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Logarithmic spiral

A logarithmic spiral, also called an equiangular spiral or growth spiral, is a self-similar plane curve whose radius increases exponentially with the angle according to the polar equation r = a e^{b \theta}, where r is the distance from the origin, \theta is the polar angle, and a > 0 and b > 0 are constants determining the scale and growth rate, respectively.^[1] This results in a constant angle \psi = \cot^{-1} b between the curve's tangent and the radial line from the origin, giving it equiangular properties that preserve its shape under scaling or rotation.^[1] The logarithmic spiral has been a subject of mathematical study since the 16th century, with early descriptions appearing in Albrecht Dürer's 1525 work Underweysung der Messung, followed by René Descartes' analysis in 1638 and Jacob Bernoulli's extensive exploration in the 1690s, where he highlighted its self-similarity with the phrase "eadem mutata resurgo" ("changed but the same") inscribed on his tombstone.^[2]^[1] Key properties include its parametric equations x = a e^{b \theta} \cos \theta and y = a e^{b \theta} \sin \theta, an arc length formula s(\theta) = \frac{a \sqrt{1 + b^2}}{|b|} (e^{b \theta} - 1) for b \neq 0, and its relation to the golden spiral—a special case where b \approx 0.30635, corresponding to quarter-turn expansion by the golden ratio \phi \approx 1.618.^[1]^[2] Logarithmic spirals frequently appear in natural and physical phenomena due to their modeling of exponential growth and optimal packing.^[3] In biology, they describe the coiling of nautilus and snail shells,^[3]^[4] the whorls of scalp hair,^[5] phyllotaxis in sunflowers and pinecones,^[5] and ram horns.^[4] Astronomically, spiral galaxies like the Milky Way exhibit logarithmic arm structures,^[5]^[3] while in meteorology, hurricane eyes approximate this form.^[3] Applications extend to engineering, such as antenna design^[6] and computer graphics for simulating natural patterns, and to fields like crystallography where self-organized logarithmic spirals emerge in material growth.^[3]

Mathematical Foundations

Definition in Polar Coordinates

The logarithmic spiral is defined in polar coordinates (r, \theta) by the parametric equation

r(\theta) = a \, e^{b \theta},

where a > 0 is a scale factor determining the initial radius at \theta = 0, b \neq 0 is the growth factor that controls the tightness and direction of coiling (positive b for counterclockwise expansion, negative for clockwise), and \theta is the polar angle measured from a reference direction.^[7] This form describes a curve that emanates from the origin (the pole) and spirals outward (or inward if considering negative \theta), with the radius growing exponentially as the angle increases.^[8] The exponential dependence ensures that the spiral expands proportionally with angular rotation: increments in \theta produce multiplicative changes in r, rather than additive ones as in Archimedean spirals. Taking the natural logarithm of the equation yields

\ln\left(\frac{r}{a}\right) = b \theta,

which explicitly reveals the logarithmic relationship between radius and angle, underscoring the spiral's name and its property of logarithmic growth.^[9] This growth pattern implies self-similarity: scaling the radius by e^{b \alpha} and rotating the angle by \alpha maps the curve onto itself, preserving its shape at all scales.^[8] A defining geometric feature is the equiangular property, where the angle between the radius vector and the tangent to the curve remains constant regardless of \theta. To derive this, consider the polar representation; the differential change satisfies \frac{dr}{d\theta} = b r, so the tangent direction forms an angle \phi with the radius vector such that

\tan \phi = \frac{r}{\frac{dr}{d\theta}} = \frac{1}{b},

hence \phi = \cot^{-1}(b), or equivalently b = \cot \phi.^[9] This constant \phi (the spiral angle) characterizes the spiral's "openness"—larger \phi (smaller |b|) yields looser coils, while smaller \phi (larger |b|) produces tighter spirals—and geometrically interprets the self-similarity, as radial scaling aligns with tangential progression at a fixed angle.^[8]

