We study the construction of substitution tilings of the plane based on certain simplicial configurations of tangents of the deltoid with evenly distributed orientations. The random tiling ensembles are obtained a sa result of tile rearrangements in the substitution rules associated to edge flips. Special types of random tilings for Euclidean, spherical and hyperbolic three-manifolds are also considered.

We consider connected and simply connected seven-dimensional Lie groups whose Lie algebras have nilradical ${\frak g}_{5,2}$ of Dixmier. First, we give geometric descriptions of the maximal-dimensional orbits in the coadjoint representation of all considered Lie groups. Next, we prove that, for each considered group, the family of the generic coadjoint orbits forms a measurable foliation in the sense of Connes. Finally, the topological classification of all these foliations is also provided.

Despite the longstanding interest in the shapes of the eggs the available parametric descriptions in the modern literature are given only via purely empirical formulas without any clear relationships with their measurable parameters. Here we present geometrical models of the eggs based on Perseus spirics and Cassinian ovals which were known since the ancient time but their analytical parameterization was also absent in the meantime. Such ones have been found recently and the present work is based on the idea to use spirics or Cassinians as geometrical models of the eggs shapes. New explicit formulas for the volumes, surface areas and the curvatures of the avian eggs have been derived from the first principles and these have been compared with the available experimental data.

Nonlinear dynamical systems can be studied in many different directions: i) finding integrable cases and their analytical solutions, ii) investigating the algebraic nature of the integrability, iii) topological analysis of integrable systems, and so on. The aim of the present paper is to find integrable cases of a dynamical system describing the rider and the swing pumped (from the seated position) as a compound pendulum. As a result of our analytical calculations, we can conclude that this system has two integrable cases when: 1) the dumbbell lengths and point-masses meet a special condition, 2) the gravitational force is neglected.