The Subtle Symmetry of Golden Spirals

Year: 2000 Authors: Alvin Swimmer

Core claim

Golden spirals arise from iterative Castor and golden-rectangle constructions whose Φ-map generates tangent-smooth spirals with a shared eye and harmonic line structure.

Topics

golden ratio geometry, iterative spiral construction, harmonic division, symmetry and transformation

Domains

Euclidean geometry, golden ratio, projective geometry, transformational geometry, mathematical art, visual pattern design, architecture

Methods

geometric construction, iterative mapping, ratio analysis, diagrammatic proof

Media

triangles, golden rectangles, spiral arcs, figures

Paper text

The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.

BRIDGES Mathematical Connections in Art, Music, and Science

The Subtle Symmetry of Golden Spirals

Alvin Swimmer Department of Mathematics Arizona State University Tempe, AZ 85287-1804, USA E-mail: aswimmer@asu.edu

Abstract

The beautiful symmetries of golden spirals are intuitively evident to everyone that sees them even for the first time. The purpose here is to show how these symmetries are manifestations of the special nature of the eyes of those spirals.

1. The Golden Spiral Circumscribing Golden Triangles

Following the ancient Greek geometers, I call every isosceles triangle whose base angles each measure twice the vertex angle a Castor Triangle. That is, the vertex angle measures 36 degrees or $\frac{\pi }{5}$ radians and each base angle measures 72 degrees or $\frac{2\pi }{5}$ radians. This is the triangle that occurs in the construction of a regular decagon as the central triangle determined by two radii of the circle being subdivided into 10 equal parts and a side of the decagon. It is the triangle determined by two diagonals of a regular pentagon emanating from the same vertex and the side opposite that vertex. It is also evident in the ubiquitous 5 pointed star that occurs in the flags of many countries as well as many, many other places, as each of the 5 triangles protruding from the regular pentagon that forms the interior of the star. The ratio of a side length to the base length of a Castor Triangle is the golden ratio $\varphi =\frac{\sqrt{5}+1}{2}$ and that is the reason that I call the spiral it determines a Golden Spiral.

The construction of this spiral proceeds as follows. (We use $\dot{=}$ as an abbreviation of ‘defined as’.) Let ${\mathcal{C}}_0\doteq [T_0,T_1,T_2]$ denote the Castor Triangle with vertex $T_0$ and base $[T_1,T_2]$. We bisect the base angle with vertex $T_1$ and call $T_3$, the intersection of this bisector with the opposite side $[T_2,T_0]$. Then, because of the angle relationship ${\mathcal{C}}_1\doteq [T_1,T_2,T_3]$ is another Castor Triangle. The other triangle thus formed, ${\mathcal{P}}_1\doteq [T_3,T_0,T_1]$ is also isosceles. The vertex angle in ${\mathcal{P}}_1$ measures $\frac{3\pi }{5}$ radians and each base angle measures $\frac{\pi }{5}$ radians, the same as the vertex angle of a Castor Triangle. Still following those ancient Greek geometers that adorn my list of mathematical heroes, I call all isosceles triangles with a vertex angle measuring three times each base angle, a Pollux Triangle. In a Pollux Triangle the ratio of the base length to the side length is $\varphi$. It is the triangle formed by adjacent sides of two adjacent protruding Castor Triangles of a 5 pointed star and the line joining the two vertices. A pair of triangles, one a Castor and the other a Pollux situated so that, like ${\mathcal{C}}_1,{\mathcal{P}}_1$, they share a common side and together make a larger Castor Triangle, I call Gemini Twins. In this context, the ancient Greek geometers would say that the Pollux Triangle acts as a gnomon to transform its Castor Twin into the Mother Castor.

We can now draw the first arc of our golden spiral, from $T_0$ to $T_1$ using the vertex $T_3$ of ${\mathcal{P}}_1$ as the center and its side length as the radius.

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The whole process is now repeated beginning with ${\mathcal{C}}_1$ as the mother Castor Triangle. The angle with vertex $T_2$ is bisected. The endpoint of this bisector on $[T_3,T_1]$, we call $T_4$. Then ${\mathcal{C}}_2\doteq [T_2,T_3,T_4]$ is the new Castor Triangle and its twin is ${\mathcal{P}}_2\doteq [T_4,T_1,T_2]$ and the arc drawn circumscribing the base of ${\mathcal{P}}_2$ from center $T_4$ is the second arc of our golden spiral. Note that since the centers of the two arcs are collinear with $T_1$, the tangents to the two arcs at $T_1$ coincide. Thus the spiral curve is tangent smooth at the transition point.

