# Shrinkage Points of Golden Rectangle, Fibonacci Spirals, and Golden Spirals

Authors: Jun-Sheng Duan
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## Abstract

We investigated the golden rectangle and the related Fibonacci spiral and golden spiral. The coordinates of the shrinkage points of a golden rectangle were derived. Properties of shrinkage points were discussed. Based on these properties, we conduct a comparison study for the Fibonacci spiral and golden spiral. Their similarities and differences were looked into by examining their polar coordinate equations, polar radii, arm-radius angles, and curvatures. The golden spiral has constant arm-radius angle and continuous curvature, while the Fibonacci spiral has cyclic varying arm-radius angle and discontinuous curvature.


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## Body

## 1. Introduction

A golden rectangle is such one that if we cut off a square section whose side is equal to the shortest side, the piece that remains has the same ratio of side lengths with the original rectangle.Let a golden rectangle has the side lengthsa and b b<a, then the ratio λ=b/a satisfies(1)λ=ba=a−bb=1−b/ab/a=1−λλ,that is(2)λ2=1−λ.Its positive root is the golden ratio:(3)λ=5−12≈0.618.The golden ratio is also known as the golden section, golden proportion, and golden mean [1]. The golden ratio has been found incorporated almost in all natural or organic structures, such as the bone structure of human beings [2–4], the seed pattern and geometry of plants [5], the spiral of a sea shell [6], and spiral galaxy [7]. The golden ratio was found in art and architecture as it produces pleasing shapes [8], even in special relativity [9]. Also, the golden ratio and the golden section method were applied to optimal design and search problems in different fields [10–13].As a development of the Fibonacci numbers, Stakhov and Rozin [1, 14] proposed new continuous functions based on the golden ratio: the symmetric Fibonacci sine and cosine, symmetric Lucas sine and cosine, and quasisine Fibonacci function. In particular, a new equation of the three-dimensional surface, golden shofar, was presented in [14]. These concepts may lead to new cosmological theories [14]. In [15], the relation between Fibonacci sequences with arbitrary initial numbers and the damped oscillation equation was established.A golden rectangle has the golden ratio of side lengths. It has been applied to the fields of architecture, drawing, photography, etc., as a representative of art beauty [8, 16]. In Section 2, we consider the shrinkage points of a golden rectangle and their properties. In Section 3, we compare the Fibonacci spiral and golden spiral, including their equations, polar radii, arm-radius angles, and curvatures.We note that in some references, the value1/λ is called the golden ratio and denoted as φ. Throughout this paper, we use λ to denote the golden ratio 5−1/2.


