Fibonacci Points and Inverse Stereographic Projection

Fibonacci Points and Inverse Stereographic Projection Baris Ates1, Selcan Ates2, Alp Kustepeli3 1Department of Mathematics, Izmir Institute of Technology, Türkiye 2Department of Chemical Engineering, Izmir Institute of Technology, Türkiye 3Department of Electrical & Electronics Engineering, Izmir Institute of Technology, Türkiye Corresponding author: [email protected] ORCID IDs: First Author: 0000-0002-8138-7360 Second Author: 0000-0002-0078-0647 Third Author: 0000-0002-8995-6628 DOI : 10.5281/zenodo.18011963 Abstract In the present study, we investigate the correspondence between Fibonacci structures in the Euclidean plane and their spherical counterparts through the inverse stereographic projection (ISP). The ISP provides a one-to-one mapping that enables the transfer of algebraic and geometric relations from the plane to the sphere. By applying this mapping to Fibonacci points, one can define spherical Fibonacci identities and explore their geometric manifestations on the sphere. Moreover, the sphere can be continuously deformed while preserving a bijective relation between the original and the deformed surfaces. This deformation framework allows the extension of planar Fibonacci identities to generalized spherical geometries, including spheroidal and irregular surfaces. Such deformations can represent naturally occurring shapes, thus establishing a potential connection between Fibonacci-based mathematical identities and biological or physical pattern formation. The proposed approach not only provides a geometric interpretation of Fibonacci identities on curved manifolds but also opens new perspectives for modeling spatial structures in nature through analytical deformations of spherical surfaces. Keywords: Fibonacci Points, Stereographic Projection, Deformed Spheres, Spirals. 1 Introduction The Fibonacci sequence, despite its deceptively simple recursive definition, is known to appear in a remarkable variety of natural and physical systems [ 1 , 2 ]. Classical examples such as phyllotactic arrangements in sunflowers, pine cones, and other plants demonstrate how Fibonacci numbers can model the organization of spirals and growth mechanisms in nature [ 3 , 4 ]. Owing to its intrinsic connection with the golden ratio, the sequence also arises in art, architecture, and engineering [ 5 , 6 ]. Furthermore, recent developments indicate that Fibonacci structures continue M7-1 KOSC-2025 Proceedings to play a role in emerging areas: for instance, laser-driven quantum states arranged according to Fibonacci patterns enable enhanced information storage capabilities in quantum systems [ 7 ]. These observations collectively highlight the relevance of studying Fibonacci-based constructions within broader geometric and analytical contexts. One of the most effective tools for transferring planar structures to curved geometries is the inverse stereographic projection (ISP). Stereographic projection is a classical conformal mapping between the sphere and the plane, preserving angles and providing a one-to-one correspondence except at the projection point [ 8 , 9 ]. ISP acts consistently on both discrete point sets and continuous curves, it becomes a natural framework for exploring how algebraic relations—such as those defined by Fibonacci numbers—transform when transferred