The $n$th step in this iterative process begins with the Castor Triangle ${\mathcal{C}}_n\doteq [T_n,T_{n+1},T_{n+2}]$. The base angle at $T_{n+1}$ is bisected thus creating a pair of Gemini Twins ${\mathcal{C}}_{n+1},{\mathcal{P}}_{n+1}$. A tangent smooth arc of the associated Golden Spiral is then drawn circumscribing the base of ${\mathcal{P}}_{n+1}$ using its vertex as the center.

From a transformational viewpoint we map ${\mathcal{C}}_n$ into ${\mathcal{C}}_{n+1}$ by rotating the first Castor through $\frac{3\pi }{5}$ radians and simultaneously shrinking its sides by a factor of $\varphi$. I call this transformation the $\Phi$-map. We can think of it as the point transformation that maps each vertex $T_n$ into the ‘next’ vertex $T_{n+1}$ and describe it by saying that “the $\Phi$-map maps the vertices sequentially”.

The nested sequence $\{{\mathcal{C}}_n\}$ of Castor Triangles arising by the $\Phi$-map converge to a special point $E$ called the eye of the spiral. It can be located as the intersection of the medians $[T_{n+2},L_{n+2}]$ where $L_{n+2}$ is the midpoint of $[T_n,T_{n+1}]$. $E$ has a number of fascinating properties. (For proofs of these properties see [1]).

(1) $E$ divides each median $[T_{n+2},L_{n+2}]$ in the same ratio: $\frac{[T_{n+2}E]}{[E{L}_{n+2}]}=\frac{{\varphi }^2}{2}$.

(2) $E$ is the only fixed point of the $\Phi$-map. That is, its location in every Castor Triangle remains the same. In fact the areas of the three triangles into which ${\mathcal{C}}_n$ is subdivided when $E$ is joined to the three vertices are in the ratio (where, in general, we use $[ABC]$ to denote the area of triangle $[A,B,C]$)

(3) The $\Phi$-map also maps the midpoint sequence $\{L_{n+2}\}$ sequentially. Indeed each $[L_{n+2},L_{n+3},L_{n+4}]$ is a Castor Triangle and thus this sequence determines another golden spiral. This spiral intertwines the original spiral since it has the same eye $E$.

(4) It is easy to construct the image $A_1$ under the $\Phi$-map of any point $A_0$ in the plane $\overline{T_0{T}_1{T}_2}$. Simply join $A_0$ to $T_0$ and to $T_1$ and then construct lines parallel to these lines passing through $T_1$ and $T_2$ respectively. The intersection of these two lines is $A_1$. Similarly join $A_1$ to $T_1$ and $T_2$ and then construct lines parallel to these passing through $T_2$ and $T_3$ respectively. This will yield $A_2$, the image of $A_1$. Of necessity then $[A_0,A_1,A_2]$ will be a Castor Triangle and thus will generate a new golden spiral intertwining the other golden spirals since its eye is also $E$.

Thus $E$ is simultaneously the eye of an infinite family of golden spirals all of which can be generated from any one of them by the $\Phi$-map.

One other fascinating property of the Castor Sequence is that in each Castor Triangle ${\mathcal{C}}_n$, the interval $[T_n,T_{n+3}]$ on the side $[T_n,T_{n+2}]$ is divided harmonically by $M_{n+1}$, the midpoint of $[T_n,T_{n+2}]$ and $L_{n+4}$, the midpoint of $[T_{n+2},T_{n+3}]$ (which is a side of ${\mathcal{C}}_{n+1}$). In other words, $[T_n{T}_{n+3}]$ is the harmonic mean of $[T_n{M}_{n+1}]$ and $[T_n{L}_{n+4}]$. Thus the set $\{T_n,M_{n+1},T_{n+3},L_{n+4}\}$ is the geometric counterpart of a major musical chord.

The Subtle Symmetry of Golden Spirals 115

2. The Golden Spiral Inscribed in Golden Rectangles

A golden rectangle is one for which the ratio of the length of the long side to that of the short side is $\varphi$. This automatically makes the long side length the geometric mean between the semiperimeter and the length of the short side.

Let ${\mathcal{R}}_1\doteq [R_1,R_{-2},R_0,R_2]$ denote a golden rectangle as in Figure 3 with vertical side lengths $[R_1{R}_{-2}]=[R_0{R}_2]=1$ and horizontal side lengths $[R_{-2}{R}_0]=[R_2{R}_1]=\varphi$.