## 2. Shrinkage Points of Golden Rectangle

From the golden rectangleOACB in Figure 1, the square is cut on the right, and the remaining rectangle is also a golden rectangle. We continue the operation as the following . The squares on the top, left and bottom are cut away in the follow-up three procedures. The cutting process in the counterclockwise direction can be ongoing. By the theorem of interval nest, there is a singleton xs,ys, denoted by a dot in Figure 1, belonging to all of the golden rectangles. We call this singleton the shrinkage point of the original golden rectangle.Figure 1
Shrinkage point of golden rectangle.Next, we determine the location of the shrinkage point on the golden rectangleOACB in Figure 1. Set up the rectangular coordinate system OAB. Points A and B have the coordinates a and b on axes OA and OB, respectively. Dashed lines represent the cut lines. Successively, they cut the axis OA at the points d1,c1,d2,c2,…. The coordinate values are calculated as(4)c1=λ1−λa,d1−c1=1−λ2a,c2=λ1−λa+λ1−λ3a,d2−c2=1−λ4a,c3=λ1−λa+λ1−λ3a+λ1−λ5a,….By induction, we have the general expression as follows:(5)cn=λ1−λa1+1−λ2+1−λ4+⋯+1−λ2n−1=λ1−λa1−1−λ2n1−1−λ2.The limitation leads to the abscissa of the shrinkage point:(6)xs=limn⟶∞cn=1−λ2−λa.Inserting the value ofλ, it has the form(7)xs=5−510a≈0.276a.Similarly, the vertical coordinate of the shrinkage point is(8)ys=1−λ2−λb=5−510b≈0.276b.About the shrinkage point, we prove the following properties:Property 1.
In the golden rectangleOACB in Figure 2, the points O and C are connected, and the vertical line is made such that BS⊥OC with the foot point S; then, the point S is just the shrinkage point.Figure 2
Geometric approach of the shrinkage point.Proof.
We supposeOA=a and OB=b=λa. Set up the rectangular coordinate system OAB. Then, the equations of straight lines OC and BS are(9)y=λx,y=−1λx+b,respectively. Solving the equations and using the relation λ2=1−λ, we obtain(10)x=1−λ2−λa,y=1−λ2−λb.
From equations (6) and (8), the proof is completed.Property 2.
Let in Figure3, OACB be a golden rectangle and S be the shrinkage point, and CE=CA, SF//OA. Then, SE⊥SA, SE/SA=λ, and ∠FSA≔α=arctan2λ−1.Figure 3
The schematic diagram for Property 2.Proof.
DenoteOA=a. Set up the Cartesian coordinate system OAB. Then, we have the following rectangular coordinates:(11)Aa,0,Ea−aλ,aλ,Sxs,ys,Fa,ys.
Considering the vectors(12)SE⟶=a−aλ−xs,aλ−ys,SA⟶=a−xs,−ys,and calculating the scalar product(13)SE⟶⋅SA⟶=a−aλ−xsa−xs−ysaλ−ys,yields(14)SE⟶⋅SA⟶=0.
This meansSE⟶⊥SA⟶. From(15)SE⟶=a−aλ−xs2+aλ−ys2,SA⟶=a−xs2+ys2,we calculate that(16)SE⟶SA⟶=λ.
Finally, the acute angle∠FSA satisfies(17)tan∠FSA=ysa−xs=2λ−1.
So, we have(18)∠FSA≔α=arctan2λ−1.In Section3, the values of xs,ys,andα will be used. The length of the long side of a golden rectangle is denoted as a.By symmetry of rectangles, there are four shrinkage points on a golden rectangle, shown by dots in Figure4. As a comparison, we display the trisection lines and the golden lines of the same golden rectangle. The golden lines divide the width or the height at the golden ratio. These dots and lines are important references for drawing and photography.Figure 4
The shrinkage points, trisection lines (solid lines), and golden lines (dashed lines) of a golden rectangle.


## 3. Fibonacci Spirals and Golden Spirals

In this section, the shrinkage point in the lower left of a golden rectangle is served as the pole of the polar coordinate system and the origin of the rectangular coordinate system whenever a coordinate system is introduced in a golden rectangle.