onto curved surface. The geometric picture becomes even richer when one considers deformations of the sphere. Deforming the sphere through arbitrary single-valued functions in the normal direction enables the construction of spheroidal or even non-differentiable surfaces that still remain in bijective correspondence with the plane. This generalized setting allows planar structures to be transported onto a wide range of geometries, including those resembling naturally occurring shapes that deviate from perfect spherical symmetry. In the present work, we build upon these ideas to investigate the correspondence between Fibonacci configurations in the Euclidean plane and their spherical or deformed spherical counterparts. Starting with planar points whose coordinates are determined by Fibonacci numbers, we apply the inverse stereographic projection to construct their images on S 2 and derive spherical analogues of well-known Fibonacci identities. Moreover, by introducing analytic deformations of the sphere, we generalize these constructions to more complex surfaces. This framework not only offers a geometric reinterpretation of Fibonacci identities on curved manifolds but also suggests potential links with biological and physical systems in which Fibonacci-like patterns emerge. 2 Inverse Stereographic Projection of Fibonacci Points In this section, a short review of the previously obtained results will be presented. The Fibonacci points introduced in [ 10 ] are the points such that, the absolute value of each coordinate of the point is a Fibonacci number. Thus, such points can be written in the form P(x, y)=P(c1Fn, c2Fm)(1) where c1 and c2 are either +1 or − 1. The stereographic projection is a mapping that relates the points on the sphere with the points on the plane. This mapping is one to one and invertible. For a sphere located on the ζ axis and described by ξ2 + η2 + ( ζ−R ) 2 = R2 , stereographic projection of the point Ps ( ξ, η, ζ )on the sphere is mapped to a unique point P ( x, y )on the x−y plane via stereographic projection as Ps(ξ, η, ζ)=Ps 4R2x 4R2+x2+y2,4R2y 4R2+x2+y2,2R(x2+y2) 4R2+x2+y2!(2) M7-2 2nd Kocaeli Science Congress, November 19-21, 2025 2.1 Rotation of Fibonacci Points and conversely, the inverse stereographic projection of a point P(x, y)is found by [8], [9]: P(x, y) = P2Rξ 2R−ζ,2Rη 2R−ζ.(3) The ISP of a Fibonacci point is given by the following relation: Ps(ξF n,m, ηF n,m, ζF n,m)=Ps 4R2c1Fn 4R2+F2 n+F2 m ,4R2c2Fm 4R2+F2 n+F2 m ,2R(F2 n+F2 m) 4R2+F2 n+F2 m!.(4) Let us consider the special Fibonacci points Pn ( Fn, 0) on the x -axis, whose first coordinates are Fibonacci numbers, i.e., P0 (0 , 0) , P1,2 (1 , 0) , P3 (2 , 0) , . . . , Pn ( Fn, 0). The coordinates of the ISP of the points Pn(Fn,0) are obtained as PsξF n, ηF n, ζF n=Ps 4R2Fn 4R2+F2 n ,0,2RF 2 n 4R2+F2 n!.(5) In order to find a relation for the ξF n , one can consider the first coordinates and rearrange them as ξF nF2 n−4R2Fn+ 4R2ξF n= 0 which yields Fn=2R2+ 2RλnqR2−(ξF n)2 ξF n ,(6) where λn is used to denote λn ( R )whose value is simply equals to ± 1depending on the values of nand R[10]. Considering the Equation (2), one can conclude that the coordinates of the ISP of the sequential points Pn(Fn,0) satisfy the following nonlinear addition relations: 2R2+ 2λnRqR2−(ξF n)2 ξF n +2R2+ 2λn+1RqR2−(ξF n+1)2 ξF n+1 =2R2+ 2λn+2RqR2−(ξF