M A T H E M A T I C A L E C O N O M I C S No. 11(18) 2015 VORTICES AND COMPLEX NUMBERS Arkadiusz Maciuk, Antoni Smoluk Abstract. 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 radiuses – the 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. Keywords: complex number, vortex. JEL Classification: C02, C60. DOI: 10.15611/me.2015.11.07. 1. Introduction All number systems originate from an additive semigroup of natural numbers . The simplest extension of this semigroup is  – a ring of integer numbers. Its further extension without the ring that has no zero divisors is – a field of rational numbers. A continuous extension of  a field is the set of real numbers , typically thought of as points on  a straight line. Natural numbers serve as counters of discrete objects, integer numbers assist in comparing discrete sets: more or less, and how much more or less, while rational numbers help to divide a whole into parts, and real numbers – to measure distances. Hence, each number system is destined for some practical application. Most people are satisfied with those practical number systems because complex numbers, once called imaginary numbers, are regarded as a strange idea of a mathematician who created it merely to have a new object of interest. However, nobody has any doubts that in nature there exist disks such as round cakes and the circular sectors of such Arkadiusz Maciuk, Antoni Smoluk Department of Mathematics and Cybernetics, Wrocław University of Economics [email protected] 78 Antoni Smoluk, Arkadiusz Maciuk disks – slices of cakes. These portions of cake shapes that we call here circular triangles represent precisely complex numbers. Complex numbers are not pure inventions of unworldly mathematicians, but more practical and ordinary than real numbers. The most frequent definition of a complex number is equivalent to a pair of real numbers. Hence, a set of complex numbers is frequently, and for a very good reason, identified with a plane  2, i.e. a Cartesian product of two straight lines. Writing a complex number c∈ in the form z = a + bi actually means that z =(a,b)=a(1,0)+b(0,1). The vector (1,0) is treated as a unit in the field of complex numbers, while the vector i =(0,1) is called a complex unit. A linear algebra  of dimen- sion 2 over a field becomes a field of complex numbers once we assume  one relation defining that i2 = −(1,0). The idea of a circle commonly occurring in nature can be derived from a field of real numbers. Namely, a circle is a quotient group of a field of  real numbers  by a subgroup 2π of integer multiples of a number 2π. Thus, it holds = /2π. 2.Archimedes and complex numbers A circle implies angles, therefore angles are equivalence classes of real numbers translated by 2πn, where n∈. Archimedes’ constant π used to define a circle has merely a normalizing character that comes from the claim that the circumference of a circle with the radius 1 equals 2π. Since Ar- chimedes introduced the notion of an arc whose essence is an angle (Figure 1), hence he can be regarded as the pioneer and even as the father of com- plex numbers. He did not invent them, but discovered them – as they exist in nature as circular objects such as trees, stones, shapes of planets and their orbits. Complex numbers can be interpreted as linear transformations of a −b two-dimensional space 2. Linear map   is an analogue of com- b a  plex number z =a+bi. The equality of isomorphic objects does not imply their sameness, but their correspondence, equivalence. This isomorphism identifies complex numbers with orthogonal transformations of a given shape; orthogonality here means that rows are orthogonal vectors. If the radius of a complex number r = a2 +b2 is strictly greater than 1, then the Vortices and complex numbers 79 above-given linear operation is the composition of rotation by an angle represented by z and of dilation represented by the radius r. On the other hand, if r is strictly less than 1, then the linear operation is reduced to a contraction, i.e. a shrinkage. The equality of a unitary radius denotes a pure rotation. A dilation implies that successive compositions of linear operations represented by r diverge from the point of equilibrium, whereas the radius that is less than unity secures the convergence to the point of equilibrium. All vortices observed in the plane, such as flowing water or windswept leaves, have this nature. Thus, complex numbers can be observed in nature because they model what we typically see, i.e. the states of equilibrium or its perturbation – shifting away from equilibrium. 