arXiv:2003.00852v1 [math.NT] 27 Feb 2020 olwn manner: following hr r orsc crlso iia oncint Fibona to Fo connection similar of of tangency + of such four points 1.1. are Figure the there See through concyclic! passes are (1.1) so-defined points that out turns It The tlosadfes“-iesoa” u hr sahde cir fo hidden the a with points is and there pairs, consecutive But of ratios “1-dimensional”. Consider feels and looks It Let Introduction 1. F x eateto ahmtc,Suhr liosUniversity, Illinois Southern , of Department n MSC: c Keywords: that ratio. circles golden four the circ form to Ford They related of points points numbers. tangency Fibonacci involve certain Namely, and numb shown. Fibonacci is circles, ratio Ford between connection amusing An codntso 11 r aispoue rmFbncinu Fibonacci from produced ratios are (1.1) of -coordinates eteFbncisqec aee sfollows: as labeled sequence Fibonacci the be ioac ubr n odcircles Ford and numbers Fibonacci 13,5C6 88,51M15. 28A80, 52C26, 11B39, 32 34 21 13 8 5 3 2 1 1 0 1 0 ioac ubr,Fr ice,gle ratio. golden circles, Ford numbers, Fibonacci F 0 = 0 , 2 1 F 1 = 1  [email protected] , ac ,2020 3, March ez Kocik Jerzy F F 5 3 2 Abstract n 2 F n + n 1 + 1 , 1 F = 2 1 n F + 13 8 1 n  + F n − 1 , abnae IL62901 Carbondale, lwn coordinates: llowing c numbers. cci 34 21 n u hr smr.The more. is there But ∈ e r concyclic are les l nti sequence. this in cle dcrls Actually, circles. rd ttexai at x-axis the ut r,adgolden and ers, Z ... br nthe in mbers (1.1) JerzyKocik FibonaccinumbersandFordcircles 2

0 1 1, 1 1  

1, 1 1 2 2 2  

3, 1 1 5 5 5  8 1 1 13, 13 13 1   02 1 1 ϕ−

Figure 1.1: Fibonacci points

2. Ford circles

Let us recall some basic facts about Ford circles.

Proposition 2.1. [L. Ford] Define K p, m to be a circle of radius 1 2m2 to = p [ ] / x-axis at point x m . Then

p q det = 1 K p, m and K q, n are tangent (2.1) m n ± ⇒ [ ] [ ]   and otherwise are disjoint. Moreover, the Ford circle at p q p + q x = (2.2) n ⊕ m ≡ m + n is inscribed in the triangle-like region between the two at p n and q m, / / and the x-axis.

For the proof see [2].

Remark: We shall also include the horizontal line y = 1 among the Ford circles, as = 1 a line over the point at infinity, x 0 . JerzyKocik FibonaccinumbersandFordcircles 3

0 1 1 1 1 2 1 3 2 3 4 1 1 6 5 4 3 5 2 5 3 4 5 1

Figure 2.1: Ford circles

Proposition 2.2. The point of tangency of two tangent Ford circles (2.1) is

pm + qn 1 K p, m K q, n = , (2.3) [ ] ∩ [ ] m2 + n2 m2 + n2   Proof: Simple similarity of triangles leads to q x : x p = R : r with the ( n − ) ( − m ) R = 1 r = 1 x  radii being 2n2 and 2m2 , which easily solves for .

Remark: Remark: Note that formula (2.3) differs from the one given in [1] and reported in [5]; it is symmetric and does not depend on the order of circles.

R r

p q m x n

Figure 2.2: Derivation of formula (2.3) JerzyKocik FibonaccinumbersandFordcircles 4

3. Fibonacci numbers

By the extended Fibonacci sequence we understand a bilateral sequence Fn defined by: F0 = 0, F1 = 1, Fn+1 = Fn + Fn 1, n Z − ∈ It looks like this ... 5 3 2 1 1 0 1 1 2 3 58132134 ... − − ↑ ↑ ↑↑↑↑↑↑↑ F 3 F 2 F 1 F0 F1 F2 F3 F4 F5 − − − The extended Fibonacci sequence satisfies the basic identities valid for the standard Fibonacci sequence. Among them are the following:

i Fn 1Fn + FnFn+1 = F2n ( ) − 2 2 ii F + F = F n+ (3.1) ( ) n n+1 2 1 2 = n iii Fn+1Fn 1 Fn 1 ( ) − − (− ) 4. Ford and Fibonacci

1 1 0 1 3 2 1 - 1 - 2 1 2 5 3 1 ϕ ϕ 1 − −

Proposition 4.1. Let Fn be the extended Fibonacci sequence: Then the sequence of points with coordinates F 1 2n , (4.1) F + F +  2n 1 2n 1  lie on a circle centered at 1, 0 of radius √5 2. Moreover, the above points are (− 2 ) / among points of tangency of Ford circles. The circle cuts the x-axis at the golden cut x = ϕ 1 and negative golden ratio, x = ϕ. − − JerzyKocik FibonaccinumbersandFordcircles 5

Proof: First, let us prove the the points are concyclic. Denote 2n = m. We need to verify the quadratic circle equation

F 1 2 1 2 5 m + + = . F + 2 F + 4  m 1   m 1  After squaring, collecting like terms, and factoring out the common 4, we get

2 2 F + FmFm+ = F + 1 . m 1 − 1 m

Next, factor the common Fm+1:

2 Fm+ Fm+ Fm = F + 1 1 ( 1 − ) m and use the Fibonacci’s defining property to get:

= 2 + Fm+1Fm 1 Fm 1, − which is one of the standard identities true for even m, see Eq. (3.1) (iii).