Representation in Cartesian Coordinates

The parametric representation of the logarithmic spiral in Cartesian coordinates is obtained by substituting the polar form r = a e^{b \theta} into the standard conversions x = r \cos \theta and y = r \sin \theta, yielding

\begin{align*} x &= a e^{b \theta} \cos \theta, \\ y &= a e^{b \theta} \sin \theta, \end{align*}

where a > 0 is a scaling parameter, b determines the rate of growth or decay (with b \neq 0), and \theta serves as the parameter, typically ranging over (-\infty, \infty) to trace the full spiral.^[1] This form facilitates algebraic manipulation and numerical computation, as \theta can be varied to generate points along the curve. For instance, with a = 1 and b = 0.2, at \theta = 0, the point is (1, 0); at \theta = \pi/2, it is approximately (0, 1.374); and at \theta = \pi, it is approximately (-1.874, 0), illustrating the spiral's outward expansion as \theta increases. An implicit relation in Cartesian coordinates can be derived from the polar definition by expressing r = \sqrt{x^2 + y^2} and \theta = \arctan(y/x), leading to \ln \sqrt{x^2 + y^2} = b \arctan(y/x), or equivalently \frac{1}{2} \ln(x^2 + y^2) = b \arctan(y/x). However, this equation faces significant challenges: the arctangent function is multi-valued and periodic with period $2\pi, requiring adjustments like adding $2\pi k for integer k to capture the entire spiral, while the principal branch limits coverage to a single arm. Using the two-argument arctangent function \atantwo(y, x) helps address quadrant issues but still results in a transcendental relation that cannot be rearranged into a simple algebraic form f(x, y) = 0 without retaining the parametric or multi-branch structure, complicating direct algebraic analysis or closed-form solutions for intersections and other geometric queries.^[10]

Historical Development

Early Discoveries

The earliest mathematical explorations of spiral curves date back to ancient Greece, where Archimedes described a spiral in which the radius increases linearly with the angle of rotation, known today as the Archimedean spiral. This curve, detailed in his work On Spirals around 225 BCE, served as a foundational precursor to later spiral studies by demonstrating rotational growth in a plane, though it differed fundamentally from the logarithmic form due to its arithmetic rather than exponential expansion.^[11] In the 16th century, Albrecht Dürer provided the first explicit description of a curve resembling the logarithmic spiral in his 1525 treatise Underweysung der Messung mit dem Zirckel und Richtscheyt, referring to it as the "eternal line" (ewige Linie) for its infinite, self-similar windings in both directions. Dürer illustrated the spiral through freehand drawings and geometric constructions, connecting it to artistic proportions and natural forms such as nautilus shells, which were emerging as curiosities in European collections during the Renaissance. Nautilus shells, prized for their chambered, coiled structures, inspired early geometric interest in spiral patterns observed in conchology, though Dürer's focus remained on practical applications in art and measurement rather than analytic properties.^[2] The 17th century saw further developments in spiral geometry, particularly through René Descartes, who in 1638 identified the equiangular spiral—a form where the angle between the tangent and radius vector remains constant—while investigating mechanical problems like falling bodies. Descartes described this curve in correspondence and early analytic geometry, distinguishing it from linear spirals and employing rudimentary polar representations to plot its path. Concurrently, others like Evangelista Torricelli independently examined similar spirals, computing their arc lengths, amid broader advancements in conchological observations where coiled shell growth prompted geometric modeling.^[12]^[13] These explorations transitioned toward exponential characterizations in the late 17th century, facilitated by John Napier's invention of logarithms in 1614 and subsequent algebraic innovations that enabled descriptions of growth via exponential functions. This algebraic framework laid the groundwork for more precise formulations, culminating in later refinements by mathematicians like Jacob Bernoulli.^[14]

