Pseudospheric shells in the construction

The architects working with the shell use well-established geometry forms, which make up about 5-10 % of the number of known surfaces, in their projects. However, there is such a well-known surface of rotation, which from the 19th century to the present is very popular among mathematicians-geometers, but it is practically unknown to architects and designers, there are no examples of its use in the construction industry. This is a pseudosphere surface. For a pseudospherical surface with a pseudosphere rib radius, the Gaussian curvature at all points equals the constant negative number. The pseudosphere, or the surface of the Beltram, is generated by the rotation of the tracersis, evolvent of the chain line. The article provides an overview of known methods of calculation of pseudospherical shells and explores the strain-stress state of thin shells of revolution with close geometry parameters to identify optimal forms. As noted earlier, no examples of the use of the surface of the pseudosphere in the construction industry have been found in the scientific and technical literature. Only Kenneth Becher presented examples of pseudospheres implemented in nature: a gypsum model of the pseudosphere made by V. Martin Schilling at the end of the 19th century.

Example of Bianchi Transformation of Kuen’s Surface

The work is devoted to the study of the Bianchi transformation for surfaces of constant negative Gaussian curvature. The surfaces of rotation of constant negative Gaussian curvature are the Minding top, the Minding coil, and the pseudosphere (Beltrami surface). Surfaces of constant negative Gaussian curvature also include Kuen’s surface and the Dini’s surface. Studying the surfaces of constant negative Gaussian curvature (pseudospherical surfaces) is of great importance for the interpretation of Lobachevsky planimetry. Geometric characteristics of pseudospherical surfaces are found to be related to the theory of networks, the theory of solitons, nonlinear differential equations, and sin-Gordon equations. The sin-Gordon equation plays an important role in modern physics. Bianchi transformations make it possible to obtain new pseudospherical surfaces from a given pseudospherical surface. The Bianchi transformation for the Kuen’s surface is constructed using a mathematical software package.

Human Eye Optics within a Non-Euclidian Geometrical Approach and Some Implications in Vision Prosthetics Design

An analogy with our previously published theory on the ionospheric auroral gyroscope provides a new perspective in human eye optics. Based on cone cells’ real distribution, we model the human eye macula as a pseudospherical surface. This allows the rigorous description of the photoreceptor cell densities in the parafoveal zones modeled further by an optimized paving method. The hexagonal photoreceptors’ distribution has been optimally projected on the elliptical pseudosphere, thus designing a prosthetic array counting almost 7000 pixel points. Thanks to the high morphological similarities to a normal human retina, the visual prosthesis performance in camera-free systems might be significantly improved.

Transformation of Bianchi for Minding Top

The work is devoted to the study of the Bianchi transform for surfac­es of revolution of constant negative Gaussian curvature. The surfaces of rotation of constant negative Gaussian curvature are the Minding top, the Minding coil, the pseudosphere (Beltrami surface). The study of surfaces of constant negative Gaussian curvature (pseudospherical surfaces) is of great importance for the interpretation of Lobachevsky planimetry. The connection of the geometric characteristics of pseudospherical surfaces with the theory of networks, with the theory of solitons, with nonlinear differential equations and sin-Gordon equations is established. The sin-Gordon equation plays an important role in modern physics. Bianchi transformations make it possible to obtain new pseudospherical surfaces from a given pseudospherical surface. The Bianchi transform for the Minding top is constructed. Using a mathematical package, Minding's top and its Bianchi transform are constructed.

II. On the non-euclidian plane geometry

1. I consider the hyperbolic or Lobatschewskian geometry: this is a geometry such as that of the imaginary spherical surface x 2 + y 2 + z 2 = —1; and the imaginary surface may be bent (without extension or Contraction) into the real surface considered by Beltrami, and which I will call the Pseudosphere, viz., this is the surface of revolution defined by the equations x = log cot ½θ—cosθ, √y 2 + z 2 = sinθ. We have on the imaginary spherical surface imaginary points corresponding to real points of the pseudosphere, and imaginary lines (arcs of great circle) corresponding to real lines (geodesics) of the pseudosphere, and, moreover, any two such imaginary points or lines of the imaginary spherical surface have a real distance or inclination equal to the corresponding distance or inclination on the pseudosphere. Thus the geometry of the pseudo­sphere, using the expression straight line to denote a geodesic of the surface, is the Lobatschewskian geometry; or rather I would say this in regard to the metrical geometry, or trigonometry, of the surface; for in regard to the descriptive geometry, the statement requires (as will presently appear) some qualification. 2. I would remark that this realisation of the Lobatschewskian geometry sustains the opinion that Euclid’s twelfth axiom is undemonstrable. We may imagine rational beings living in a two-dimensional space, and conceiving of space accordingly, that is having no conception of a third dimension of space; this two-dimensional space need not however be a plane, and taking it to be the pseudospherical surface, the geometry to which their experience would lead them would be the geometry of this surface, that is, the Lobatschewskian geometry. With regard to our own two-dimensional space, the plane, I have, in my Presidential Address (B. A., Southport, 1883) expressed the opinion that Euclid’s twelfth axiom in Playfair’s form of it does not need demonstration, but is part of our notion of space, of the physical space of our experience; the space, that is, which we become acquainted with by experience, but which is the representation lying at the foundation of all physical experience.