There are a number of different types of spirals. There are flat spirals, 3-D spirals, right-handed spirals, left-handed spirals, equiangular spirals, geometric spirals, logarithmic spirals and rectangular spirals. The most well known spiral is that of the nautilus shell. All spirals have two things in common: expansion and growth. They are symbols of infinity.

![[img/sg_img-095.png|A equiangular spiral and its secants.]]

History

Want to learn differential equations? Visit Differential Equations, Mechanics, and Computation.

The investigation of spirals began at least with the ancient Greeks. The famous Equiangular Spiral was discovered by Rene Descartes in 1638, who started from the property $s = a\cdot r$. Evangelista Torricelli, who died in 1647, worked on it independently and used for a definition the fact that the radii are in geometric progression if the angles increase uniformly. From this he discovered the relation $s = a \cdot r$ – that is, he found the rectification of the curve. Jacob Bernoulli (1654-1705), some fifty years later, found all the “reproductive” properties of the curve; and these almost mystic properties of the “wonderful” spiral made him request to have the curve be engraved on his tomb: with the phrase ― Eadem mutata resurgo ― “I shall arise the same, though changed.” [source: E. H. Lockwood (1961); Robert C. Yates (1952)]

Description

Equiangular spiral describes a family of spirals of one parameter. It is defined as a curve that cuts all radial line at a constant angle. It also called logarithmic spiral, Bernoulli spiral, and logistique.

Explanation

1. Let there be a spiral (that is, any curve $r==f[θ]$ where f is a monotonic increasing function)
2. From any point $P$ on the spiral, draw a line toward the center of the spiral. (this line is called the radial line)
3. If the angle formed by the radial line and the tangent for any point $P$ is constant, the curve is a equiangular spiral.
   ![[img/sg_img-096.png]]

A special case of equiangular spiral is the circle, where the constant angle is 90°.

![[img/sg_img-102a.png|Equiangular spirals with 40°, 50°, 60°, 70°, 80° and 85° (left to right)]]

Formulas

Let $α$ be the constant angle.
Polar: $r == E^{(θ * \cot[α])}$
Parametric: $E^{(t* \cot[α])} \text{ }\{Cos[t],Sin[t]\}$
Cartesian: $x^2 + y^2 == E^{(ArcTan[y/x] \cdot \cot[α] )}$

Properties

Point Construction and Geometric Sequence

Length of segments of any radial ray cut by the curve is a geometric sequence, with a multiplier of $E^{(2 π Cot[α])}$.

Lengths of segments of the curve, cut by equally spaced radial rays, is a geometric sequence.

![[img/sg_img-103.png]]

The curve cut by radial rays. The length of any green ray’s segments is geometric sequence. The lengths of red segments is also a geometric sequence. In the figure, the dots are points on a 85°
equiangular spiral.

Catacaustic

[Catacaustic] of a equiangular spiral with light source at center is a equal spiral.

Proof: Let $O$ be the center of the curve. Let $α$ be the curve’s constant angle. Let $Q$ be the reflection of $O$ through the tangent normal of a point $P$ on the curve. Consider Triangle $[O,P,Q]$. For any point $P$, $Length[Segment$[O,P]] == Length[Segment[P,Q]]$ and $Angle[O,P,Q]$ is constant. $Angle[O,P,Q]$ is constant because the curve’s constant angle definition.) Therefore, by argument
of similar triangle, then for any point $P$, $$Length[Segment[O,Q]]==Length[Segment[O,P]] * s$$for some constant $s$. Since scaling and rotation around its center does not change the curve, thus the locus of $Q$ is a equiangular spiral with constant angle $α$, and $$Angle[O,Q,P] == α$$$Line[P,Q]$ is the tangent at $Q$.

![[img/sg_img-104.png|[Equiangular Spiral Caustic]]]

Curvature

• The [evolute] of a equiangular spiral is the same spiral rotated.
• The [involute] of a equiangular spiral is the same spiral rotated.

![[img/sg_img-105a.png]]

Left: Tangent circles of a 80° equiangular spiral. The white dots are the centers of tangent circles, the lines are the radiuses. Right: Lines are the tangent normals, forming the evolute curve by [envelope]. [Equiangular Spiral Evolute]

Radial

The radial of a equiangular spiral is itself scaled. The figure on the left shows a 70° equiangular spiral and its radial. The figure on the right shows its [involute], which is another equiangular spiral.

![[img/sg_img-107a.png]]

Inversion

The [inversion] of a equiangular spiral with respect to its center is a equal spiral.

![[img/sg_img-109.png]]

Pedal

The pedal of a equiangular spiral with respect to its center is a equal spiral.

![[img/sg_img-110.png]]

Pedal of a equiangular spiral. The lines from center to the red dots is perpendicular to the
tangents (blue lines). The blue curve is a 60° equiangular spiral. The red dots forms its pedal.

Pursuit Curve

Persuit curves are the trace of a object chasing another. Suppose there are $n$ bugs each at a corner of a $n$ sided regular polygon. Each bug crawls towards its next neighbour with uniform speed. The trace of these bugs are equiangular spirals of $(n-2)/n * π/2$ radians (half the angle of the polygon’s corner).

