An introduction to the unpublished book "Reflections on a Tube" by Mitchell J. Feigenbaum.

This paper is an adaptation of the introduction to a book project by the late Mitchell J. Feigenbaum (1944-2019). While Feigenbaum is certainly mostly known for his theory of period doubling cascades, he had a lifelong interest in optics.

Solid-body trajectoids shaped to roll along desired pathways.

In everyday life, rolling motion is typically associated with cylindrical (for example, car wheels) or spherical (for example, billiard balls) bodies tracing linear paths. However, mathematicians have, for decades, been interested in more exotically shaped solids such as the famous oloids, sphericons, polycons, platonicons and two-circle rollers that roll downhill in curvilinear paths (in contrast to cylinders or spheres) yet indefinitely (in contrast to cones, Supplementary Video 1). The trajectories traced by such bodies have been studied in detail, and can be useful in the context of efficient mixing and robotics, for example, in magnetically actuated, millimetre-sized sphericon-shaped robots, or larger sphericon- and oloid-shaped robots translocating by shifting their centre of mass.

General Theory of Specific Binding: Insights from a Genetic-Mechano-Chemical Protein Model.

Proteins need to selectively interact with specific targets among a multitude of similar molecules in the cell. However, despite a firm physical understanding of binding interactions, we lack a general theory of how proteins evolve high specificity. Here, we present such a model that combines chemistry, mechanics, and genetics and explains how their interplay governs the evolution of specific protein-ligand interactions.

Imaging in reflecting spheres.

We study the formation of images in a reflective sphere in three configurations using caustics on the field of light rays. The optical wavefront emerging from a source point reaching a subject following passage through the optical system is, in general, a Gaussian surface with partial focus along the two principal directions of the Gaussian surface; i.e.

Dimensional reduction in complex living systems: Where, why, and how.

The unprecedented prowess of measurement techniques provides a detailed, multi-scale look into the depths of living systems. Understanding these avalanches of high-dimensional data-by distilling underlying principles and mechanisms-necessitates dimensional reduction. We propose that living systems achieve exquisite dimensional reduction, originating from their capacity to learn, through evolution and phenotypic plasticity, the relevant aspects of a non-random, smooth physical reality.

Colony entropy-Allocation of goods in ant colonies.

Allocation of goods is a key feature in defining the connection between the individual and the collective scale in any society. Both the process by which goods are to be distributed, and the resulting allocation to the members of the society may affect the success of the population as a whole. One of the most striking natural examples of a highly successful cooperative society is the ant colony which often acts as a single superorganism.

Green function of correlated genes in a minimal mechanical model of protein evolution.

The function of proteins arises from cooperative interactions and rearrangements of their amino acids, which exhibit large-scale dynamical modes. Long-range correlations have also been revealed in protein sequences, and this has motivated the search for physical links between the observed genetic and dynamic cooperativity. We outline here a simplified theory of protein, which relates sequence correlations to physical interactions and to the emergence of mechanical function.

Desert ants achieve reliable recruitment across noisy interactions.

We study how desert ants, Cataglyphis niger, a species that lacks pheromone-based recruitment mechanisms, inform each other about the presence of food. Our results are based on automated tracking that allows us to collect a large database of ant trajectories and interactions. We find that interactions affect an ant's speed within the nest.

Leaders of neuronal cultures in a quorum percolation model.

We present a theoretical framework using quorum percolation for describing the initiation of activity in a neural culture. The cultures are modeled as random graphs, whose nodes are excitatory neurons with k(in) inputs and k(out) outputs, and whose input degrees k(in) = k obey given distribution functions p(k). We examine the firing activity of the population of neurons according to their input degree (k) classes and calculate for each class its firing probability Φ(k)(t) as a function of t.

Ergodicity and slowing down in glass-forming systems with soft potentials: no finite-temperature singularities.

The aim of this paper is to discuss some basic notions regarding generic glass-forming systems composed of particles interacting via soft potentials. Excluding explicitly hard-core interaction, we discuss the so-called glass transition in which a supercooled amorphous state is formed, accompanied by a spectacular slowing down of relaxation to equilibrium, when the temperature is changed over a relatively small interval. Using the classical example of a 50-50 binary liquid of N particles with different interaction length scales, we show the following.

Thermal rectification in billiardlike systems.

We study the thermal rectification phenomenon in billiard systems with interacting particles. This interaction induces a local dynamical response of the billiard to an external thermodynamic gradient. To explain this dynamical effect we study the steady state of an asymmetric billiard in terms of the particle and energy reflection coefficients.

Entropy of dialogues creates coherent structures in e-mail traffic.

We study the dynamic network of e-mail traffic and find that it develops self-organized coherent structures similar to those appearing in many nonlinear dynamic systems. Such structures are uncovered by a general information theoretic approach to dynamic networks based on the analysis of synchronization among trios of users. In the e-mail network, coherent structures arise from temporal correlations when users act in a synchronized manner.

Curvature of co-links uncovers hidden thematic layers in the World Wide Web.

Beyond the information stored in pages of the World Wide Web, novel types of "meta-information" are created when pages connect to each other. Such meta-information is a collective effect of independent agents writing and linking pages, hidden from the casual user. Accessing it and understanding the interrelation between connectivity and content in the World Wide Web is a challenging problem [Botafogo, R.