3.Isomorphic models of complex numbers A multiplicative group * of nonzero complex numbers is isomorphic to a product group ×*, where * denotes a multiplicative group of real + + positive numbers. Thus, each complex number is a pair: an angle and a radius (Figure 1). So it holds z = ru, where r∈ is a radius, while u + belonging to a group is identical to an angle φ since it holds  u =cosφ+isinφ, 0≤φ<2π. In this paper, isomorphic structures are re- garded as identical by default – they are not distinguished. A complex number therefore is a circular triangle with the area σ= 1rα, where α is an arc of the length rφ. Hence the area of a circular 2 triangle equals σ= 1r2φ. Only those complex numbers that are also real, 2 strictly positive numbers have the zero area. Numbers with a zero imaginary part, i.e. those with φ=0, have the zero area, as they are line segments. Complex numbers are also bivectors; the number z =a+bi is represented by a pair of orthogonal vectors (a,b),(−b,a) (Figure 2). These vectors determine a square whose area is radius squared, i.e. S =r2 =a2 +b2. In the latter case each non-zero complex number has a strictly positive area. Only the number zero has a zero area. A twofold emphasis on the area of a com- plex number indicates that these numbers are flat entities – they represent angles and arcs. The modulus of a complex number therefore is a root of its area S . 80 Antoni Smoluk, Arkadiusz Maciuk Fig. 1. A complex number – a piece of cake Fig. 2. Bivector – a complex number Source: own elaboration. Complex numbers are entities represented by arcs and circles or by lin- ear transformations being actually bivectors – squares. So, a complex num- ber can be interpreted as a circular triangle – a piece of cake, or as a square – a linear transformation. These are tangible entities, at least to a degree like real numbers are tangible. Real numbers measure just lengths, while com- plex numbers measure both lengths and arcs, hence they are said to have an area. 4.Vortices and spirals A subgroup n of nth roots of unity of a group  can be identified with a gear. A set of such subgroups represents a mechanical clock. These sub- groups contain cyclicality of time. Following a number of rotations the cogs of a gear mesh together like they did at the beginning. Periodic functions occur in nature in the same way as complex numbers; complex numbers and periodic functions model phenomena called vortices. What is a periodic function? Each function whose domain is the circle group is said to be  periodic. The most important periodic function is the cosine and other trigo- nometric functions naturally related with the cosine. The idea of periodicity is fundamental in natural sciences. Life can be defined as the continuous renovation of an organism. Linear maps a −b A(z)= Az, where A=  b a  are a matrix representation of a complex number a+ib, in a complex do- main are reduced to composition of rotations and real linear functions. Rotation is the essence of periodicity. Vortices and complex numbers 81 If matrix A≠0, then a group of linear transformations A (z)=(a+bi)tu, with a composition satisfying a natural condition t A °A = A , t∈,u∈, is connected with the above linear operation. As t t t+t 1 2 1 2 a consequence, a group of operators generated by matrix A is obtained. An orbit of the point u∈ is the set O(u)={A (u):t∈} that is a spiral t function of the parameter t with equation r (t)=ρte(ψ+tφ)i, where A A=ρeiφ, u =eiψ and t denotes time. Exactly one orbit passes through every point of the group *. An orbit is a subset of a field . A function f speci- fying the orbit O(u) is a subset of ×, because f ={(t,Atu):t∈}, u thus f (t)= Atu. A function f is the trajectory of the point u moving u u along the orbit O(u). In order to obtain all possible orbits, it suffices to consider only the numbers belonging to the circle group . A vortex of the  number A denoted by Vor(A) is a family of orbits Vor(A)={O(u):u∈T}. Obviously, orbits of different points in the circle are disjointed. The  orbits of points in the circle with the radius ρ∈* are formed by the orbits + of points in the circle : it suffices to multiply the latter by ρ. The scale is  changed; the radius ρ becomes a unity. Fig. 3. Vortices generated by the number 2i and the number – 2i demonstrated respectively by three orbits of points being the cube roots of 1 Source: own elaboration. A family of spirals {r : A∈*} is the image of the vortex of the num- A ber A (Figure 3). A spiral O(u) is the orbit of the point u∈*, the image 82 Antoni Smoluk, Arkadiusz Maciuk of a half-line z =tu, t∈* , using the isomorphism of the additive group +  and the multiplicative group * The image of a circle centered at zero is + also a circle with a center at zero, with a suitably modified radius when | A|≠1. If a modulus of Ais greater than 1, then spirals diverge; if it is equal to 1, then the motion is purely circular; when a modulus of A is less than 1, spirals converge in the point 0. Spiral orbits are actually exponential spirals with an equation in polar coordinates r =aebt where t∈. The angle φ of the number A determines the intensity of convolutions and the direction of the spiral’s rotations. Conjugation of the number A establishes a spiral with the same intensity of convolutions, but with the opposite rotation. If the motion of the point in the orbit is clockwise in the first case, then it is counter clockwise in the other (Figure 3). If the number is real, or the real part is zero, then the orbit is a meridian – the point does not rotate. The orbit of the point zero is a single point – zero. A vortex is actual- ly a product of exponential growth t∈→ρt∈ and of rotation of the circle t∈→eit∈. A vortex is motion (t,ψ)∈×[0,2π]→ρte(φt+ψ)i, where A=ρeiφ and ρ2 =a2 +b2. 