List of spirals

Last updated

This list of spirals includes named spirals that have been described mathematically.

ImageNameFirst describedEquationComment
circle ${\displaystyle r=k}$The trivial spiral
Archimedean spiral c.320 BC${\displaystyle r=a+b\cdot \theta }$
Euler spiral also called Cornu spiral or polynomial spiral
Fermat's spiral (also parabolic spiral)1636 [1] ${\displaystyle r^{2}=a^{2}\theta }$
hyperbolic spiral 1704${\displaystyle r=a/\theta }$also reciprocal spiral
lituus 1722${\displaystyle r^{2}\theta =k}$
logarithmic spiral 1638 [2] ${\displaystyle r=a\cdot e^{b\theta }}$approximations of this are found in nature
Fibonacci spiral circular arcs connecting the opposite corners of squares in the Fibonacci tilingapproximation of the golden spiral
golden spiral ${\displaystyle r=\varphi ^{\theta {\frac {2}{\pi }}}\,}$special case of the logarithmic spiral
Spiral of Theodorus (also Pythagorean spiral)an polygonal spiral composed of contiguous right triangles, that approximates the Archimedean spiral
involute 1673
helix ${\displaystyle r(t)=1,\,}$${\displaystyle \theta (t)=t,\,}$${\displaystyle h(t)=t.\,}$a 3-dimensional spiral
Rhumb line (also loxodrome)type of spiral drawn on a sphere
Cotes's spiral 1722
Poinsot's spirals ${\displaystyle r=a\operatorname {csch} (n\theta ),\,}$
${\displaystyle r=a\operatorname {sech} (n\theta )}$
Nielsen's spiral 1993 [3] ${\displaystyle x(t)=\operatorname {ci} (t),\,}$
${\displaystyle y(t)=\operatorname {si} (t)}$
A variation of Euler spiral, using sine integral and cosine integrals
Polygonal spiral special case approximation of logarithmic spiral
Fraser's Spiral 1908Optical illusion based on spirals
Conchospiral ${\displaystyle {\begin{cases}r=\mu ^{t}a\\\theta =t\\z=\mu ^{t}c\end{cases}}}$three-dimensional spiral on the surface of a cone.
Calkin–Wilf spiral
Ulam spiral (also prime spiral)1963
Sack's spiral 1994variant of Ulam spiral and Archimedean spiral.
Seiffert's spiral spiral curve on the surface of a sphere
Tractrix spiral1704 [4] ${\displaystyle {\begin{cases}r=A\cos(t)\\\theta =\tan(t)-t\end{cases}}}$
Pappus spiral 1779${\displaystyle {\begin{cases}r=a\theta \\\psi =k\end{cases}}}$3D conical spiral studied by Pappus and Pascal [5]
doppler spiral ${\displaystyle {\begin{cases}x=a(t\cos(t)+kt)\\y=at\sin(t)\end{cases}}}$2D projection of Pappus spiral [6]
Atzema spiral ${\displaystyle {\begin{cases}x=\sin(t)/t-2\cos(t)-t\sin(t)\\y=-\cos(t)/t-2\sin(t)+t\cos(t)\end{cases}}}$The curve that has a catacaustic forming a circle. Approximates the Archimedean spiral. [7]
Atomic spiral 2002${\displaystyle r=\theta /(\theta -a)}$This spiral has two asymptotes; one is the circle of radius 1 and the other is the line ${\displaystyle \theta =a}$ [8]
Galactic spiral 2019${\displaystyle {\begin{cases}dx=R(y/({\sqrt {(}}x^{2}+y^{2}))d\theta \\dy=R(\rho (\theta )-x/{\sqrt {(}}x^{2}+y^{2}))d\theta \end{cases}}{\begin{cases}x=\sum dx\\y=\sum dy+R\end{cases}}}$The differential spiral equations were developed to simulate the spiral arms of disc galaxies, have 4 solutions with three different cases:${\displaystyle \rho <1,\rho =1,\rho >1}$, the spiral patterns are decided by the behavior of the parameter ${\displaystyle \rho }$. For ${\displaystyle \rho <1}$, spiral-ring pattern; ${\displaystyle \rho =1,}$ regular spiral; ${\displaystyle \rho >1,}$ loose spiral. R is the distance of spiral starting point (0, R) to the center. The calculated x and y have to be rotated backward by (${\displaystyle -\theta }$) for plotting. Please check the references for the detail [9]

Related Research Articles

In mathematics, a unary operation is an operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is the function f : AA, where A is a set. The function f is a unary operation on A.

In mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contains an axiomatic system and all its derived theorems. An axiomatic system that is completely described is a special kind of formal system. A formal theory is an axiomatic system that describes a set of sentences that is closed under logical implication. A formal proof is a complete rendition of a mathematical proof within a formal system.

