A Fermat's spiral or parabolic spiral is a
plane curve
In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic ...
with the property that the
area between any two consecutive full turns around the spiral is invariant. As a result, the distance between turns grows in
inverse proportion to their distance from the spiral center, contrasting with the
Archimedean spiral (for which this distance is invariant) and the
logarithmic spiral
A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige Linie"). More ...
(for which the distance between turns is proportional to the distance from the center). Fermat spirals are named after
Pierre de Fermat.
[Anastasios M. Lekkas, Andreas R. Dahl, Morten Breivik, Thor I. Fossen]
"Continuous-Curvature Path Generation Using Fermat's Spiral"
In: ''Modeling, Identification and Control''. Vol. 34, No. 4, 2013, pp. 183–198, .
Their applications include curvature continuous blending of curves,
[ modeling ]plant growth Important structures in plant development are buds, shoots, roots, leaves, and flowers; plants produce these tissues and structures throughout their life from meristems located at the tips of organs, or between mature tissues. Thus, a living plant a ...
and the shapes of certain spiral galaxies, and the design of variable capacitors, solar power reflector arrays, and cyclotron
A cyclotron is a type of particle accelerator invented by Ernest O. Lawrence in 1929–1930 at the University of California, Berkeley, and patented in 1932. Lawrence, Ernest O. ''Method and apparatus for the acceleration of ions'', filed: Jan ...
s.
Coordinate representation
Polar
The representation of the Fermat spiral in polar coordinates is given by the equation
for . The two choices of sign give the two branches of the spiral, which meet smoothly at the origin. If the same variables were reinterpreted as Cartesian coordinates, this would be the equation of a parabola
In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exact ...
with horizontal axis, which again has two branches above and below the axis, meeting at the origin.
Cartesian
The Fermat spiral with polar equation can be converted to the Cartesian coordinates by using the standard conversion formulas and . Using the polar equation for the spiral to eliminate from these conversions produces parametric equations for the curve:
which generate the points of one branch of the curve as the parameter ranges over the positive real numbers.
For any generated in this way, dividing by cancels the parts of the parametric equations, leaving the simpler equation . From this equation, substituting by
(a rearranged form of the polar equation for the spiral) and then substituting by (the conversion from Cartesian to polar) leaves an equation for the Fermat spiral in terms of only and :
Because the sign of is lost when it is squared, this equation covers both branches of the curve.
Geometric properties
Division of the plane
A complete Fermat's spiral (both branches) is a smooth double point free curve, in contrast with the Archimedean and hyperbolic spiral. It divides the plane (like a line or circle or parabola) into two connected regions. But this division is less obvious than the division by a line or circle or parabola. It is not obvious to which side a chosen point belongs.
Polar slope
From vector calculus in polar coordinates one gets the formula
:
for the ''polar slope'' and its angle between the tangent of a curve and the corresponding polar circle (see diagram).
For Fermat's spiral one gets
:
Hence the slope angle is monotonely decreasing.
Curvature
From the formula
:
for the curvature of a curve with polar equation and its derivatives
:
one gets the ''curvature'' of a Fermat's spiral:
At the origin the curvature is 0. Hence the complete curve has at the origin an inflection point and the -axis is its tangent there.
Area between arcs
The area of a ''sector'' of Fermat's spiral between two points and is
:
After raising both angles by one gets
:
Hence the area of the region ''between'' two neighboring arcs is
only depends on the ''difference'' of the two angles, not on the angles themselves.
For the example shown in the diagram, all neighboring stripes have the same area: .
This property is used in electrical engineering for the construction of variable capacitors.
Special case due to Fermat
In 1636, Fermat wrote a letter [''Lettre de Fermat à Mersenne du 3 juin 1636, dans Paul Tannery.'' In: ''Oeuvres de Fermat.'' T. III, S. 277, ]
Lire en ligne.
' to Marin Mersenne which contains the following special case:
Let ; then the area of the black region (see diagram) is , which is half of the area of the circle with radius . The regions between neighboring curves (white, blue, yellow) have the same area . Hence:
* The area between two arcs of the spiral after a full turn equals the area of the circle .
Arclength
The length of the arc of Fermat's spiral between two points can be calculated by the integral:
:
This integral leads to an elliptical integral, which can be solved numerically.
Circle inversion
The inversion at the unit circle has in polar coordinates the simple description .
* The image of Fermat's spiral under the inversion at the unit circle is a lituus
The word ''lituus'' originally meant a curved augural staff, or a curved war-trumpet in the ancient Latin language. This Latin word continued in use through the 18th century as an alternative to the vernacular names of various musical instruments ...
spiral with polar equation When , both curves intersect at a fixed point on the unit circle.
* The tangent (-axis) at the inflection point (origin) of Fermat's spiral is mapped onto itself and is the asymptotic line of the lituus spiral.
The golden ratio and the golden angle
In disc phyllotaxis
In botany, phyllotaxis () or phyllotaxy is the arrangement of leaves on a plant stem. Phyllotactic spirals form a distinctive class of patterns in nature.
Leaf arrangement
The basic arrangements of leaves on a stem are opposite and alterna ...
, as in the sunflower and daisy, the mesh of spirals occurs in Fibonacci number
In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
s because divergence (angle of succession in a single spiral arrangement) approaches the golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
. The shape of the spirals depends on the growth of the elements generated sequentially. In mature-disc phyllotaxis
In botany, phyllotaxis () or phyllotaxy is the arrangement of leaves on a plant stem. Phyllotactic spirals form a distinctive class of patterns in nature.
Leaf arrangement
The basic arrangements of leaves on a stem are opposite and alterna ...
, when all the elements are the same size, the shape of the spirals is that of Fermat spirals—ideally. That is because Fermat's spiral traverses equal annuli in equal turns. The full model proposed by H. Vogel in 1979 is
:
where is the angle, is the radius or distance from the center, and is the index number of the floret and is a constant scaling factor. The angle 137.508° is the golden angle which is approximated by ratios of Fibonacci number
In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
s.
The resulting spiral pattern of unit disks should be distinguished from the Doyle spiral
In the mathematics of circle packing, a Doyle spiral is a pattern of non-crossing circles in the plane in which each circle is surrounded by a ring of six tangent circles. These patterns contain spiral arms formed by circles linked through opposi ...
s, patterns formed by tangent disks of geometrically increasing radii placed on logarithmic spiral
A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige Linie"). More ...
s.
Solar plants
Fermat's spiral has also been found to be an efficient layout for the mirrors of concentrated solar power plants.
See also
* List of spirals
This list of spirals includes named spirals that have been described mathematically.
See also
* Catherine wheel (firework)
* List of spiral galaxies
* Parker spiral
* Spirangle
* Spirograph
Spirograph is a geometric drawing device that ...
* Patterns in nature
Patterns in nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be modelled mathematically. Natural patterns include symmetries, trees, spirals, meanders, waves, ...
* Spiral of Theodorus
In geometry, the spiral of Theodorus (also called ''square root spiral'', ''Einstein spiral'', ''Pythagorean spiral'', or ''Pythagoras's snail'') is a spiral composed of right triangles, placed edge-to-edge. It was named after Theodorus of Cyre ...
References
Further reading
*
External links
*
Online exploration using JSXGraph (JavaScript)
Fermat's Natural Spirals, in sciencenews.org
{{Pierre de Fermat
Spirals