n+2)2 ξF n+2 , (7) ηF n=ηF n+1 =ηF n+2 = 0,(8) sζF n 2R−ζF n +v u u t ζF n+1 2R−ζF n+1 =v u u t ζF n+2 2R−ζF n+2 ,(9) 2R ξF n 2R−ζF n +2R ξF n+1 2R−ζF n+1 =2R ξF n+2 2R−ζF n+2 (10) where short hand notation λn=λn(R)is used and they are equal to ±1for any n. 2.1 Rotation of Fibonacci Points The rotation of the special Fibonacci points Pn ( Fn, 0) about the origin by an angle according to the transformation rule x′= cos αx −sin αy y′= sin αx + cos αy (11) 2nd Kocaeli Science Congress, November 19-21, 2025 M7-3 KOSC-2025 Proceedings allows one to obtain new points P′ ( x′, y′ )such that their distance to the origin is a Fibonacci number. ISP of those rotated Fibonacci points also satisfy following nonlinear addition relations: 2R2cos α+ 2λ′ nRqR2cos2α−(ξ′ n)2 ξ′ n +2R2cos α+ 2λ′ n+1RqR2cos2α−(ξ′ n+1)2 ξ′ n+1 =2R2cos α+ 2λ′ n+2RqR2cos2α−(ξ′ n+2)2 ξ′ n+2 ,(12) 2R2sin α+ 2λ′ nRqR2sin2α−(η′ n)2 η′ n +2R2sin α+ 2λ′ n+1RqR2sin2α−(η′ n+1)2 η′ n+1 =2R2sin α+ 2λ′ n+2RqR2sin2α−(η′ n+2)2 η′ n+2 ,(13) sζ′ n 2R−ζ′ n +v u u t ζ′ n+1 2R−ζ′ n+1 =v u u t ζ′ n+2 2R−ζ′ n+2 (14) where λ′ n s are equal to ± 1[ 10 ]. The relations between the mixed coordinates of the images of the points on the rotated line can also be obtained from Equation (3) as ξ′ n 2R−ζ′ n +ξ′ n+1 2R−ζ′ n+1 =ξ′ n+2 2R−ζ′ n+2 (15) η′ n 2R−ζ′ n +η′ n+1 2R−ζ′ n+1 =η′ n+2 2R−ζ′ n+2 .(16) For the most general Fibonacci numbers P(c1Fn, c2Fm) . Inverse stereographic projection allows to find the following relations between the coordinates of the sequential points [10]: ξn,m 2R−ζn,m +ξn+1,m 2R−ζn+1,m =ξn+2,m 2R−ζn+2,m (17) ηn,m 2R−ζn,m +ηn,m+1 2R−ζn,m+1 =ηn,m+2 2R−ζn,m+2 .(18) 2.2 Generalization of Inverse Stereographic Projection The correspondence between the points on the sphere and the points on the plane governed by the stereographic projection can be generalized to the deformed spheres [ 8 ], [ 11 ], [ 12 ]. The position vector of a point on the sphere centered at the origin is given in spherical coordinates as rsph = Rˆr. Deformation of this sphere in radial direction is achieved by the following relation [13] rd=R(1+βf(θ, φ)) ˆr(19) where f ( θ, φ )is arbitrary single valued function and β is the deformation parameter. In order to find images of plane points on these deformed spheres via ISP, one may first transform the point P0(x, y) onto the sphere by using (2) and obtain a point P1 = ( ξ1, η1, ζ1 ). Since M7-4 2nd Kocaeli Science Congress, November 19-21, 2025 the origin of this sphere is located in the ζ axis, by a translation one can obtain a new point P2 = ( ξ2, η2, ζ2 ) = ξ1, η1, ζ1−R on the sphere whose center is located at the origin. The deformation function depends on the angular variables θ and φ and thus the amount of the radial deviation depends on the position of the point on the sphere. In order to use (19), which governs the surface deformation, one has to determine the spherical angles. The values of the θ and φcan be found from the following definitions: θ=Arccos(ζ2 R), φ =Arctan(y x).(20) Once the spherical angles are found, according to the deformation (19), one can obtain a new point P3(t)=(ξ3, η3, ζ3)on deformed sphere with the following relations: ξ3=ξ2+βRfθ, φsin θcos φ(21) η3=η2+βRfθ, φsin θsin φ(22) ζ3=ζ2+βRfθ, φcos θ.(23) The final point P3 is the inverse stereographic projection of P0 onto the deformed sphere. For the uniqueness of the transformation one may consult the work [11]. 