### 3.1. Equations

The Fibonacci spiral can be generated from a golden rectanglePACB in Figure 5. It is made of quarter-circles tangent to the interior of each square as follows. We draw the quarter-circle AE^, center D, through two corners of the square DACE such that the sides of the square are tangent to the arc. Succeedingly, the quarter-circle EG^ with the center F in the square GFEB, the quarter-circle GI^ with the center H in the square PIHG, the quarter-circle IK^ with the center J in the square IDKJ, and so on.Figure 5
The Fibonacci spiral.For the sake of following comparison with the golden spiral, we derive the equation in polar coordinates for the Fibonacci spiral.First, for the arcAE^, we take a point on it with the polar coordinates Mr,θ. The ordinary rectangle coordinates are Mrcosθ,rsinθ. The center of the arc has the rectangle coordinates Da−aλ−xs,−ys. According to definition, the distance is a constant, MD=aλ, i.e.,(19)rcosθ−a+aλ+xs2+rsinθ+ys2=a2λ2.It is rearranged in the powers ofr as(20)r2+2aλ32−λsinθ−λcosθr−λ26λ−22−λ2a2=0.The positive root ofr is the equation in polar coordinates for the arc AE^:(21)r=−aλ32−λsinθ−λcosθ+aλ2−λλ4sinθ−λcosθ2+6λ−2,−α≤θ≤−α+π2.Polar coordinate equations for other quarter-circles can be given by similarity. For example, for the quarter-circleEG^,(22)r=−aλ42−λsinθ−π2−λcosθ−π2+aλ22−λλ4sinθ−π2−λcosθ−π22+6λ−2,−α+π2≤θ≤−α+π,and for the quarter-circle GI^,(23)r=−aλ52−λsinθ−π−λcosθ−π+aλ32−λλ4sinθ−π−λcosθ−π2+6λ−2,−α+π≤θ≤−α+3π2.The Fibonacci spiral does not have continuous curvature, and is an approximation for the golden spiral. The golden spiral is a special type of the logarithmic spiral. Using the polar coordinates the logarithmic spiral has the equation:(24)r=cekθ,c>0,k<0.The golden spiral has the special property such that for every incrementπ/2 of θ, the distance from the center of the spiral multiplies the golden ratio λ. That is,(25)ekπ/2=λ.It follows that(26)k=2πlnλ.The polar coordinate equation of the golden spiral is derived as follows:(27)r=ce2/πlnλθ,or equivalently,(28)r=cλ2/πθ.In the golden rectanglePACB in Figure 6, since the modulus of the vector OA⟶ is(29)OA⟶=a−xs2+ys2=a2−λ1+λ6,and the intersection angle between the polar axis and vector OA⟶ is α=arctan2λ−1, we have the following property.Figure 6
The Fibonacci spiral (solid line) and golden spiral (dashed line).Property 3.
In the golden rectanglePACB in Figure 6, the golden spiral with the shrinkage point O as the pole, through the point A can be given by the equation:(30)r=a2−λ1+λ6λ2/πθ+α.
In Figure6, we show the Fibonacci spiral in solid line and golden spiral in dashed line. In order to display their distinction, the difference of polar radii of the Fibonacci spiral and golden spiral, rf−rg, is plotted in the ordinary rectangle coordinate system in Figure 7, where we take a=1. In Figure 6, the two spirals overlap at each corners A,E,G,I,…. Within each of quarter-circles, the two spirals intersect exactly once.Figure 7
The difference of polar radii of the Fibonacci spiral and golden spiral (a=1).


### 3.2. Arm-Radius Angles

The arm-radius angle at a pointM on a spiral is the acute angle between the tangent line at the point M and the polar radius OM. It is well known that for the logarithmic spiral r=cekθ,c>0,k<0, the arm-radius angle is constant, and it satisfies cotβ=−k. So, the logarithmic spiral is also called equiangular spiral. As a special case of logarithmic spiral, the golden spiral r=ce2/πlnλθ has the equiangular properity, i.e., the arm-radius angle, independent of c and θ:(31)βg=arccot−2πlnλ≈1.2735radianor72.97∘,as shown in Figure 8.Figure 8
Arm-radius angle of the golden spiral.For the Fibonacci spiral in Figure5, the arm-radius angle βf is not constant, but periodic variation such that βfθ+π/2=βfθ by similarity. For the quarter-circle AE^ in Figure 5, the parameter equation is(32)x=rθcosθ,y=rθsinθ,where rθ is given in equation (21). From the vectors(33)OM⟶=rθcosθ,rθsinθ,MT⟶=x′θ,y′θ,we express the arm-radius angle for the Fibonacci spiral:(34)βf=arccosOM⟶⋅MT⟶OM⟶⋅MT⟶,−α≤θ≤π2−α.The arm-radius angleβf is a complicated function of θ, independent of a. By means of the software MATHEMATICA, the curves of arm-radius angles βf and βg versus polar angle θ are shown in the ordinary rectangle coordinate system in Figure 9, where we limit −α≤θ≤2π−α. The arm-radius angle βg is a constant, while βf oscillates continuously around βg. At each corners, θ=−α, π/2−α, π−α, …, the arm-radius angle βf varies unsmoothly. MATHEMATICA code generating Figure 9 is attached in Appendix.Figure 9
Arm-radius angles for the Fibonacci spiral (solid line) and golden spiral (dashed line).