Jacob Bernoulli and the Spira Mirabilis

In 1692, Jacob Bernoulli published a seminal paper in the journal Acta Eruditorum titled "Spira mirabilis," in which he derived the form of the logarithmic spiral by starting from its defining equiangular property—the characteristic that the angle between the tangent and the radius vector remains constant regardless of the point on the curve.^[15] This work highlighted the spiral's remarkable invariance under transformations such as inversion and reflection, which Bernoulli explored through geometric constructions and differential analysis.^[16] So profound was Bernoulli's admiration for this curve's perpetual self-similarity and growth pattern, where each turn enlarges proportionally without altering its shape, that he bestowed upon it the name Spira mirabilis, or "wonderful spiral," underscoring its aesthetic and mathematical elegance as a symbol of unending renewal.^[15] This nomenclature reflected his view of the spiral as a natural embodiment of continuous expansion, distinct from earlier spiral forms like the Archimedean or hyperbolic varieties that had been studied sporadically in antiquity and the Renaissance.^[17] Bernoulli's fascination extended to his personal legacy; upon his death in 1705, he requested that a logarithmic spiral be engraved on his tombstone in Basel's St. Peter's Cathedral, accompanied by the Latin epitaph Eadem mutata resurgo—"Although changed, I shall arise the same"—symbolizing the spiral's transformative yet invariant nature as a metaphor for resurrection and eternal recurrence.^[18] His insights into the spiral's properties reverberated through the mathematical community, inspiring expansions by contemporaries and successors, notably Leonhard Euler, who in the 1720s and beyond incorporated Bernoulli's foundational ideas into broader studies of polar curves and series representations in his correspondence and treatises.^[19]

Key Properties

Geometric Characteristics

The logarithmic spiral is distinguished by its equiangular property, where the angle \phi between the tangent line and the radius vector remains constant at every point along the curve. This fixed angle, typically denoted as \phi = \cot^{-1} b for the polar equation r = a e^{b \theta}, arises directly from the exponential form of the spiral's radius as a function of the polar angle \theta, ensuring geometric uniformity regardless of position.^[1] The relation d\theta/dr = \tan \phi / r from the polar representation underscores this constancy, highlighting how the spiral maintains proportional growth in angular and radial directions. A defining geometric feature is the spiral's self-similarity, meaning it appears identical when viewed at any scale after appropriate rotation and magnification. This invariance holds under similarity transformations, such that rotating the curve by an angle \alpha and scaling it by e^{b \alpha} superimposes it onto itself, preserving the overall shape indefinitely.^[20] As a result, the logarithmic spiral lacks a characteristic length scale, with finer details mirroring the global structure—a property that sets it apart from non-self-similar curves. The spiral's logarithmic scaling manifests in its radial progression, where the distance from the origin doubles after an angular advancement of \Delta \theta = (\ln 2)/b radians. This exponential expansion ensures that linear dimensions, such as the lengths of successive spiral arms between full rotations, grow proportionally to the local radius, unlike the Archimedean spiral where arm spacings remain constant.^[1] For instance, in a spiral with b \approx 0.306 (corresponding to the golden ratio growth), the radius increases by a factor of \phi \approx 1.618 every $90^\circ, illustrating the balanced geometric expansion.^[21]

Analytic and Growth Properties

The logarithmic spiral, defined in polar coordinates by r(\theta) = a e^{b \theta} where a > 0 and b \neq 0, exhibits distinctive analytic properties derived from its parametric equations x(\theta) = a e^{b \theta} \cos \theta and y(\theta) = a e^{b \theta} \sin \theta. These properties arise from the exponential relationship between the radius and the polar angle \theta, leading to rates of change that reflect self-similar expansion. The constant angle \phi between the radius vector and the tangent, satisfying b = \cot \phi, serves as a foundational parameter for these derivations.^[1] The arc length s(\theta) from the origin (as \theta \to -\infty) to a point at angle \theta is obtained by integrating the differential arc length element in polar coordinates, ds = \sqrt{r^2 + \left( \frac{dr}{d\theta} \right)^2} \, d\theta. Substituting r(\theta) = a e^{b \theta} yields \frac{dr}{d\theta} = b r, so

ds = r \sqrt{1 + b^2} \, d\theta.

