been an essential skill through the times going back to early prehistoric life. The image of rope is easily discernible and could perhaps be said to be iconic. It has also been important for early symbolic meaning and for creed. Examples are the spinning seidh [1], the shimenawa prayer rope [2], the shen ring, and the cartouche in hieroglyphs [3]. Scenes from Egyptian tombs display advanced ropemaking [4, 5]. Figure 1 shows a scene from the tomb of Akhethotep and Ptahhotep, and it appears to be depicted that the ropes are held under tensile stress while being laid. The round tool hanging from the rope is perhaps a stone helping the ropemakers to gauge that sufficient tensile stress is present; to maintain a nearly straight rope requires the presence of adequate tensile strength depending on the weight of the stone. In a scene from the tomb of Rekh-mi-Re a special belt is depicted which help the ropemakers to apply tensile stress by the use of their body weight [5]. Large quantities of ancient Egyptian rope has been found in a cave at the Red Sea coast [6], also found at the site are two limestones with holes now described as anchors [7, 8]. We believe that another possible use of these stones might have been as weights used during the rope production akin to the tomb scene depicted in Figure 1. Classical ropes appear with an easy discernible geometrical structure, even though they have been fabricated in different human cultures from a large variety of fibrous materials with diverse physical properties. Relatively recently, Zhang et al. has demonstrated that yarn formation can be performed with the use of nano-sized strands as well [9]. Why does the resulting geometry of rope appear so similar, as if it depends little on the material used, and why are ropes inextensible? Here we show, that these properties of rope are due to a universal behavior of helical structures which depends on geometry. It arXiv:1004.0814v3 [physics.pop-ph] 8 Aug 2010 Figure 1: Ropemaking in ancient egypt. Tomb of Akhethotep and Ptahhotep, about 2300 BC. The round tool hanging from the rope close to the person to the left is perhaps a stone helping the ropemakers to gauge that sufficient tensile stress is present. stems from the observation, derived below, that there is a maximum number of rotations that can be added to a helical structure (or more precisely an N-helix, where N ≥2). One consequence is that a tightly laid rope, where each of the strands are rotated to their maximum in one direction while being helically laid with the maximum number of rotations in the opposite direction will be interlocked, unable to unwind, and hence a functional rope. Mathematical aspects of the helical geometry of yarns are described by Treloar [10] and by Fraser et al. [11], and a comprehensive review for wire rope is given by Costello [12]. The counter-twisting of the strands and the rope, respectively, has been discussed from a mathematical perspective [11], and ropemaking in a historical context [13]. From the point of view of topology it is important to consider the amount of writhing, see Thompson and Campneys [14] and Stump et al. [15]. Thompson et al. have suggested that double helices will kinematically lock-up at 45◦[16], Gonzales and Maddocks have introduced the global curvature and investigated the significance hereof when understanding helical structure formation [17]. Przyby l and Piera´nski have derived the conditions for self-contacts for single helices [18], and Neukirch and van der Heijden the condition for inter-strand contacts in an N-ply [19]. Olsen and Bohr have determined the close-packed helical structures from a calculation of the volume fraction for a helix as a function of the pitch angle [20]. It is surprisingly simple to see that there is a geometrical limit to the number of rotations on a helix, in the following we show that helical structures can be maximally rotated. A helical curve is uniquely described by two independent variab

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