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.
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 lengths a and b , then the ratio satisfiesthat is
Its positive root is the golden ratio:
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 value is called the golden ratio and denoted as φ. Throughout this paper, we use λ to denote the golden ratio .
2. Shrinkage Points of Golden Rectangle
From the golden rectangle 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 , 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.
Next, we determine the location of the shrinkage point on the golden rectangle in Figure 1. Set up the rectangular coordinate system . Points A and B have the coordinates a and b on axes and , respectively. Dashed lines represent the cut lines. Successively, they cut the axis at the points . The coordinate values are calculated as
By induction, we have the general expression as follows:
The limitation leads to the abscissa of the shrinkage point:
Inserting the value of λ, it has the form
Similarly, the vertical coordinate of the shrinkage point is
About the shrinkage point, we prove the following properties:
Property 1. In the golden rectangle in Figure 2, the points O and C are connected, and the vertical line is made such that with the foot point S; then, the point S is just the shrinkage point.
Proof. We suppose and . Set up the rectangular coordinate system . Then, the equations of straight lines and arerespectively. Solving the equations and using the relation , we obtainFrom equations (6) and (8), the proof is completed.
Property 2. Let in Figure 3, be a golden rectangle and S be the shrinkage point, and , . Then, , , and .
Proof. Denote . Set up the Cartesian coordinate system . Then, we have the following rectangular coordinates:Considering the vectorsand calculating the scalar productyieldsThis means . Fromwe calculate thatFinally, the acute angle satisfiesSo, we have
In Section 3, the values of 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 Figure 4. 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.
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 rectangle in Figure 5. It is made of quarter-circles tangent to the interior of each square as follows. We draw the quarter-circle , center D, through two corners of the square such that the sides of the square are tangent to the arc. Succeedingly, the quarter-circle with the center F in the square , the quarter-circle with the center H in the square , the quarter-circle with the center J in the square , and so on.
For the sake of following comparison with the golden spiral, we derive the equation in polar coordinates for the Fibonacci spiral.
First, for the arc , we take a point on it with the polar coordinates . The ordinary rectangle coordinates are . The center of the arc has the rectangle coordinates According to definition, the distance is a constant, i.e.,
It is rearranged in the powers of r as
The positive root of r is the equation in polar coordinates for the arc :
Polar coordinate equations for other quarter-circles can be given by similarity. For example, for the quarter-circle ,and for the quarter-circle ,
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:
The golden spiral has the special property such that for every increment of θ, the distance from the center of the spiral multiplies the golden ratio λ. That is,
It follows that
The polar coordinate equation of the golden spiral is derived as follows:or equivalently,
In the golden rectangle in Figure 6, since the modulus of the vector isand the intersection angle between the polar axis and vector is we have the following property.
Property 3. In the golden rectangle in Figure 6, the golden spiral with the shrinkage point O as the pole, through the point A can be given by the equation:In Figure 6, 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, , is plotted in the ordinary rectangle coordinate system in Figure 7, where we take . In Figure 6, the two spirals overlap at each corners . Within each of quarter-circles, the two spirals intersect exactly once.
3.2. Arm-Radius Angles
The arm-radius angle at a point M on a spiral is the acute angle between the tangent line at the point M and the polar radius . It is well known that for the logarithmic spiral , the arm-radius angle is constant, and it satisfies . So, the logarithmic spiral is also called equiangular spiral. As a special case of logarithmic spiral, the golden spiral has the equiangular properity, i.e., the arm-radius angle, independent of c and θ:as shown in Figure 8.
For the Fibonacci spiral in Figure 5, the arm-radius angle is not constant, but periodic variation such that by similarity. For the quarter-circle in Figure 5, the parameter equation iswhere is given in equation (21). From the vectorswe express the arm-radius angle for the Fibonacci spiral:
The arm-radius angle is a complicated function of θ, independent of a. By means of the software MATHEMATICA, the curves of arm-radius angles and versus polar angle θ are shown in the ordinary rectangle coordinate system in Figure 9, where we limit . The arm-radius angle is a constant, while oscillates continuously around . At each corners, , , , , the arm-radius angle varies unsmoothly. MATHEMATICA code generating Figure 9 is attached in Appendix.
3.3. Curvatures
We rewrite the golden spiral in equation (30) to the parametric equation:
Inserting the curvature formulawe obtain the curvatures of the golden spiral
We take and limit . The Fibonacci spiral has discontinuous curvatures : , and 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 are shown in an ordinary rectangle coordinate system.
4. Conclusion
We considered the golden rectangle and the related Fibonacci spiral and golden spiral. In Section 2, 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.
Appendix
MATHEMATICA code for Figure 9:
la = (Sqrt[5] − 1)/2; betag = ArcCot[−2 Log[la]/Pi]; r = −a la ^ 3/(2 − la) (Sin[th] − la Cos[th]) + a la/(2 − la) Sqrt[la ^ 4 (Sin[th] − la Cos[th]) ^ 2 + 6 la − 2]; x = r Cos[th]; y = r Sin[th]; OM = {x, y}; MT = {D[x, th], D[y, th]}; betaf = ArcCos[Abs[OM.MT]/Norm[OM]/Norm[MT]]; al = ArcTan[2 la − 1]; f1 = Plot[{betaf, betag}, {th, −al, −al + Pi/2}, PlotStyle ⟶ {{}, {Dashed}}]; betaf2 = betaf/. th ⟶ (th − Pi/2); f2 = Plot[{betaf2, betag}, {th, −al + Pi/2, −al + Pi}, PlotStyle ⟶ {{}, {Dashed}}]; betaf3 = betaf/. th ⟶ (th − Pi); f3 = Plot[{betaf3, betag}, {th, −al + Pi, −al + 3 Pi/2}, PlotStyle ⟶ {{}, {Dashed}}]; betaf4 = betaf/. th ⟶ (th − 3 Pi/2); f4 = Plot[{betaf4, betag}, {th, −al + 3 Pi/2, −al + 2 Pi}, PlotStyle ⟶ {{}, {Dashed}}]; Show[f1, f2, f3, f4, AxesOrigin ⟶ {0, 1.23}, PlotRange ⟶ {{−al, 6.15}, {1.2, 1.37}}, Ticks ⟶ {None}].
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The author declares that he has no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (no. 11772203) and the Natural Science Foundation of Shanghai (no. 17ZR1430000).