Integrating gives

s(\theta) = \frac{a \sqrt{1 + b^2}}{|b|} e^{b \theta},

assuming the appropriate limits and sign convention for b > 0; this form highlights the hyperbolic growth proportional to the radius r, as s(\theta) = \frac{r \sqrt{1 + b^2}}{|b|}. This exponential scaling of arc length with angle underscores the spiral's unbounded yet proportionally increasing extent.^[1] The curvature \kappa of the logarithmic spiral, which measures the rate of change of the tangent direction with respect to arc length, simplifies to a form inversely proportional to the radius. Using the polar curvature formula

\kappa = \frac{\left| r^2 + 2 \left( \frac{dr}{d\theta} \right)^2 - r \frac{d^2 r}{d\theta^2} \right|}{\left[ r^2 + \left( \frac{dr}{d\theta} \right)^2 \right]^{3/2}},

substitution of \frac{dr}{d\theta} = b r and \frac{d^2 r}{d\theta^2} = b^2 r results in the numerator r^2 (1 + b^2) and denominator r^3 (1 + b^2)^{3/2}, yielding

\kappa = \frac{1}{r \sqrt{1 + b^2}}.

Since \sqrt{1 + b^2} = \csc \phi, this further simplifies to \kappa = \frac{\sin \phi}{r}, indicating that curvature decreases inversely with radial distance while remaining tied to the fixed angle \phi. This property distinguishes the logarithmic spiral from circles (constant curvature) or other spirals with varying rates.^[1] The speed of parametrization, or the magnitude of the velocity vector with respect to \theta, is given by \frac{ds}{d\theta} = \sqrt{r^2 + \left( \frac{dr}{d\theta} \right)^2} = r \sqrt{1 + b^2}. Relating to the geometric angle, \sqrt{1 + b^2} = \frac{1}{\sin \phi}, so

\frac{ds}{d\theta} = \frac{r}{\sin \phi}.

This expression reveals that the rate of arc length accumulation per unit angle is directly proportional to the current radius, scaled by the constant \sin \phi, facilitating uniform angular progression amid radial expansion.^[1] The growth rate of the logarithmic spiral manifests in the exponential increase of both radius and arc length with \theta, governed by \frac{dr}{d\theta} = b r, which integrates to the defining equation and implies continuous proportional enlargement. This mirrors the mechanics of continuous compounding in finance, where principal grows as P(t) = P_0 e^{k t}, analogous to the spiral's radius r(\theta) = a e^{b \theta} with angle playing the role of time; such models describe phenomena like population growth or decay where rates are proportional to current size.^[22]

Variations and Approximations

Special Cases

The logarithmic spiral admits several notable special cases defined by particular choices of the growth parameter b in its polar equation r = a e^{b \theta}. The equiangular spiral serves as a synonym for the logarithmic spiral, highlighting its characteristic property that the angle \phi between the tangent line and the radial line from the origin remains constant for all points on the curve, with \phi = \cot^{-1}(b).^[1] When b = 0, this angle \phi = 90^\circ, resulting in a degenerate case where the spiral reduces to a circle of fixed radius a; however, non-degenerate equiangular spirals typically feature b \neq 0.^[23] A prominent special case is the golden spiral, in which the radius increases by the golden ratio \phi \approx 1.618 over every quarter-turn (an angular increment of \pi/2 radians), yielding the specific parameter b = \frac{\ln \phi}{\pi/2}.^[24] This configuration closely approximates the chambered growth patterns in nautilus shells and other natural structures exhibiting self-similar expansion tied to Fibonacci sequences.^[25] Poinsot's spiral arises as a special instance within generalized logarithmic spirals, where the parameter b is selected to describe trajectories traversed at constant speed under a central inverse-cube force field, producing spiral orbits that spiral inward or outward toward the center.^[26] The limiting behaviors of the logarithmic spiral further illustrate its versatility: as b \to 0, the curve approaches a circle of radius a; conversely, as b \to \infty, the spiral tightens and approximates a straight ray extending radially from the origin.^[23]