For the second part, consider the Ford circles above the Fibonacci fractions:

Fn 2 1 1 0 1 1 2 3 5 : ..., , − , , , , , , , , ... Fn+ 1 1 0 1 1 2 3 5 8 1 − The consecutive Ford circles over this sequence are clearly mutually tangent, since

Fn 1 Fn n det − = 1 Fn Fn+1 (− )   thus (2.1) is satisfied. What remains is to check what are their points of tangency. For this, we use Proposition 2.2. Under substitution we get:

Fn 1Fn + FnFn+1 1 F2n 1 K F , F K F , F + = − , = , n 1 n n n 1 2 2 2 2 [ − ] ∩ [ ] F + F F + F F n+ F n+ n n+1 n n+1 !  2 1 2 1  where we used (3.1) (i) and (ii). This concludes the proof.  JerzyKocik FibonaccinumbersandFordcircles 6

D A B C

3 2 1 0 1 2 3 − − − ϕ2 ϕ τ τ2 τ2 τ ϕ ϕ2 − − − − Figure 5.1: Four golden circles upon Ford circles

5. Four concyclic sequences

There are four instances of similar correspondence between Fibonacci sequence and Ford circles. Here they are denoted as (A), (B), (C), and (D), together with the defining recipe: 3 1 0 1 3 8 21 A: − 2 − 1 1 2 5 13 34

... 5 3 2 1 1 0 1 1 2 3 5 8 13 21 34 ... − −

B: 3 1 0 1 3 8 21 − 5 − 2 1 1 2 5 13

5 2 1 1 2 13 C: 2 1 5 1 2 5 13 34

... 5 3 2 1 1 0 1 1 2 3 5 8 13 21 34 ... − − 5 5 2 1 1 2 13 1 2 5 − 13 21 D: − 2 − 1 − − − − Sequence (A) is the one discussed in the previous sections. Each sequence corespondents to a circle of radius √5 2 and intersects ford circles at points the / coordinates of which are ratios of Fibonacci numbers. The centers lie on the x-axis at the following points: A : x = 1 2, B : x = 1 2, C : x = 3 2, D : x = 3 2 − / / / − /

The circles cut the x-axis at different values of the golden ratio, as show in Figure 5.1. We use the following notation:

√5 + 1 √5 1 ϕ = 1.618, τ = − 0.618, ϕτ = 1 2 ≈ 2 ≈

Proof: One may start with sequence (A) and add 1 to the xvariable in every term:

F 1 F + 1 2n + 1, = 2n 2 , F + F + F + F +  2n 1 2n 1   2n 1 2n 1  JerzyKocik FibonaccinumbersandFordcircles 7 which leads to sequence (B). Shifting again by 1 produces

F + 1 F + 1 2n 2 + 1, = 2n 3 , F + F + F + F +  2n 1 2n 1   2n 1 2n 1  which defines the sequence and circle (C). Shifting (A) by a unit to the left produces

F2n 1 F2n 1 1 1, = − − , F + − F + F + F +  2n 1 2n 1   2n 1 2n 1  which produces thesequence and circle (D). 

Sequences (A) (B) and (C) are genuine ratios of the Fibonacci numbers, while sequence (D) has negative signs of the x-coefficients imposed artificially. Note that A and B are negatives of each other, and so are C and D. The interval (0,1) contains only fragments of (A) and (B).

Remark: Note that the open circle y = 1 contributes to the points in the sequences (A), (B), (C), and (D).

We finish with an image showing magnification of the circle system over the interval 0, 1 (Figure 5.2). [ ]

0 1 1 1 1 2 1 3 2 3 4 1 1 6 5 4 3 5 2 5 3 4 5 1 1 ϕ2 1 ϕ / / Figure 5.2: The indicated points of tangency between the circles are of form of fractions involving the Fibonacci numbers:

1 3 8 1 √5 1 right sequence: , , , ... = − 2 5 13 −→ ϕ 2 1 2 5 1 3 √5 left sequence: , , , ... = − 2 5 13 −→ ϕ2 2 JerzyKocik FibonaccinumbersandFordcircles 8

References

[1] Apostol, T. M., Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, 1997 (Âğ5.5, p. 101). [2] Lester Ford, Fractions, AMM, 45, No. 9 (1938), pp. 586-601. [3] Jerzy Kocik, A theorem on circle configurations, arXiv:0706.0372. [4] Steven Vajda, Fibonacci and Lucas Numbers, and the Golden Section, Dover, 1989. [5] Weisstein, Eric W. "Ford Circle." From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/FordCircle.html