The paper emphasizes that complex numbers are objects with their equivalents commonly occurring in nature. Just like real numbers measure lengths in a physical world, complex numbers measure vortices observed in nature. The spiral orbits in this paper are exponential spirals (also called logarithmic spirals). A vortex is identified by determining a complex number that generates it. To determine this number, we need two snap-reading observations that provide the argument of a complex number, while the ratio of radiusesthe modulus of a complex number. Therefore, we also deal with the area of a complex number. Complex numbers involve a meaningful description of the laws of nature, i.e. of vortices and of equilibrium.

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One of the main problems of practical applications of degressively proportional allocations of goods and burdens is lack of uniqueness of this principle. Even under given boundary conditions of allocation, i.e. determined minimal and maximal amounts of a good that can be assigned in a given allocation, there are usually many feasible solutions. The lack of formal rules of allocation is the reason why the allocation is typically a result of negotiations among its agents. A number of allocations favor some of agents or their groups, therefore other agents cannot accept them. The aim of this paper is to indicate a way of reducing the set of all feasible solutions exclusively to those that are neutral to all agents. As a result of the term of lexicographic preference of allocation agents defined on the basis of the relation theory followed by a numerical analysis of sets of all feasible solutions, it is possible to determine a core of this set in the form of a subset of all feasible solutions that are acceptable by all agents. In addition, this subset can be further divided into smaller subsets with regard to the degree of acceptance of their elements. Theoretical analysis is complemented by case studies, one of which is application of this idea to the allocation of seats in the European Parliament among the member states of the European Union.

The principle of degressively proportional apportionment of goods, being a compromise between equality and proportionality, facilitates the application of many different allocation rules. Agents with smaller entitlements are more interested in an allocation that is as close to equality as possible, while those with greater entitlements prefer an allocation as close to proportionality as possible. Using relative entropy to quantify the inequity of allocation, this paper indicates an allocation that neutralizes these two contradictory approaches by symmetrizing the inequities perceived by the smallest and largest agents participating in the apportionment. First, based on some selected properties, the set of potential allocation rules was reduced to those generated by power functions. Then, the existence of the power function whose exponent is determined so as to generate the allocation that symmetrizes the relative entropy with respect to equal and proportional allocations was shown. As a result, all agents of the apportionment are more inclined to accept the proposed allocation regardless of the size of their entitlements. The exponent found in this way shows the significant relationship between the problem under study and the well-known Theil indices of inequality. The problem may also be seen from this viewpoint.