Everything about Spiral totally explained
In
mathematics, a
spiral is a
curve which emanates from a central point, getting progressively farther away as it revolves around the point. The concise mathematical definition is "The locus of a point moving at constant speed whose distance from a fixed point increases at a specific rate."
Spiral or helix
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. Mathematically this is incorrect but the terms are increasing in common usage.
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 spiral
Image:Logarithmic spiral.svg|logarithmic spiral
Image:Fermat's spiral.png|Fermat's spiral
Image:Hyperspiral.png|hyperbolic spiral
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 (math).
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
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, for example keeping the same
bearing. The curve has an
infinite number of
revolutions, 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 curve progresses across the surface of the sphere. Therefore, this line has finite length. Notice that this isn't the same thing as the
rhumb line described earlier.
As a symbol
The spiral plays a certain 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's 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).
The spiral is the most ancient symbol found on every civilized continent. Due to its appearance at burial sites across the globe, the spiral most likely represented the "life-death-rebirth" cycle. Similarly, the spiral symbolized the sun, as ancient people thought the sun was born each morning, died each night, and was reborn the next morning..
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're dizzy or dazed.
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 pirpendicular to the axis and the shell will form a planer discoid shape. In others it follows a skew path forming a
helico-spiral pattern.
Thompson also studied spirals occurring in
horns,
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
»
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.
Further Information
Get more info on 'Spiral'.
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