5.Spirals on a sphere The field of complex numbers compactified by one point becomes  the two-dimensional sphere, the added point is called the North Pole ∞, the point zero is called the South Pole 0, and the group becomes the Equator  of this sphere. For the sake of the subsequent inquiry, it is convenient to assume that 0⋅z =0,∞⋅z =∞, where z is any point of the sphere. As a consequence, the product of zero and infinity is not commutative, because it holds 0⋅∞=0, but ∞⋅0=∞ The southern hemisphere can be represented parametrically in the form x =rcosφ,y =rsinφ, z =θ, where r∈[0,1], S φ∈[0,2π); and parameter θ satisfies (ρ−1)2 +r2 =1, so θ=1− 1−r2 . The equation of the northern hemisphere is given by analogous formulas: x = rcosφ,y = rsinφ, z = 2−z , where r∈[0,1], φ∈[0,2π). Therefore, N S the North Pole has the coordinates (0,0,2), and the South Pole (0,0,0) Complex numbers are made equivalent to a sphere with the radius 1 and the Vortices and complex numbers 83 center (0,0,1), naturally exclusive of the north pole. After compactification, a field develops into a sphere represented by the equation  x2 + y2 +(z−1)2 =1. A parametric representation of a spiral on this sphere has the form: x =ρ−t cos(φt+ψ)   y =ρ−t sin(φt+ψ)  z =1+sing(t) 1−ρ−2t , where ρ is the modulus of the complex number A=ρeiφ=a+bi, φ – the angle generated by this number, ψ – the angle related to the number u∈ and t∈. If ρ>1, then a vortex is the outflow from the South Pole and moves northbound, whereas if ρ=1 then a vortex is a stable rotation whose orbits are parallels (circles of latitude). If ρ<1, then a vortex moves south- bound from the north –the point zero, i.e. the South Pole is the equilibrium point. For ρ≠1 poles are skew-symmetric points: when a point on the orbit moves about one pole to the left, then its motion around the other pole is in the opposite direction. The poles are the limit points of these orbits (Figure 4). Fig. 4. A spiral on a cone and its graphical projection on the XZ plane Source: own elaboration. 84 Antoni Smoluk, Arkadiusz Maciuk One may otherwise explain that complex numbers dance around poles. The following kinds of orbits are distinguished: clockwise and counterclockwise spiral orbits, meridian (semicircular) orbits and parallels. Parallel orbits separate the set of clockwise spiral orbits from the set of counterclockwise spiral orbits, whereas meridian orbits separate the set of spiral orbits of complex numbers with the strictly negative imaginary part from the set of spiral orbits of complex numbers with the strictly positive imaginary part. A family of spiral orbits of the corresponding type is stable. On the other hand, families of parallel and meridian orbits are not stable – they can be regarded as catastrophic states, because any trivial shift in the number gen- erating a vortex induces a completely different family of orbits. Using the above mentioned cogs  , let us introduce a general defini- n tion of a periodic function. Periodicity is related to automorphic functions. If G is any group, and H – a normal subgroup G, then the function f :G →Y is periodic with respect to a subgroup H , if f (x)= f (x+h), in an additive notation, for x∈G, h∈H . Thus, the function f is automorphic with respect to a subgroup H . Its translations remain un- changed. A group is a set of objects called transpositions in elementary  teaching. The quotient group P()=/D, where D is a two-element 1 group {−1,1}, is a family of directions on a plane. What are the continuous ( ) functions from a linear space C(P  ,)? They are periodic continuous 1 ( ) functions belonging to a space of continuous functions C T, , whose diagrams are symmetric with respect to any plane passing through the axis of the cylinder in which these diagrams are positioned. The existence of the group P() is obvious in terms of algebra. However, this is a topological 1 group. Is it possible to attach opposite points of the circle so as to change the direction of rotation? In practice it is not possible – there is no model of such a group in nature. It is worth noting besides that P()=/D =* /* and D =* /* . A topology on P() is the 1 + 1 strongest topology in which a projection Ψ:T → P() defined by the 1 formula Ψ(u)=u/D is still a