![[img/sg_img-111a.png]]

Left: shows the trace of four bugs, resulting four equiangular spirals of 45°. Above right: six objects forming a chasing chain. Each line is the direction of movement and is tangent to the equiangular spirals so formed.

Spiral in nature

Spiral is the basis for many natural growths.

![[img/sg_img-113b.png]]

Seashells have the geometry of equiangular spiral.

![[img/sg_img-117.png|A cauliflower (Romanesco broccoli) exhibiting equiangular spiral and fractal geometry. (Photo by Dror Bar-Natan. Source)]]

Spirals

• Belousov’s Brew: A recipe for making spiraling patterns in chemical reactions.

• Equiangular spiral. Properties of Bernoulli’s logarithmic ‘spiralis mirabilis’.

• Fermat’s spiral and the line between Yin and Yang

• Fermat’s spiral and the line between Yin and Yang. Taras Banakh, Oleg Verbitsky, and Yaroslav Vorobets argue that the ideal shape of the dividing line in a Yin-Yang symbol is formed, not from two semicircles, but from Fermat’s spiral.

• Fourier series of a gastropod. L. Zucca uses Fourier analysis to square the circle and to make an odd spiral-like shape.

• The golden bowls and the logarithmic spiral.

• Golden spiral flash animation, Christian Stadler.

• Graphite with growth spirals on the basal pinacoids]. Pretty pictures of spirals in crystals. (A pinacoid, it turns out, is a plane parallel to two crystallographic axes.)

• Helical Gallery. Spirals in the work of M. C. Escher and in X-ray observations of the sun’s corona.

• Mathematical imagery by Jos Leys. Knots, Escher tilings, spirals, fractals, circle inversions, hyperbolic tilings, Penrose tilings, and more. -

• Log-spiral tiling, and other radial and spiral tilings, S. Dutch.

• Looking at sunflowers. In this abstract of an undergraduate research paper, Surat Intasang investigates the spiral patterns formed by sunflower seeds, and discovers that often four sets of spirals can be discerned, rather than the two sets one normally notices.

• Modeling mollusc shells with logarithmic spirals, O. Hammer, Norsk Net. Tech. Also includes a list of logarithmic spiral links.

• Pi curve. Kevin Trinder squares the circle using its involute spiral. See also his quadrature based on the 3-4-5 triangle.

• Pictures of various spirals, Eric Weeks.

• Polyform spirals.

• Ram’s Horn cardboard model of an interesting 3d spiral shape bounded by a helicoid and two nested cones.

• Research: spirals, Mícheál Mac an Airchinnigh. Presumably this connects to his thesis that “there is a geometry of curves which is computationally equivalent to a Turing Machine”.

• Seashell spirals. Xah Lee examines the shapes of various real seashells, and offers prize money for formulas duplicating them.

• Soddy Spiral. R. W. Gosper calculates the positions of a sequence of circles, each tangent to the three previous ones.

• Spidron, a triangulated double spiral shape tiles the plane and various other surfaces. With photos of related paperfolding experiments.

• Spira Mirabilis logarithmic spiral applet by A. Bogomily.

• Spiral generator, web form for creating bitmap images of colored logarithmic spirals.
  ![[img/sg_img-118.png]]

• Spiral in a liquid crystal film.

• Spiral minaret of Samara.

• A spiral of squares with Fibonacci-number sizes, closely related to the golden spiral, Keith Burnett. See also his hand-painted Taramundi spiral.

• Spiral tea cozy, Kathleen Sharp.

• Spiral tilings. These similarity tilings are formed by applying the exponential function to a lattice in the complex number plane.
  ![[img/sg_img-119.png]]

• Spiral tower. Photo of a building in Iraq, part of a web essay on the geometry of cyberspace.

• Spiral triangles, Eric Weeks.

• Spiraling Sphere Models. Bo Atkinson studies the geometry of a solid of revolution of an Archimedean spiral.

• Spirals]. Mike Callahan and Larry Shook use a spreadsheet to investigate the spirals formed by repeatedly nesting squares within larger squares.

• Spirals and other 2d curves, Jan Wassenaar.

• Spring into action. Dynamic origami. Ben Trumbore, based on a model by Jeff Beynon from Tomoko Fuse’s book Spirals.

• These two pictures by Richard Phillips are from the now-defunct Maths with Photographs website. The chimney is (Phillips thinks) somewhere in North Nottinghamshire, England. A similar collection of Phillips’ mathematical photos is now available on CD-ROM.
  ![[img/sg_img-120a.png]]

• Three spiral tattoos from the Discover Magazine Science Tattoo Emporium.

• The uniform net (10,3)-a. An interesting crystal structure formed by packing square and octagonal helices.

• Wonders of Ancient Greek Mathematics, T. Reluga. This term paper for a course on Greek science includes sections on the three classical problems, the Pythagorean theorem, the golden ratio, and the Archimedean spiral.