Finite Blaschke Products and the Golden Ratio

It is known that the golden ratio $\alpha =\frac{1+\sqrt{5}}{2}$ has many applications in geometry. In this paper we consider some geometric properties of finite Blaschke products related to the golden ratio.


Introduction
The golden ratio α = 1+ (1.1) The golden ratio appears in modern research in many fields. For example, in [13] the golden ratio is used in graphs, in [8] it is proved that in any dimension all solutions between unity and the golden ratio to the optimal spherical code problem for N spheres are also solutions to the corresponding DLP (the densed local packing problem) problem.
In this paper we give a connection between geometric properties of Blaschke products and the golden ratio.
The rational function is called a finite Blaschke product of degree n for the unit disc where |β| = 1 and |a i | < 1, 1 ≤ i ≤ n. We call the finite Blaschke products of the following form as canonical: z − a j 1 − a j z , |a j | < 1 for 1 ≤ j ≤ n − 1. (1.2) Note that the canonical Blaschke products correspond to finite Blaschke products vanishing at the origin. It is well-known that every Blaschke product B of degree n with B(0) = 0, is associated with a unique Poncelet curve (for more details see [2], [3] and [6]). From [2] we know that the Poncelet curve associated with a Blaschke product of degree 3 is an ellipse.
Here we investigate the relationships between the zeros of these canonical finite Blaschke products and the golden ratio for n = 2, 3, 4. Also we give some examples for the cases n = 5, 10.

Blaschke Products of Degree Two
Let AB be a line segment and C be a point on the line segment AB such that AC is the greater part of AB. Recall that we say the point C divides the line segment AB in the golden ratio if AC BC = α [9]. In this section we consider a finite Blaschke product B of degree two of the form with a = 0, |a| < 1. From [2], we know that there exist two distinct points z 1 and z 2 on ∂D that B a (z) maps to λ, for any point λ on the unit circle ∂D, and that the line joining z 1 and z 2 passes through a, the nonzero zero of B a . Conversely, let L be any line through the point a, then for the points z 1 and z 2 at which L intersects ∂D we have B a (z 1 ) = B a (z 2 ). Now we ask the following questions: 1) Does the point a divide the line segment [z 1 , z 2 ] joining z 1 and z 2 in the golden ratio?
2) If it does, what is the number of these line segments?
The answers of these questions are given in the following theorem.
Theorem 2.1. Let B a (z) = z z−a 1−az be a Blaschke product with a = 0, |a| < 1. There are infinitely many values of a such that there is a line segment with endpoints on the unit circle divided by a in the golden ratio. Furthermore the number of such line segments is at most two for a fixed a. Then we get 1 + |a| 1 − |a| > α.
If α = 1+|a| 1−|a| , then the line passing through the points z 1 , z 2 and a is the diameter of the unit circle.
For this reason, as long as 1+|a| 1−|a| ≥ α, there is a segment divided by a in the golden ratio. Now we find the number of the segments divided by a in the golden ratio for a such a.
Let a be chosen such that 1+|a| 1−|a| ≥ α and z 1 be chosen such that the point a divides the line segment [z 1 , z 2 ] in the golden ratio. Then by definition we have Using the fact that |z| = 1 for z ∈ ∂D we can write Also we know that B(z 1 ) = B(z 2 ) and so we obtain and from the equation (2.5) we find After substitute (2.7) into (2.6) we get the equation Clearly the last equation (2.8) has at most two roots with respect to z 2 . Hence there are at most two line segments [z 1 , z 2 ] passing through the point a and divided by a in the golden ratio. This fact can be also seen by some geometric arguments.

