In
mathematics, a
spiral is a
curve which
emanates from a central point, getting progressively farther away
as it revolves around the point.
Spiral or helix
An Archimedean spiral, a helix, and a conic spiral.
A "spiral" and a "
helix" are two terms that
are easily confused, but represent different objects.
A spiral is typically a
planar
curve (that is, flat), like the groove on a
record or the arms of a
spiral galaxy. A helix, on the other hand, is
a three-dimensional coil that runs along the surface of a cylinder,
like a
screw. There are many instances where
in
colloquial usage spiral is used as a
synonym for helix, notably
spiral staircase and
spiral binding
of books.
In the side picture, the black curve at the bottom is an
Archimedean spiral, while the green curve
is a helix. A cross between a spiral and a helix, such as the curve
shown in red, is known as a conic helix. An example of a conic
helix is the spring used to hold and make contact with the negative
terminals of AA or AAA batteries in remote controls.
Two-dimensional spirals
A
two-dimensional spiral may be
described most easily using
polar
coordinates, where the
radius r
is a
continuous monotonic function of angle θ. The circle would be
regarded as a
degenerate
case (the function not being strictly monotonic, but rather
constant).
Some of the more important sorts of two-dimensional spirals
include:
Image:Archimedean spiral.svg|Archimedean spiralImage:Cornu
Spiral.svg|Cornu spiralImage:Fermat's spiral.svg|Fermat's
spiralImage:Hyperspiral.svg|hyperbolic
spiralImage:Lituus.svg|lituusImage:Logarithmic
spiral.svg|logarithmic spiralImage:Spiral of Theodorus.svg|spiral
of Theodorus
Three-dimensional spirals
For simple 3-d spirals, a third variable,
h (height), is
also a continuous,
monotonic
function of θ. For example, a conic
helix
may be defined as a spiral on a conic surface, with the distance to
the apex an exponential function of θ.
The
helix and
vortex can
be viewed as a kind of
three-dimensional
spiral.
For a helix with thickness, see
spring
.
Another kind of spiral is a conic spiral along a circle. This
spiral is formed along the surface of a
cone whose axis is bent and restricted to a
circle:
This image is reminiscent of a
Ouroboros
symbol and could be mistaken for a torus with a
continuously-increasing diameter:
Spherical spiral
Archimedean Spherical Spiral
A
spherical spiral (
rhumb line
or loxodrome, left picture) is the curve on a sphere traced by a
ship traveling from one pole to the other while keeping a fixed
angle (unequal to 0° and to 90°) with respect
to the meridians of
longitude, i.e.
keeping the same
bearing. The
curve has an
infinite number of
revolution, with the distance between
them decreasing as the curve approaches either of the poles.
The gap between the curves of an
Archimedean spiral (right picture)
remains constant as the radius changes and is hence not a
rhumb line.
As a symbol
The Newgrange entrance slab
The spiral
plays a specific role in symbolism, and
appears in megalithic art, notably in the
Newgrange tomb or in many Galician petroglyphs such as the
one in Mogor. See also
triple
spiral.
While scholars are still debating the subject, there is a growing
acceptance that the simple spiral, when found in Chinese art, is an
early symbol for the sun.
Roof tiles dating back to the Tang Dynasty with this symbol have been found
west of the ancient city of Chang'an (modern-day
Xian).
Spirals are also a symbol of
hypnosis,
stemming from the
cliché of people and
cartoon characters being hypnotized by staring into a spinning
spiral (One example being
Kaa in Disney's
The Jungle
Book). They are also used as a symbol of
dizziness, where the eyes of a cartoon character,
especially in
anime and
manga, will turn into spirals to show they are dizzy
or dazed. The spiral is also a prominent symbol in the anime
Gurren Lagann, where it
symbolizes the
double helix structure
of
DNA, representing biological
evolution, and the spiral structure of a
galaxy, representing
universal evolution.
In nature
The study of spirals in
nature have a long
history,
Christopher Wren observed
that many
shells form a
logarithmic spiral.
Jan Swammerdam observed the common
mathematical characteristics of a wide range of shells from
Helix to
Spirula and
Henry Nottidge Moseley described the
mathematics of
univalve shells.
D’Arcy Wentworth Thompson's
On
Growth and Form gives extensive treatment to these spirals. He
describes how shells are formed by rotating a closed curve around a
fixed axis, the
shape of the curve remains
fixed but its size grows in a
geometric progression. In some shell
such as
Nautilus and
ammonites the generating curve revolves in a plane
perpendicular to the axis and the shell will form a planar discoid
shape. In others it follows a skew path forming a
helico-spiral pattern.
Thompson also studied spirals occurring in
horn,
teeth,
claws and
plants.
Spirals in plants and animals are frequently described as
whorls.
A model for the pattern of
florets in the
head of a
sunflower was proposed by H
Vogel. This has the form
- \theta = n \times 137.5^{\circ},\ r = c \sqrt{n}
where n is the index number of the floret and c is a constant
scaling factor, and is a form of
Fermat's spiral. The angle 137.5° is related
to the
golden ratio and gives a close
packing of florets.
In art
The spiral has inspired artists down the ages. The most famous
piece of 60s Land Art was Robert Smithson's Spiral Jetty, at the
Great Salt Lake in Utah. The theme continues in David Wood's Spiral
Resonance Field at the Balloon Museum in Albuquerque.
References
See also
External links
- SpiralZoom.com, an educational website about
the science of pattern formation, spirals in nature, and spirals in
the mythic imagination.