The Hidden Wheel-Within

Abstract

There is this old, eternal question: Why don't animals have wheels? In this perspective we show that they actually do. And they do so in a physically extraordinary way -- by combining incompatible elasticity, differential geometry and dissipative self-organization. Nature's wheel -- the ``wheel-within'' -- has been mysteriously concealed in plain sight, yet it spins in virtually every slender-body organism: in falling cats, crocodilians spinning to subdue their prey, rolling fruit-fly larvae, circumnutating plants and even in some of our own body movements. Flying somehow under the radar of our cognition, in recent years the wheel-within also tacitly entered the field of soft robotics, finally opening our eyes for its ubiquitous role in Nature. We here identify its underlying physical ingredients, namely the existence of a neutrally-stable, shape-invariant and actively driven elastic mode. We then reflect on various man-made realizations of the wheel-within and outline where it could be spinning from here.

Figures (5)

• Figure 1: The "wheel-within" is a universal tool of Nature and utilized by many slender organisms in fauna and flora: (A) A Drosophila larva rolling on a substrateDrosophila_prep. (B) A crocodile's death roll when catching a preycroco_youtube. (C) A wrestler performing the "bridge", also called "upa" in Brazilian jiu-jitsuBridge. (D) A falling cat, spinning around to land on the feetcatbook. (E) Circumnutation, i.e. spinning around the gravity axis during growth, of Arabidopsis thalianaAgostinelli_nutationMugnai_nutation. (F) The flagellar hook: the left panel shows a sketch of the hook on top of the rotary motor; the other panels show cryo-EM pictures (modified from Ref.Bact-Hook). (G) Two artificial, self-organized wheels-within ("fiberboids"): on the left a stroboscopic image of a spaghetti driven by osmosis to roll on a humidified kitchen towel, on the right a thermally driven PDMS fiber rolling on a hot substrateBazirBaumann.
• Figure 2: A-C Neutrally stable/Zero-elastic energy modes (ZEEM). D-G Isoskinning objects with neutral modes. (A) A torus formed by bending a rod and glueing the ends together as the simplest structure displaying a circular ZEEMBaumann. (B) Shell structure made of two different layersguest_zero-stiffness_2011. (C) Neutrally stable mode in a "kinked" arc-shaped shellKok_HerderVdLans_Radaelli: the inflection point can be moved along the crest without energy cost, see the flat energy region in the sketch below. (D) The rigid wheel-axle system: the wheel turns (red arrow) round the axle while keeping the outer skin (dashed) invariant. (E) A neutrally stable torus rotating around its curved centerline. (F) A neutrally stable self-buckled shell reorienting by keeping its outer skin fixed. (G) A neutrally stable twist deformation of a Möbius strip moving along its center-line within the same skin.
• Figure 3: Active driving, crankshaft analogy and dynamic frustration. (A) A falling cat, spinning to get on its feet, seen as part of a torus. The analogy to the piston engine: the spine/ribs correspond to the crankshaft and the contractile muscles (which are actuated by the animal) to the pistons. (B) In the fiberdrive, a closed torus heated from below, one can consider parts of material, e.g. as indicated, as the pistons. The crankshaft arises due to the material's connectivity. (C) Dynamic frustration/incompatibility of equilibria: every piston has a current length (left) that differs from its thermodynamically preferred length (right) at the given temperature. This leads to a situation where every piston is "frustrated", creating an overall torque.
• Figure 4: Man-made examples of the wheel-within. (A) A photo-thermally driven fiberdrive (torus) swimming in a liquid in the low Reynolds number regimelightdrivenTorus. (B) A (helical) fiberboid utilizing body heat to roll along a human armbodyheat. The helical shape slightly breaks the perfect ZEEM, but also has advantages like a better adaptation to the surrounding. (C) An Archimedean spiral can be seen as several fiberdrives mounted in series, adding up the torques they can exert to lift a weightBaumann. (D) A Möbius strip made of a composite material (anisotropic hydrogel with plasmonic nanoparticles) that rotates under static light illuminationMoebius3. (E) A "knotbot", performing two modes of motion: poloidal ("rolling") and toroidal ("rotating") under static light illuminationknotbots. (F) A confined blister that rotates when heated along its circumference; modified from Ref.blister.
• Figure 5: Degrees of freedom of the wheel-within family. A) The fiberdrive (torus) and the Möbius strip as "mono-wheels" with a single ZEEM (Z). B) Fiberboids (open fibers) and crumpled sheets have two degrees of freedom: one ZEEM (Z) and one auxiliary bending mode (M) with finite stiffness that can modulate the active dynamics of the ZEEM. All known wheels-within in Nature are of this type, with the exception of the flagellar hook. C) Any elastic knot (knotbot) has two ZEEMs: a poloidal ($Z_1$) and a toroidal ZEEM ($Z_2$). D) A raspberry-like deflated and crumpled elastic sphere can in principle have three ZEEMs, corresponding to three spatial rotations.