Tibor Tarnai

Infinitesimal and Finite Mechanisms

In the design of engineering structures, an important question is whether a structure is rigid. For conventional structures, rigidity is a fundamental requirement. However, there are cases where just the opposite is required. In order to answer the question of rigidity we have to know the static-kinematic properties of the structure. In the forthcoming, these properties will be investigated for bar-and-joint assemblies, that is, for structures composed of straight bars and frictionless pin joints. Firstly, we survey the basic terms to be used in the analysis.

From spherical circle coverings to the roundest polyhedra

Abstract The problem treated here is: amongst the convex polyhedra that can be circumscribed about the unit sphere and have faces, which has the minimum surface area? A new optimization method based on mechanical analogies is worked out to solve this problem. By using this method, new computer-generated solutions are presented for and . The second of these two conjectured roundest polyhedra has icosahedral symmetry. The relation of the results of this problem to the minimum coverings of the sphere with equal circles is discussed.

Polymorphism in Multisymmetric Close Packings of Equal Spheres on a Spherical Surface

Locally optimum, high-symmetry solutions are sought to the Tammes problem: to determine the maximum angular diameter of n equal circles, which can be packed on a sphere without overlapping. By using triangular surface lattices, in the range 1 ≤ n ≤ 500, those values of n are determined, for which the circles can be close packed in arrangements different from each other with tetrahedral, octahedral, and icosahedral symmetry. Tables are given of the results.

Partial covering of a circle by equal circles. Part I: The mechanical models

How must n equal circles of given radius be placed so that they cover as great a part of the area of the unit circle as possible? To analyse this mathematical problem, mechanical models are introduced. A generalized tensegrity structure is associated with a maximum area configuration of the n circles, whose equilibrium configuration is determined numerically with the method of dynamic relaxation, and the stability of equilibrium is investigated by means of the stiffness matrix of the tensegrity structure. In this Part I, the principles of the models are presented, while an application will be shown in the forthcoming Part II.

Circle Packings and the Sacred Lotus

How must n non-overlapping equal circles be packed in a given circle so that the diameter of the circles will be as large as possible? This paper presents an account of this problem and its putative solutions and related configurations in lotus receptacles, classical Japanese mathematics (wasan) and traditional Japanese design. Particular emphasis is placed on the connection between the conjectural solutions of this discrete geometrical problem and the fruit arrangements in the receptacles of lotuses, because in most cases the actual fruit arrangements are identical to the mathematical solutions. As the lotus is an important symbol in Buddhism and lotus decorations are quite common in Japanese Buddhist art, packings of circles in a circle have been represented in Japanese art for centuries.

Arrangement of 23 points on a sphere (on a conjecture of R. M. Robinson)

The problem of determining the largest angular diameter dn of n equal circles which can be packed on the surface of a sphere without overlapping is investigated. It is known that the best packing of 5 (11) circles on a sphere is obtained if one circle is removed from the best packing of 6 (12) circles. Robinson has suggested that perhaps there are some other cases also where this property holds, possibly n = 24, 48, 60, 120 are the circle numbers for which dn–1 = dn. In this paper it is proved that this property does not hold for n = 24, thus d23 > d24, and it is conjectured that dn–1 > dn for n = 48, 60,120. A new packing construction is presented for 23 circles on a sphere with circle diameter 43.709964° and for 29 circles with circle diameter 38.677079°.