A remarkable connection between soliton theory and an important and beautiful branch of the theory of graphical statics developed by Maxwell and his contemporaries is revealed. Thus, it is demonstrated that reciprocal triangles which constitute the simplest pair of reciprocal figures representing both a framework and a self-stress encapsulate the integrable… (More)
It is shown that the integrable discrete Schwarzian KP (dSKP) equation which constitutes an algebraic superposition formula associated with, for instance, the Schwarzian KP hierarchy, the classical Darboux transformation and quasi-conformal mappings encapsulates nothing but a fundamental theorem of ancient Greek geometry. Thus, it is demonstrated that the… (More)
A novel class of discrete integrable surfaces is recorded. This class of discrete O surfaces is shown to include discrete analogues of classical surfaces such as isothermic, 'linear' Weingarten, Guichard and Petot surfaces. Moreover, natural discrete analogues of the Gaußian and mean curvatures for surfaces parametrized in terms of curvature coordinates are… (More)
The conformal geometry of the Schwarzian Davey-Stewartson II hierarchy and its discrete analogue is investigated. Connections with discrete and continuous isothermic surfaces and generalised Clifford configurations are recorded. An interpretation of the Schwarzian Davey-Stewartson II flows as integrable deformations of conformally immersed surfaces is given.
It is established that there exists an intimate connection between isomet-ric deformations of polyhedral surfaces and discrete integrable systems. In particular, Sauer's kinematic approach is adopted to show that second-order infinitesimal isomet-ric deformations of discrete surfaces composed of planar quadrilaterals (discrete conjugate nets) are determined… (More)
A novel class of integrable surfaces is recorded. This class of O surfaces is shown to include and generalize classical surfaces such as isothermic, constant mean curvature, minimal, 'linear' Weingarten, Guichard and Petot surfaces and surfaces of constant Gaußian curvature. It is demonstrated that the construction of a Bäcklund transformation for O… (More)
A nonlinear coupled system descriptive of multi-ion electrodiffusion is investigated and all parameters for which the system admits a single-valued general solution are isolated. This is achieved via a method initiated by Painlevé with the application of a test due to Kowalevski and Gambier. The solutions can be obtained explicitly in terms of Painlevé… (More)