Equiangular Spirals
Kaz Uyehara
What is a spiral?
 A curve that starts from a point of
origin whose radius of curvature
continually increases
 Spirals that are more of a state rather
than a form are not as interesting (a
chameleon coiling its tail)
Why might we see spirals?
 If something wants to get bigger without
changing how it grows, why not just keep
adding on in the same way?
 Each increment is similar to its predecessor.
 It grows only at one end, but the form is
constant.
 For the equiangular spiral there is always an
element of time. Different parts will have a
different age.
The Spiral of Archimedes
 “If the radius vector revolve uniformly about the pole,
a point (P) travel with uniform velocity along it, the
curve described will be that called the equable spiral
,or spiral of Archimedes”
 The radius, r = OP, will increase in arithmetic
progression and will be equal to a constant (a)
multiplied by the number of whorls (or angle b). So r
= ab.
The Equiangular Spiral (also called
logarithmic)
 We don’t really see Archimedes’ Spirals around in nature, but
we do see equiangular ones.
 If the point moves along the radius vector with an increasing
velocity from the pole, the path is an equiangular spiral.
 The radius vector will increase in length in geometric
progression as it sweeps through successive equal angles. So,
r=a^b
Some properties of the
equiangular spiral
It grows in size, but never changes shape,
So it has constant similarity of form.
A magnified small spiral is the same as a
large one.
(The angle between the
Tangent and radial line
At the point (r, theta) is constant.
Some quick examples
 Hawks approach their prey in an equiangular spiral, because
they see best at an angle form their direction of flight
 Some insects approach a light in an equiangular spiral because
they have compound eyes (they do not see straight in front of
them). So they keep readjusting their path by a constant angle.
 Spiral arms of a galaxy are roughly logarithmic.
 The arms of tropical cyclones are also roughly logarithmic.
 But perhaps most interesting for us… the shape of dead tissue
in certain parts of organisms.
The Molluscan Shell
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Different types of shells have different constant angle and the angle of
retardation - “the retardation in growth of the inner as compared with
the outer part of each whorl (how much they are separated from each
other), and the enveloping angle of the cone - “the angle which a
tangent to the whorl makes with the axis”
The shell is often times made by adding on shapes like that of the
mouth of the shell at some angle to the axis, such that the shape of
the shell is maintained (called a gnomon).
Small angle
Large angle
The Angle
 Well… it turns out that a lot of times the “constant” angle
changes sometimes. Or it isn’t a perfect equiangular spiral.
 As organisms get older the angle sometimes increases with age
(whorls get closer)
Horns!
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Horns are less symmetrical and harder to measure than shells, but they
have an element of time similar to that of shells.
Horns are closed curves unlike shells.
Rhino horns are logarithmic spirals, but the angle is small, so it is hard
to see the spiral. But, some horns are just not log spirals.
The lack of symmetry in horns may have to do with them being bent
by their own weight or the horn will actually become more stiff (and
straighten out). Plus… Thompson says that they different parts of the
horn have different rates of growth.
More Thompson, MORE!
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Since horns on a lot of animals are not along a line of symmetry, it is
not surprising that they do not grow symmetrically. Thompson did a
bunch of stuff with horns to prove that a lot of the horns can be
explained by log spirals, unequal growth rates, and torsion (think the
transformation chapter).
Thompson doesn’t really talk much about the natural selection of the
horn shape, but he does note that weight of the horns often perfectly
balance the head of the animal so that there is no natural strain.
Teeth, beaks, and claws also grow in our classic add on fashion. We
don’t see the spirals as easily because they do not grow long enough.
But we start to see them when they grow abnormally long. But normal
wear and tear or limits on length make it less apparent.
Not really sure what’s up with Narwhal’s tusk… It’s a screw.
Thompson gives a kind of sketchy argument for this.
What’s the deal with
Fibonacci?
The sequence:
The Golden Ratio is the limit in the ratios of successive
Terms in the Fibonacci sequence.
The Golden Ratio is approximately 1.618132.
Let c be the circumference of a circle, that is
divided into two arcs (a and b) such that:
C/a = a/b. The angle of this arc is the…
Golden Angle! So called because a/b = Golden ratio
Approx = 137.51 degrees
Fibonacci in nature
 A lot more hyped up then actually present.
 A lot of things are close to the golden angle, but not as many as
some would have you believe.
 But the sequence is related to our spirals. If you make squares
with side lengths equal to successive terms in the sequence,
each time you add a square you will get a similar rectangle.
 If we draw a line connecting the corners of the squares we get
our spiral. These are Fibonacci rectangles (there are also
Fibonacci triangles and all sorts of other stuff).
More Fibonacci (I am not that impressed)
 Many flowers have Fibonacci numbers for petals.
 Seed heads, pine cones, and veggies can also have Fibonacci
numbers of spirals.
Slightly More Impressive
 So it is thought that the Fibonacci spirals are
geometrically/mathematically favorable to
reduce elastic energy.
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
These experiments suggest that stressing inorganic material
can produce Fibonacci spirals. The results were fairly robust.
They suggest that Fibonacci spirals appear when spheres are stressed
And there is available conical support.
Sources
 D. W. (D'Arcy Wentworth) Thompson, On
Growth and Form
 Wikipedia ftw!
 Li, Zhang, and Cao. Triangular and Fibonacci
Number Patterns Driven by Stress on
Core/Shell Microstructures. 2005.
 Li, Ji, and Cao. Stressed Fibonacci Spiral
Patterns of Definite Chirality. 2007.