### 3.3. Curvatures

We rewrite the golden spiral in equation (30) to the parametric equation:(35)x=a2−λ1+λ6λ2/πθ+αcosθ,y=a2−λ1+λ6λ2/πθ+αsinθ.Inserting the curvature formula(36)Kg=x′θy″θ−x″θy′θx′2θ+y′2θ3/2,we obtain the curvatures of the golden spiral(37)Kg=π2−λa1+λ6π2+4ln2λλ2/πθ+α.We takea=1 and limit −α≤θ≤2π−α. The Fibonacci spiral has discontinuous curvatures Kf: λ−1,λ−2,λ−3, and λ−4 for four quarter-circles, respectively, while the golden spiral has the continuous curvature in eqaution (37). In Figure 10, curvatures of the Fibonacci spiral and golden spiral versus θ on the interval −α≤θ≤2π−α are shown in an ordinary rectangle coordinate system.Figure 10
Curvatures of the Fibonacci spiral (solid line) and golden spiral (dashed line).


## 3.1. Equations

The Fibonacci spiral can be generated from a golden rectanglePACB in Figure 5. It is made of quarter-circles tangent to the interior of each square as follows. We draw the quarter-circle AE^, center D, through two corners of the square DACE such that the sides of the square are tangent to the arc. Succeedingly, the quarter-circle EG^ with the center F in the square GFEB, the quarter-circle GI^ with the center H in the square PIHG, the quarter-circle IK^ with the center J in the square IDKJ, and so on.Figure 5
The Fibonacci spiral.For the sake of following comparison with the golden spiral, we derive the equation in polar coordinates for the Fibonacci spiral.First, for the arcAE^, we take a point on it with the polar coordinates Mr,θ. The ordinary rectangle coordinates are Mrcosθ,rsinθ. The center of the arc has the rectangle coordinates Da−aλ−xs,−ys. According to definition, the distance is a constant, MD=aλ, i.e.,(19)rcosθ−a+aλ+xs2+rsinθ+ys2=a2λ2.It is rearranged in the powers ofr as(20)r2+2aλ32−λsinθ−λcosθr−λ26λ−22−λ2a2=0.The positive root ofr is the equation in polar coordinates for the arc AE^:(21)r=−aλ32−λsinθ−λcosθ+aλ2−λλ4sinθ−λcosθ2+6λ−2,−α≤θ≤−α+π2.Polar coordinate equations for other quarter-circles can be given by similarity. For example, for the quarter-circleEG^,(22)r=−aλ42−λsinθ−π2−λcosθ−π2+aλ22−λλ4sinθ−π2−λcosθ−π22+6λ−2,−α+π2≤θ≤−α+π,and for the quarter-circle GI^,(23)r=−aλ52−λsinθ−π−λcosθ−π+aλ32−λλ4sinθ−π−λcosθ−π2+6λ−2,−α+π≤θ≤−α+3π2.The Fibonacci spiral does not have continuous curvature, and is an approximation for the golden spiral. The golden spiral is a special type of the logarithmic spiral. Using the polar coordinates the logarithmic spiral has the equation:(24)r=cekθ,c>0,k<0.The golden spiral has the special property such that for every incrementπ/2 of θ, the distance from the center of the spiral multiplies the golden ratio λ. That is,(25)ekπ/2=λ.It follows that(26)k=2πlnλ.The polar coordinate equation of the golden spiral is derived as follows:(27)r=ce2/πlnλθ,or equivalently,(28)r=cλ2/πθ.In the golden rectanglePACB in Figure 6, since the modulus of the vector OA⟶ is(29)OA⟶=a−xs2+ys2=a2−λ1+λ6,and the intersection angle between the polar axis and vector OA⟶ is α=arctan2λ−1, we have the following property.Figure 6
The Fibonacci spiral (solid line) and golden spiral (dashed line).Property 3.
In the golden rectanglePACB in Figure 6, the golden spiral with the shrinkage point O as the pole, through the point A can be given by the equation:(30)r=a2−λ1+λ6λ2/πθ+α.
In Figure6, we show the Fibonacci spiral in solid line and golden spiral in dashed line. In order to display their distinction, the difference of polar radii of the Fibonacci spiral and golden spiral, rf−rg, is plotted in the ordinary rectangle coordinate system in Figure 7, where we take a=1. In Figure 6, the two spirals overlap at each corners A,E,G,I,…. Within each of quarter-circles, the two spirals intersect exactly once.Figure 7
The difference of polar radii of the Fibonacci spiral and golden spiral (a=1).