7 (seven) is the natural number following 6 and preceding 8. It is the only prime number preceding a cube, and is often considered lucky in Western culture, and is often seen as highly symbolic.

6 (six) is the natural number following 5 and preceding 7. It is a composite number and the smallest perfect number.

MathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at Urbana–Champaign.

Wolfram Research is an American multinational company that creates computational technology. Wolfram's flagship product is the technical computing program Wolfram Mathematica, first released on June 23, 1988. Other products include Wolfram Alpha, Wolfram SystemModeler, Wolfram Workbench, gridMathematica, Wolfram Finance Platform, webMathematica, the Wolfram Development Platform, and the Wolfram Programming Lab. Wolfram Research founder Stephen Wolfram is the CEO. The company is headquartered in Champaign, Illinois, United States.

Eric Wolfgang Weisstein is an encyclopedist who created and maintains MathWorld and Eric Weisstein's World of Science (ScienceWorld). He is the author of the CRC Concise Encyclopedia of Mathematics. He works for Wolfram Research, Inc.

In the differential geometry of curves, a roulette is a kind of curve, generalizing cycloids, epicycloids, hypocycloids, trochoids, epitrochoids, hypotrochoids, and involutes.

An antimagic square of order n is an arrangement of the numbers 1 to n2 in a square, such that the sums of the n rows, the n columns and the two diagonals form a sequence of 2n + 2 consecutive integers. The smallest antimagic squares have order 4. Antimagic squares contrast with magic squares, where each row, column, and diagonal sum must have the same value.

In geometry, the gyrobifastigium is the 26th Johnson solid (J26). It can be constructed by joining two face-regular triangular prisms along corresponding square faces, giving a quarter-turn to one prism. It is the only Johnson solid that can tile three-dimensional space.

In mathematics, a mathematical object is said to satisfy a property locally, if the property is satisfied on some limited, immediate portions of the object.

In mathematics, an interprime is the average of two consecutive odd primes. For example, 9 is an interprime because it is the average of 7 and 11. The first interprimes are:

Edgar van Tuyll was the chief quantitative strategist of Pictet & Cie, where he worked from 1995 to 2017. He has been extensively quoted by the media for his prediction of the 2000 Dot-com bubble crash and of the bull market beginning in March 2003. He is among the minority of strategists expecting a US recession in 2007-2008. His website Links to unsolved problems, prizes and research is top ranked by Google for list of unsolved problems in mathematics and physics. He is the author of several entries in the "CRC Concise Encyclopedia of Mathematics", Chapman & Hall, 2002. He is the grandson of Antti Johannes Rantamaa.

In mathematics, an operation is a function which takes zero or more input values to a well-defined output value. The number of operands is the arity of the operation.

A golden triangle, also called a sublime triangle, is an isosceles triangle in which the duplicated side is in the golden ratio to the base side:

In geometry, a polytope of dimension 3 or higher is isohedral or face-transitive when all its faces are the same. More specifically, all faces must be not merely congruent but must be transitive, i.e. must lie within the same symmetry orbit. In other words, for any faces A and B, there must be a symmetry of the entire solid by rotations and reflections that maps A onto B. For this reason, convex isohedral polyhedra are the shapes that will make fair dice.

Differential equations, in particular Euler equations, rose in prominence during World War II in calculating the accurate trajectory of ballistics, both rocket-propelled and gun or cannon type projectiles. Originally, mathematicians used the simpler calculus of earlier centuries to determine velocity, thrust, elevation, curve, distance, and other parameters.

In geometry, an edge is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope. In a polygon, an edge is a line segment on the boundary, and is often called a side. In a polyhedron or more generally a polytope, an edge is a line segment where two faces meet. A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal.

In geometry, a Devil's curve is a curve defined in the Cartesian plane by an equation of the form

In mathematics, the Ernst equation is an integrable non-linear partial differential equation, named after the American physicist Frederick J. Ernst.

References

1. "Fermat spiral - Encyclopedia of Mathematics". www.encyclopediaofmath.org. Retrieved 18 February 2019.
2. Weisstein, Eric W. "Logarithmic Spiral". mathworld.wolfram.com. Wolfram Research, Inc. Retrieved 18 February 2019.
3. Weisstein, Eric W. "Nielsen's Spiral". mathworld.wolfram.com. Wolfram Research, Inc. Retrieved 18 February 2019.
4. "Tractrix spiral". www.mathcurve.com. Retrieved 2019-02-23.
5. "Conical spiral of Pappus". www.mathcurve.com. Retrieved 28 February 2019.
6. "Doppler spiral". www.mathcurve.com. Retrieved 28 February 2019.
7. "Atzema spiral". www.2dcurves.com. Retrieved 11 March 2019.
8. "atom-spiral". www.2dcurves.com. Retrieved 11 March 2019.
9. Pan, Hongjun. "New spiral" (PDF). www.arpgweb.com. Retrieved 5 March 2021.