continuous function. The real time is a mixture of circular time with linear time. Such a combination yields a spiral on a cone. If the motion along the spiral takes place to the top of the cone, then we deal with a stabilizing vortex – a sys- tem given by the spiral tends to an equilibrium. If the motion is in the oppo- site direction, away from the top of the cone, then a cybernetic system de- Vortices and complex numbers 85 fined by such a spiral becomes destabilized and as a consequence it is de- x2 y2 z2 stroyed. The equation of an elliptical cone is of the form: + − =0; a2 b2 c2 if a =b, then the cone is rotational – circular. A parametrical representation of this cone is given by the system of equations: x =atcosφ  y =btsinφ ,   z =ct where t,φ∈. So a spiral on this cone is a line given parametrically by the system of equations: x =atcos(τt+ψ)  y =btsin(τt+ψ) ,  z =ct, where t∈, and the quantities a,b,c,τ,ψ∈ are fixed. In addition, we recall that Archimedes’ spiral in polar coordinates is defined by the equation r =ρφ, where φ∈. The observed stock exchange data make it possible to identify a cone and to determine a spiral that describes a stock market performance. The estimated cone is presented in a canonical form, that is given above, suitably changing the coordinates. The canonical form provides a timely recognition of market trends: whether a stock exchange tends to equilibrium or to desta- bilization. Stock exchange data are typically observed as a curve with verti- ces (a polygonal chain) that is equivalent to a projection of a spiral placed in a cone onto a plane. Certainly, a character of this spiral varies with time. Stock exchange time can be naturally divided into homogenous intervals in which a spiral remains stable. The type of the spiral can be an indication for investors. If the examined system tends to destabilization, then one can either earn a lot or lose a lot; when the system tends to stabilization, risks are reduced, but profits are moderate. 6.Practical remarks A vortex is measured by a vortex and length by length. There exist vor- tices, perhaps not in nature, but only in theory, which cannot be measured 86 Antoni Smoluk, Arkadiusz Maciuk by complex numbers, in a similar way as there exist artificial lines without lengths. The richness of complex numbers supports the hypothesis that one can measure any natural vortex by an appropriate complex number. Measuring a vortex observed on a windy day is a straightforward activi- ty. One takes two pictures of whirling dust so as to determine the three points: the center s of the vortex, the initial point p and its image q. A ratio   of the distance r = sp to the distance r = sq is the modulus of a complex 1 2   number, and the angle between its radiuses sp, sq and its argument φ. Therefore A=ρeiφ. A cobweb model for determining the equilibrium price is a well-known economic equivalent of a vortex. If the relationship between supply and price and the relationship between demand and price are known, then a cobweb model defines a price equate at which supply and demand are bal- anced [Jakimowicz, Juzwiszyn 2012]. Galloping inflation is an example of a harmful phenomenon that results in motion leading to destabilization – excess money supply in the market causes a financial crisis marked by a general price increase [Dechert 1996]. Typical vortices are observed on a stock exchange. These are spiral motions on a cone resulting from changes in numbers of stocks and their prices over time [Juzwiszyn 2010]. Vortices do not exist without time, as they imply a change and motion. In order to describe an economic vortex by a single complex number, we need to measure it. Vortices concerning stock ex- changes should be measured in short time intervals. A measurement consists in determining an angle and a radius of a vortex, i.e. a related complex number is determined. The case in outer space is similar, however the measurements are carried out in appropriately long time intervals. The rotational motion commonly observed in outer space results from gravitation forces among material objects. Two bodies independently sent into space will establish one set of objects rotating around each other after some time. Every day we deal with a sun clock containing the three cogs – the Sun, the Earth and the Moon. The music of the celestial spheres is an ancient concept. The music of celestial and financial spheres is encoded by complex numbers. The sound of these numbers is an open question. The frequent turbulences observed in nature and in societies result from interactions among different forces. The net force is a flat spiral or a spatial spiral on a cone. Social disturbances stimulated by agitators can turn into anarchy and destroy the entire society. Therefore, political, religious and professional groupings should be monitored. Social organizations are con-