## 3.2. Arm-Radius Angles

The arm-radius angle at a pointM on a spiral is the acute angle between the tangent line at the point M and the polar radius OM. It is well known that for the logarithmic spiral r=cekθ,c>0,k<0, the arm-radius angle is constant, and it satisfies cotβ=−k. So, the logarithmic spiral is also called equiangular spiral. As a special case of logarithmic spiral, the golden spiral r=ce2/πlnλθ has the equiangular properity, i.e., the arm-radius angle, independent of c and θ:(31)βg=arccot−2πlnλ≈1.2735radianor72.97∘,as shown in Figure 8.Figure 8
Arm-radius angle of the golden spiral.For the Fibonacci spiral in Figure5, the arm-radius angle βf is not constant, but periodic variation such that βfθ+π/2=βfθ by similarity. For the quarter-circle AE^ in Figure 5, the parameter equation is(32)x=rθcosθ,y=rθsinθ,where rθ is given in equation (21). From the vectors(33)OM⟶=rθcosθ,rθsinθ,MT⟶=x′θ,y′θ,we express the arm-radius angle for the Fibonacci spiral:(34)βf=arccosOM⟶⋅MT⟶OM⟶⋅MT⟶,−α≤θ≤π2−α.The arm-radius angleβf is a complicated function of θ, independent of a. By means of the software MATHEMATICA, the curves of arm-radius angles βf and βg versus polar angle θ are shown in the ordinary rectangle coordinate system in Figure 9, where we limit −α≤θ≤2π−α. The arm-radius angle βg is a constant, while βf oscillates continuously around βg. At each corners, θ=−α, π/2−α, π−α, …, the arm-radius angle βf varies unsmoothly. MATHEMATICA code generating Figure 9 is attached in Appendix.Figure 9
Arm-radius angles for the Fibonacci spiral (solid line) and golden spiral (dashed line).


## 3.3. Curvatures

We rewrite the golden spiral in equation (30) to the parametric equation:(35)x=a2−λ1+λ6λ2/πθ+αcosθ,y=a2−λ1+λ6λ2/πθ+αsinθ.Inserting the curvature formula(36)Kg=x′θy″θ−x″θy′θx′2θ+y′2θ3/2,we obtain the curvatures of the golden spiral(37)Kg=π2−λa1+λ6π2+4ln2λλ2/πθ+α.We takea=1 and limit −α≤θ≤2π−α. The Fibonacci spiral has discontinuous curvatures Kf: λ−1,λ−2,λ−3, and λ−4 for four quarter-circles, respectively, while the golden spiral has the continuous curvature in eqaution (37). In Figure 10, curvatures of the Fibonacci spiral and golden spiral versus θ on the interval −α≤θ≤2π−α are shown in an ordinary rectangle coordinate system.Figure 10
Curvatures of the Fibonacci spiral (solid line) and golden spiral (dashed line).


## 4. Conclusion

We considered the golden rectangle and the related Fibonacci spiral and golden spiral. In Section2, we gave the coordinates of the shrinkage points of a golden rectangle. Properties of shrinkage points were presented. In Section 3, we compared the Fibonacci spiral and golden spiral by examining their equations in polar coordinates, relationship of polar radii, and differences of arm-radius angles and curvatures. The golden spiral has a constant arm-radius angle and continuous curvature. As an approximation of the golden spiral, the Fibonacci spiral has continuous and smooth polar radius, cyclic varying arm-radius angle, and discontinuous curvature.



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