Blaschke Products of Degree Three
In this section, we consider a finite Blaschke product B of degree three of the form with distinct zeros at the points 0, a 1 and a 2 . It is well-known that for any specified point λ of the unit circle ∂D, there exist 3 distinct points z 1 , z 2 and z 3 of ∂D such that B(z 1 ) = B(z 2 ) = B(z 3 ) = λ.
We know the following theorem for a Blaschke product of degree three.
Theorem 3.1. (See [2] Theorem 1) Let B be a Blaschke product of degree three with distinct zeros at the points 0, a 1 and a 2 . For λ on the unit circle, let z 1 , z 2 and z 3 denote the points mapped to λ under B. Then the lines joining z j and z k for j = k are tangent to the ellipse E with equation Conversely, every point on E is the point of tangency of a line segment joining two distinct points z 1 and z 2 on the unit circle for which B(z 1 ) = B(z 2 ).
The ellipse E in (3.1) is called a Blaschke 3-ellipse associated with the Blaschke product B(z) of degree 3. There are many studies on the ellipse E given in (3.1) (see [3], [4], [5], [7], [10] and [11] for more details). For any λ ∈ ∂D, we know that E circumscribed in the triangle ∆(z 1 , z 2 , z 3 ), where z 1 , z 2 and z 3 are the points mapped to λ under B.
A golden triangle is an isosceles triangle such that the ratio of one its lateral sides to the base is the golden ratio α = 1+ √ 5 2 . A golden ellipse is an ellipse such that the ratio of the major axis to the minor axis is the golden ratio 1+ √ 5 2 (see [9] for more details). We have the following questions: 1) Are there any Blaschke 3-ellipses which are circumscribed (at least) one golden triangle?
2) Can a Blaschke 3-ellipse be a golden ellipse? If so, what is the number of these ellipses?
We begin by the answering of the first question.
Theorem 3.2. There are infinitely many golden triangles whose three vertices lie on the unit circle.
Proof. Without loss of generality, let x and y be chosen so that x, y > 0 and such that the triangle with vertices at the points 1, −x+iy, −x−iy is inscribed in the unit circle. We try to determine the values of x and y such that x 2 + y 2 = 1. By the definition of a golden triangle it is sufficient to show that there are values of x and y on the unit circle such that 2αy = y 2 + (x + 1) 2 . (3.2) Squaring both sides of (3.2) and using the fact that x 2 + y 2 = 1, we obtain 2y 2 α 2 = x + 1. Then we have Solving this quadric equation for x and y, we obtain x = 0, 809017 and y = 0.587785 where y = √ 1 − x 2 . So we have one golden triangle such that its vertices are on the unit circle. Then there are infinitely many golden triangles with vertices on the unit circle by rotation. Now we can construct some examples using some results from [3] and [7]. Recall that two sets {z 1 , z 2 , ..., z n } and {w 1 , w 2 , ..., w n } of points from ∂D are interspersed if 0 ≤ arg(z 1 ) < arg(w 1 ) < ... < arg(z n ) < arg(w n ) < 2π (see [3] for more details).
From [4], we know that the ellipses inscribed in triangles with vertices on the unit circle are precisely Blaschke 3-ellipses.
Example 3.1. Let ∆(z 1 , z 2 , z 3 ) be a golden triangle on the unit circle. From Theorem 2.1 in [7], we know that the Steiner ellipse E inscribed in this golden triangle has foci a 1 and a 2 with the following equation : and Then this Steiner ellipse E is the Poncelet curve of the Blaschke product Example 3.2. Let z 1 , z 2 , z 3 and w 1 , w 2 , w 3 be triples of points which form the golden triangles ∆(z 1 , z 2 , z 3 ) and ∆(w 1 , w 2 , w 3 ) on the unit circle so that {z 1 , z 2 , z 3 } and {w 1 , w 2 , w 3 } are interspersed sets of the points. From Corollary 10 on page 97 in [3], we know that there exists a Blaschke product B of degree 3 which maps 0 to 0 such that B(z j ) = B(z k ) and B(w j ) = B(w k ) for all j and k (1 ≤ j, k ≤ 3). Since we can choose the triples z 1 , z 2 , z 3 and w 1 , w 2 , w 3 by infinitely many different ways then clearly there are infinitely many Blaschke ellipses each of which has at least two golden triangle circumscribing them and having the vertices on the unit circle.
We have seen examples of Blaschke products of degree three of which Poncelet curves inscribed in at least one or two golden triangles. Now we consider the answer of our second question.  b 2 α = c 2 we find a = ±c √ α. Now we consider the Blaschke product associated with this ellipse. If this ellipse is a Blaschke ellipse, it must be 2a = 1 + c 2 by the definition of a Blaschke ellipse. Hence we find c 2 ± 2 √ αc + 1 = 0. As these equations have only one positive root c = 1 2 (− 2(−1 + √ 5)+ 2(1 + √ 5)), there is one golden ellipse which is a Blaschke ellipse. Since every rotation of this golden ellipse is again golden, clearly we have infinitely many golden Blaschke ellipses in the unit disc.
We give the following definition.
and a 2 = −a 1 . By the proof of Theorem 3.3 we know that the Blaschke 3-ellipse E associated with B 1 is a golden ellipse. So B 1 (z) is a golden Blaschke product. The image of this golden Blaschke ellipse under the rotation transformation f (z) = ( 1 2 + i √ 3 2 )z is another golden Blaschke ellipse. Clearly we find the equation of f (E) as More precisely, this image ellipse f (E) is the Poncelet curve of the following Blaschke product : .