To construct the golden spiral associated with ${\mathcal{R}}_1$ we subdivide it by the line segment $[R_3,R_4]$ where $R_3$ divides $[R_{-2},R_0]$ and $R_4$ divides $[R_1,R_2]$ in the golden ratio. This subdivides ${\mathcal{R}}_1$ into a square ${\mathcal{S}}_1\doteq [R_1,R_{-2},R_3,R_4]$ each of whose sides is of length 1 and a new rectangle ${\mathcal{R}}_3\doteq [R_3,R_0,R_2,R_4]$ the ratio of whose sides is $\frac{1}{\varphi -1}=\frac{1}{\frac{1}{\varphi }}=\varphi$ so that ${\mathcal{R}}_3$ is another golden rectangle. Reversing the process, we can say if to ${\mathcal{R}}_3$ we adjoin ${\mathcal{S}}_1$ we end up with ${\mathcal{R}}_1$ so that ${\mathcal{S}}_1$ is a gnomon for ${\mathcal{R}}_3$.

Using $R_4$ as a center and $[R_4,R_1]$ as the radius we inscribe in ${\mathcal{S}}_1$, an arc from $R_1$ to $R_3$. This is the first arc of our golden spiral.

We now iterate this process using ${\mathcal{R}}_3$ as the mother rectangle subdividing it into a square ${\mathcal{S}}_3\doteq [R_3,R_0,R_5,R_6]$ and a new golden rectangle ${\mathcal{R}}_5\doteq [R_5,R_2,R_7,R_8]$. With $R_6$ as the center we inscribe an arc beginning at $R_3$ and ending at $R_5$. Since $R_3$ is collinear with both centers $R_4$ and $R_6$ the two arcs form a tangent smooth curve at $R_3$.

Each step in the iteration process leads to a subdivision of a golden rectangle ${\mathcal{R}}_{2d-1}$ by the interval $[R_{2d+1},R_{2(d+1)}]$ into a square ${\mathcal{S}}_{2d-1}$ with vertices $R_n$ with $n=2d-1,2(d-2),2(d-1),2d$ and a new golden rectangle ${\mathcal{R}}_{2d+1}$ with vertices $R_m$ where $m=2d+1,2(d-1),2d,2(d+1)$. If $d$ is even the long sides are horizontal, otherwise they are vertical. The arc of the golden spiral associated with this square connects $R_{2d-1}$ to $R_{2d+1}$ and the center is $R_{2(d+1)}$.

The nested sequence of golden rectangles converge to a point $E$ that is the eye of the associated spiral. It may be constructed as the intersection of the diagonals $[R_{-2},R_2]$ and $[R_0,R_4]$ and it divides each of those diagonals in the ratio ${\varphi }^2$. These diagonals are perpendicular.

When $R_1$, the unused vertex of ${\mathcal{R}}_1$, is joined to $E$ it intersects the opposite side at $R_5$. When $R_3$ the unused vertex of ${\mathcal{R}}_3$ is joined to $E$ it intersects the opposite side in $R_7$. These two lines are also perpendicular.

Concerning these four ‘spokes’ emanating from $E$ we have:

The Golden Harmonic Spokes Theorem

The pencil of four ‘spokes’ emanating from $E$ (the eye of the golden spiral inscribed in a nested sequence of golden rectangles), consisting of a pair of perpendicular diagonals containing all of the even indexed vertices, and a pair of perpendicular ‘non-diagonals’ containing all of the odd indexed vertices, contains all of the vertices of all the golden rectangles in the sequence. Each pair bisects the other pair (so that the 8 angles they determine with vertex $E$ are each $\frac{\pi }{4}$ radians and the two pairs together consequently form an harmonic set of lines. Thus they intersect each vertical and each horizontal side in an harmonic set of vertices.

The vertices on each of the four harmonic spokes are marvelously arranged in that any four ‘consecutive’ (i.e. their indices are of the form $(d,d+4,d+8,d+12)$) vertices on each of them is also an harmonic set of points.

References

[1] Swimmer, A., On Golden Spirals: The Subtlety of Their Symmetry. Preprint.

Alvin Swimmer

Figure 1. The Gemini Family and its Spiral

The Subtle Symmetry of Golden Spirals

Figure 2. Five Mothers and Their Intertwining Spirals

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Figure 3. The Pencil of Harmonic Spokes