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

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Hindawi Discrete Dynamics in Nature and Society Volume 2019, Article ID 3149602, 6 pages https://doi.org/10.1155/2019/3149602

Research Article Shrinkage Points of Golden Rectangle, Fibonacci Spirals, and Golden Spirals

Jun-Sheng Duan

School of Sciences, Shanghai Institute of Technology, Shanghai 201418, China

Correspondence should be addressed to Jun-Sheng Duan; [email protected]

Received 25 June 2019; Revised 11 November 2019; Accepted 6 December 2019; Published 20 December 2019

Academic Editor: Cengiz Çinar

Copyright © 2019 Jun-Sheng Duan. +is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We investigated the golden rectangle and the related Fibonacci spiral and golden spiral. +e 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. +eir similarities and diﬀerences were looked into by examining their polar coordinate equations, polar radii, arm-radius angles, and curvatures. +e 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 art and architecture as it produces pleasing shapes [8], even in special relativity [9]. Also, the golden ratio and the golden A golden rectangle is such one that if we cut off a square section method were applied to optimal design and search section whose side is equal to the shortest side, the piece that problems in different fields [10–13]. remains has the same ratio of side lengths with the original As a development of the Fibonacci numbers, Stakhov rectangle. and Rozin [1, 14] proposed new continuous functions based Let a golden rectangle has the side lengths a and b on the golden ratio: the symmetric Fibonacci sine and co- (b < a), then the ratio λ � b/a satisfies sine, symmetric Lucas sine and cosine, and quasisine b a − b 1 − b/a 1 − λ Fibonacci function. In particular, a new equation of the λ � � � � , (1) a b b/a λ three-dimensional surface, golden shofar, was presented in [14]. +ese concepts may lead to new cosmological theories that is [14]. In [15], the relation between Fibonacci sequences with λ2 � 1 − λ. (2) arbitrary initial numbers and the damped oscillation equation was established. Its positive root is the golden ratio: A golden rectangle has the golden ratio of side lengths. It √� 5 − 1 has been applied to the fields of architecture, drawing, λ � ≈ 0.618. (3) photography, etc., as a representative of art beauty [8, 16]. In 2 Section 2, we consider the shrinkage points of a golden +e golden ratio is also known as the golden section, rectangle and their properties. In Section 3, we compare the golden proportion, and golden mean [1]. +e golden ratio Fibonacci spiral and golden spiral, including their equations, has been found incorporated almost in all natural or organic polar radii, arm-radius angles, and curvatures. structures, such as the bone structure of human beings [2–4], We note that in some references, the value 1/λ is called the seed pattern and geometry of plants [5], the spiral of a sea the golden ratio and denoted as φ. +roughout√� this paper, we shell [6], and spiral galaxy [7]. +e golden ratio was found in use λ to denote the golden ratio ( 5 − 1)/2. 2 Discrete Dynamics in Nature and Society

2. Shrinkage Points of Golden Rectangle B C From the golden rectangle OACB 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 . +e squares on the top, left and bottom are cut away in the follow-up three procedures. +e 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.

Next, we determine the location of the shrinkage point O c1 c2 d2 d1 A(a) on the golden rectangle OACB in Figure 1. Set up the rectangular coordinate system OAB. Points A and B have the Figure 1: Shrinkage point of golden rectangle. 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, .... +e coordinate values are calculated as BC