3 Relations That are Satisfied by the ISP of the Fibonacci Points on Deformed Spheres For the most general Fibonacci Points P = ( c1Fn, c2Fm ), ISP of those points on the deformed spheres satisfy following identities: ξ3 n,m −Rβf cos θcos φ R(1+βf cos θ)−ζ3 n,m +ξ3 n+1,m −Rβf cos θcos φ R(1+βf cos θ)−ζ3 n+1,m =ξ3 n+2,m −Rβf cos θcos φ R(1+βf cos θ)−ζ3 n,m (24) and η3 n,m −Rβf sin θsin φ R(1+βf cos θ)−ζ3 n,m +η3 n,m+1 −Rβf sin θsin φ R(1+βf cos θ)−ζ3 n,m+1 =η3 n,m+2 −Rβf sin θsin φ R(1+βf cos θ)−ζ3 n,m+2 .(25) These two equations, together with the (19), allows one to find ξ3 n+2, η3 n+2, ζ3 n+2 when ξ3 n, η3 n, ζ3 n and ξ3 n+1, η3 n+1, ζ3 n+1 are known. The general Fibonacci numbers Pn,m ( c1Fn, c2Fm )has quite interesting properties. For the fixed m , if one considers the sequential points Pn,m , Pn+1,m , and Pn+2,m which are the points located on the line y = c2Fm , then ISP of those points satisfy the relation (24), likewise, for the fixed n , if one considers the sequential points Pn,m , Pn,m+1 , and Pn,m+2 which are the points located on the line x = c1Fn , then ISP of those points satisfy the relation (25). 2nd Kocaeli Science Congress, November 19-21, 2025 M7-5 KOSC-2025 Proceedings 4 Fibonacci Identities and Their Transformations Fibonacci numbers satisfy a wide variety of identities. In order to illustrate the transformation of the identities, we will consider following relation: F2 n−1+F2 n=F2n−1(26) which is called the Cassini identity respectively [ 14 ]. The ISP of Fibonacci points which are located on the line y=c2Fmsatisfy the Cassini identity as follows; 2R ξ3 n−1,m −Rβf cos θcos φ R(1+βf cos θ)−ζ3 n−1,m !2 + 2R ξ3 n,m −Rβf cos θcos φ R(1+βf cos θ)−ζ3 n,m !2 =ξ3 2n−1,m −Rβf cos θcos φ R(1+βf cos θ)−ζ3 2n−1,m (27) Similarly, ISP of Fibonacci points which are located on the line x = c1Fn satisfy the Cassini identity as follows; 2R ξ3 n,m−1−Rβf cos θcos φ R(1+βf cos θ)−ζ3 n,m−1!2 + 2R ξ3 n,m −Rβf cos θcos φ R(1+βf cos θ)−ζ3 n,m !2 =ξ3 n,2m−1−Rβf cos θcos φ R(1+βf cos θ)−ζ3 n,2m−1 .(28) The other identities satisfied on the deformed spheres can be found in a similar manner. References [1] Cruz N., Olivares M., Villanueva J., (2017), The golden ratio in Schwarzschild-Kottler black holes, The European Physical Journal C, 77 (2), pp. 1-6. [2] Pletser V., (2017), Fibonacci numbers and the golden ratio in biology, physics, astrophysics, chemistry and technology: A non-exhaustive review, arXiv preprint, arXiv:1801.01369. 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A., (1984), Complex analysis with applications, Courier Corporation. M7-6 2nd Kocaeli Science Congress, November 19-21, 2025 REFERENCES [10] B. Ates, S.Ates, A.Kustepeli, A Novel Approach to the Fibonacci Numbers by Inverse Stereographic Projection, TWMS J. App. and Eng. Math, Accepted, 2025. [11] B. Ates, A. Kustepeli, Generalization of Stereographic Projection, hal-05098845, (2025). [12] Karlığa, Baki, On The Generalized Stereographic Projection, Contributions To Algebra and Geometry, 37(2) (1996), 329-336. [13] H. Ahmedov, I. H. Duru (2009),Boundary shape and Casimir energy, Journal of Physics A: Mathematical and Theoretical 42 (11), 115401. [14] Koshy T., (2001), Fibonacci and Lucas numbers with applications, John Wiley & Sons. 2nd Kocaeli Science Congress, November 19-21, 2025 M7-7