Blaschke Products of Degree Four
A golden rectangle is a rectangle such that the ratio of the length x of the longer side to the length y of the shorter side is the golden ratio 1+ √ 5 2 (see [9] for more details). We give the following theorem.
Theorem 4.1. There are infinitely many golden rectangles whose four vertices lie on the unit circle.
Proof. Without loss of generality, let x and y be chosen so that x, y > 0 and such that the rectangle with vertices at the points x+iy, x−iy, −x− iy, −x+iy is inscribed in the unit circle. We try to determine the values of x and y such that x 2 + y 2 = 1. So it is sufficient to show that there are values of x and y on the unit circle such that 2x = 2αy.
So we have one golden rectangle such that its vertices are on the unit circle. Then there are infinitely many golden triangles with vertices on the unit circle by rotation.
Example 4.1. Let z 1 , z 2 , z 3 , z 4 and w 1 , w 2 , w 3 , w 4 be eight points which form the golden rectangles (z 1 , z 2 , z 3 , z 4 ) and (w 1 , w 2 , w 3 , w 4 ) on the unit circle so that {z 1 , z 2 , z 3 , z 4 } and {w 1 , w 2 , w 3 , w 4 } are interspersed sets of the points. From Corollary 10 on page 97 in [3], we know that there exists a Blaschke product B of degree 4 which maps 0 to 0 such that B(z j ) = B(z k ) and B(w j ) = B(w k ) for all j and k (1 ≤ j, k ≤ 4).
Then clearly there are infinitely many Poncelet curves associated with a finite Blaschke product of degree 4 each of which has at least two golden rectangle circumscribing them and having the vertices on the unit circle. The dashed rectangle is the golden rectangle.
Using the following lemmas, we construct examples of finite Blaschke products of degree 4 whose Poncelet curves are ellipses and each of them have at least one golden rectangle.  (See [5] Lemma 6) For four mutually distinct points z 1 , ..., z 4 on the unit circle (0 ≤ arg z 1 < arg z 2 < arg z 3 < arg z 4 < 2π), there exists an ellipse that is inscribed in the quadrilateral with vertices z 1 , ..., z 4 . Moreover, for each quadrilateral, inscribed ellipses form a real-valued one-parameter family. Now we give the following theorem. where α = −ab and β = a+b−ab(a+b) 1−|ab| 2 . Figure 6. Blaschke product B of degree 5 whose Poncelet curve inscribed in (at least) one golden pentagon. The dashed pentagon is the golden pentagon.

Blaschke Products of Higher Degree
We know that regular pentagon and regular decagon have the same properties of the golden ratio among polygons (see [9] for more details). It is not known the equation of the Poncelet curves of Blaschke products of degree 5 or 10, so we cannot obtain similar theorems to the ones given in the previous sections. In these two cases, by the similar arguments used in the Example 3.2 and Example 4.1, we can obtain finite Blaschke products of degree 5 and 10 whose Poncelet curves circumscribed by at least two regular pentagon and regular decagon, respectively.