c1 � λ(1 − λ)a, 2 d1 − c1 �(1 − λ) a, 3 c2 � λ(1 − λ)a + λ(1 − λ) a, (4) 4 d2 − c2 �(1 − λ) a, 3 5 S c3 � λ(1 − λ)a + λ(1 − λ) a + λ(1 − λ) a, ... . By induction, we have the general expression as follows: O A 2 4 2(n− 1) cn � λ(1 − λ)a�1 +(1 − λ) +(1 − λ) + · · · +(1 − λ) � Figure 2: Geometric approach of the shrinkage point. 1 − (1 − λ)2n � λ(1 − λ)a 2 . 1 − (1 − λ) Proof. We suppose OA � a and OB � b � λa. Set up the (5) rectangular coordinate system OAB. +en, the equations of straight lines OC and BS are +e limitation leads to the abscissa of the shrinkage point: y � λx, (9) − 1 1 λ y � − x + b, xs � lim cn � a. (6) n⟶∞ 2 − λ λ Inserting the value of λ, it has the form respectively. Solving the equations and using the relation 2 √� λ � 1 − λ, we obtain − 5 5 ( ) 1 − λ xs � a ≈ 0.276a. 7 x � a, 10 2 − λ ( ) Similarly, the vertical coordinate of the shrinkage point 10 1 − λ is y � b. √� 2 − λ 1 − λ 5 − 5 y � b � b ≈ 0.276b. (8) From equations (6) and (8), the proof is completed. □ s 2 − λ 10

About the shrinkage point, we prove the following Property 2. Let in Figure 3, OACB be a golden rectangle and properties: S be the shrinkage point, and CE � CA, SF//OA. +en, SE ⊥ SA, SE/SA � λ, and ∠FSA ≔ α � arctan(2λ − 1). Property 1. In the golden rectangle OACB in Figure 2, the points O and C are connected, and the vertical line is made Proof. Denote OA � a. Set up the Cartesian coordinate such that BS ⊥ OC with the foot point S; then, the point S is system OAB. +en, we have the following rectangular just the shrinkage point. coordinates: Discrete Dynamics in Nature and Society 3

E BC

F S α

Figure 4: +e shrinkage points, trisection lines (solid lines), and O D A golden lines (dashed lines) of a golden rectangle.

Figure 3: +e schematic diagram for Property 2. comparison, we display the trisection lines and the golden lines of the same golden rectangle. +e golden lines divide the width or the height at the golden ratio. +ese dots and A(a, 0), lines are important references for drawing and photography. E(a − aλ, aλ), (11) 3. Fibonacci Spirals and Golden Spirals S xs, ys �, In this section, the shrinkage point in the lower left of a F a, ys �. golden rectangle is served as the pole of the polar coordinate system and the origin of the rectangular coordinate system Considering the vectors whenever a coordinate system is introduced in a golden �→ rectangle. SE � a − aλ − xs, aλ − ys�, �→ (12) SA �a − xs, − ys �, 3.1. Equations. +e Fibonacci spiral can be generated from a golden rectangle PACB in Figure 5. It is made of quarter- and calculating the scalar product circles tangent to the interior of each square as follows. We �→ �→ draw the quarter-circle AE� , center D, through two corners of SE · SA �a − a − x �a − x � − y a − y �, (13) λ s s s λ s the square DACE such that the sides of the square are tangent to the arc. Succeedingly, the quarter-circle EG� with yields � �→ �→ the center F in the square GFEB, the quarter-circle GI with SE · SA � 0. (14) the center H in the square PIHG, the quarter-circle IK� with �→ �→ the center J in the square IDKJ, and so on. +is means SE ⊥ SA. From �� For the sake of following comparison with the golden �→ 2 2 spiral, we derive the equation in polar coordinates for the |SE| � a − aλ − xs � +aλ − ys � , �� (15) Fibonacci spiral. �→ AE� |SA| � a − x �2 + y2, First, for the arc , we take a point on it with the polar s s coordinates M(r, θ). +e ordinary rectangle coordinates are we calculate that M(r cos θ, r sin θ). +e center of the arc has the rectangle �→ coordinates D(a − aλ − xs, − ys). According to deﬁnition, |SE| the distance is a constant, MD � aλ, i.e., �→ � λ. (16) |SA| 2 2 2 2 r cos θ − a + aλ + xs� +r sin θ + ys � � a λ . (19) Finally, the acute angle ∠FSA satisﬁes It is rearranged in the powers of r as y FSA � s � − . 3 2 tan ∠ 2λ 1 (17) 2 2aλ λ (6λ − 2) 2 a − xs r + (sin θ − λ cos θ)r − a � 0. (20) 2 − λ (2 − λ)2 So, we have +e positive root of r is the equation in polar coordinates ∠FSA ≔ α � arctan(2λ − 1). (18) for the arc AE� : □ aλ3 r � − (sin θ − λ cos θ) − In Section 3, the values of x , y , and α will be used. +e 2 λ s s �� length of the long side of a golden rectangle is denoted as a. aλ π + λ4(sin θ − λ cos θ)2 + 6λ − 2, − α ≤ θ ≤ − α + . By symmetry of rectangles, there are four shrinkage 2 − λ 2 points on a golden rectangle, shown by dots in Figure 4. As a (21) 4 Discrete Dynamics in Nature and Society

E (2/π)θ B C r � cλ . (28) T In the golden��→ rectangle PACB in Figure 6, since the modulus of the vector OA is Βf �� M ��→ a �� |OA| � a − x �2 + y2 � 1 + λ6, (29) s s 2 − λ H G F O and��→ the intersection angle between the polar axis and vector J K OA is α � arctan(2λ − 1), we have the following property.

PACB P I D A Property 3. In the golden rectangle in Figure 6, the golden spiral with the shrinkage point O as the pole, through Figure 5: +e Fibonacci spiral. the point A can be given by the equation: �� a ( )( + ) r � 1 + λ6λ 2/π θ α . (30) Polar coordinate equations for other quarter-circles can 2 − λ be given by similarity. For example, for the quarter-circle � In Figure 6, we show the Fibonacci spiral in solid line and EG, golden spiral in dashed line. In order to display their dis- aλ4 π π tinction, the diﬀerence of polar radii of the Fibonacci spiral r � − �sin�θ − � − λ cos�θ − �� r − r 2 − λ 2 2 and golden spiral, f g, is plotted in the ordinary rectangle �� coordinate system in Figure 7, where we take a � 1. In aλ2 π π 2 Figure 6, the two spirals overlap at each corners + λ4�sin�θ − � − λ cos�θ − �� + 6λ − 2, A, E, G, I, .... Within each of quarter-circles, the two spirals − 2 λ 2 2 intersect exactly once. π − α + ≤ θ ≤ − α + π, 2 3.2. Arm-Radius Angles. +e arm-radius angle at a point M (22) on a spiral is the acute angle between the tangent line at the and for the quarter-circle GI�, 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 aλ5 r � − (sin(θ − π) − λ cos(θ − π)) angle is constant, and it satisﬁes cot β � − k. So, the loga- 2 − λ rithmic spiral is also called equiangular spiral. As a special �� case of logarithmic spiral, the golden spiral r � ce((2/π)ln λ)θ aλ3 + λ4(sin(θ − π) − λ cos(θ − π))2 + 6λ − 2, has the equiangular properity, i.e., the arm-radius angle, 2 − λ independent of c and θ:

3π 2 ∘ − α + π ≤ θ ≤ − α + . βg � arccot�− ln λ � ≈ 1.2735 radian() or 72.97 , (31) 2 π (23) as shown in Figure 8. +e Fibonacci spiral does not have continuous curvature, For the Fibonacci spiral in Figure 5, the arm-radius angle βf and is an approximation for the golden spiral. +e golden is not constant, but periodic variation such that � spiral is a special type of the logarithmic spiral. Using the βf (θ + (π/2)) � βf (θ) by similarity. For the quarter-circle AE polar coordinates the logarithmic spiral has the equation: in Figure 5, the parameter equation is r � cekθ, c > 0, k < 0. (24) x � r(θ)cos θ, � (32) y � r( ) , +e golden spiral has the special property such that for θ sin θ every increment π/2 of θ, the distance from the center of the where r(θ) is given in equation (21). From the vectors spiral multiplies the golden ratio λ. +at is, ��→ OM �(r(θ)cos θ, r(θ)sin θ), ekπ/2 � . ( ) λ 25 ��→ (33) MT � x′(θ), y′(θ)�, It follows that 2 k � ln λ. (26) we express the arm-radius angle for the Fibonacci spiral: π ��→ ��→ |OM · MT| π β � arccos ��→ ��→ , − α ≤ θ ≤ − α. (34) +e polar coordinate equation of the golden spiral is f 2 derived as follows: |OM| · |MT| ((2/π)ln λ)θ r � ce , (27) +e arm-radius angle βf is a complicated function of θ, independent of a. By means of the software MATHEMA- or equivalently, TICA, the curves of arm-radius angles βf and βg versus polar Discrete Dynamics in Nature and Society 5

E BC 1.35

1.30 g , β f G β O 1.25

–α π/2 – α π – α 3π/2 – α 2π – α P I D A 1.20 θ

Figure 6: +e Fibonacci spiral (solid line) and golden spiral Figure 9: Arm-radius angles for the Fibonacci spiral (solid line) (dashed line). and golden spiral (dashed line).

0.006 8

0.004 6 0.002 g g – r , K

2 – f π α f

r 4 –α π/2 – α π – α 3π/2 – α K –0.002 θ 2 –0.004 –α π/2 – α π – α 3π/2 – α 2π – α 0 θ Figure 7: +e diﬀerence of polar radii of the Fibonacci spiral and golden spiral (a � 1). Figure 10: Curvatures of the Fibonacci spiral (solid line) and golden spiral (dashed line).

a �� ⎧⎪ 6 (2/π)(θ+α) ⎪ x � 1 + λ λ cos θ, ⎨⎪ 2 − λ (35) β ⎪ �� g βg ⎪ a ( )( + ) ⎩ y � 1 + λ6 λ 2/π θ α sin θ. 2 − λ Inserting the curvature formula � � �x′(θ)y″(θ) − x″(θ)y′(θ)� Kg � , 2 2 3/2 (36) �x′ (θ) + y′ (θ)� Figure 8: Arm-radius angle of the golden spiral. we obtain the curvatures of the golden spiral π(2 − λ) angle θ are shown in the ordinary rectangle coordinate K � �� . g 6 2 (2/π)(θ+α) (37) system in Figure 9, where we limit − α ≤ θ ≤ 2π − α. +e arm- a �1 + λ ��π2 + 4 ln λ �λ radius angle βg is a constant, while βf oscillates continuously around β . At each corners, θ � − α, (π/2) − α, π − α, ..., the We take a � 1 and limit − α ≤ θ ≤ 2π − α. +e Fibonacci g − 1 − 2 − 3 arm-radius angle β varies unsmoothly. MATHEMATICA spiral has discontinuous curvatures Kf : λ , λ , λ , and f − 4 code generating Figure 9 is attached in Appendix. λ 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 3.3. Curvatures. We rewrite the golden spiral in equation spiral versus θ on the interval − α ≤ θ ≤ 2π − α are shown in (30) to the parametric equation: an ordinary rectangle coordinate system. 6 Discrete Dynamics in Nature and Society

4. Conclusion Acknowledgments We considered the golden rectangle and the related Fibo- +is work was supported by the National Natural Science nacci spiral and golden spiral. In Section 2, we gave the Foundation of China (no. 11772203) and the Natural Science coordinates of the shrinkage points of a golden rectangle. Foundation of Shanghai (no. 17ZR1430000). Properties of shrinkage points were presented. In Section 3, we compared the Fibonacci spiral and golden spiral by References examining their equations in polar coordinates, relationship of polar radii, and differences of arm-radius angles and [1] A. Stakhov and B. Rozin, “On a new class of hyperbolic curvatures. +e golden spiral has a constant arm-radius functions,” Chaos, Solitons & Fractals, vol. 23, no. 2, pp. 379–389, 2005. angle and continuous curvature. As an approximation of the [2] M. Iosa, D. De Bartolo, G. 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Stakhov, “+e golden section, secrets of the Egyptian Conflicts of Interest civilization and harmony mathematics,” Chaos, Solitons & Fractals, vol. 30, no. 2, pp. 490–505, 2006. +e author declares that he has no conflicts of interest. Advances in Advances in Journal of The Scientifc Journal of Operations Research Decision Sciences Applied Mathematics World Journal Probability and Statistics Hindawi Hindawi Hindawi Hindawi Publishing Corporation Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 http://www.hindawi.comwww.hindawi.com Volume 20182013 www.hindawi.com Volume 2018

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