Common Approximations

In the late 17th century, during Jacob Bernoulli's investigations of the logarithmic spiral, which he termed the spira mirabilis, plotting the curve relied on manual computations using logarithmic tables to evaluate the exponential growth factor in its polar equation. These tables, first developed by John Napier in 1614 and refined to base-10 by Henry Briggs in the 1620s, enabled efficient calculation of products and quotients needed to determine radii for selected angles θ, with antilogarithms yielding the values of e^{bθ}.^[15]^[27]^[28] For analytical approximations valid over small angular ranges, the radius function r(\theta) = a e^{b\theta} can be expanded using the Taylor series of the exponential around \theta = 0:

e^{b\theta} \approx 1 + b\theta + \frac{(b\theta)^2}{2!} + \frac{(b\theta)^3}{3!} + \cdots

This series provides a polynomial surrogate for the spiral's growth, useful for local curvature analysis or initial approximations in numerical methods, though convergence limits its application to modest \theta values.^[29] For discrete point generation, in computer graphics and visualization software, the logarithmic spiral is commonly approximated by sampling points at regular increments of \theta, computing r = a e^{b\theta} numerically (often using built-in exponential functions), converting to Cartesian coordinates, and interpolating with line segments or curves. This method balances computational efficiency and visual fidelity, with step size chosen to minimize aliasing in renders.^[30] For polygonal approximations, one practical method involves constructing polygonal chains using special triangles, such as Kepler triangles with side ratios approximating the growth factor, to simulate the spiral's path with straight line segments that converge to the curve as the number of triangles increases.^[31] For the golden spiral, a particular logarithmic spiral with growth factor \phi \approx 1.618 (the golden ratio), a popular polygonal approximation is the Fibonacci spiral, formed by inscribing quarter-circle arcs in squares whose side lengths follow the Fibonacci sequence (1, 1, 2, 3, 5, ...), where consecutive terms approximate scaling by \phi; this discrete construction closely simulates the continuous spiral for visualization purposes. The golden spiral serves as a common target for such approximations due to its prevalence in natural patterns.^[32]

Real-World Applications

Occurrences in Nature

Logarithmic spirals appear prominently in the cross-sections of nautilus shells, where the chambers grow outward at a constant angle, maintaining proportional enlargement with each revolution and approximating an equiangular spiral form. This structure enables the nautilus to expand its shell while preserving its overall shape, with growth factors typically around 2.94 to 3.38 over 360 degrees depending on the species, such as Nautilus pompilius or Allonautilus scrobiculatus.^[33] In plant phyllotaxis, the arrangement of leaves, seeds, and florets often follows logarithmic spirals to optimize space packing and sunlight exposure, as seen in sunflower heads where parastichies form intersecting spirals. These patterns emerge from the golden angle of approximately 137.5 degrees between successive organs, leading to efficient radial distribution related to Fibonacci numbers, with spirals visible in both clockwise and counterclockwise directions.^[34]^[35] Spiral galaxy arms, including those in the Milky Way, are modeled as logarithmic spirals driven by density waves that propagate through the galactic disk, concentrating stars, gas, and dust in elongated patterns. Observations of HII regions, giant molecular clouds, and methanol masers support four-arm logarithmic spiral structures connecting major arms like Norma, Scutum-Centaurus, Sagittarius-Carina, and Perseus, with pitch angles varying gradually to fit tracer distributions.^[36] Animal horns and tusks exhibit logarithmic spiral coiling for structural integrity and efficient growth, as in ram horns where the curve maintains geometric similarity across scales, and elephant tusks that widen proportionally with length following a power law akin to logarithmic expansion. This form supports load-bearing without excessive material use, with slopes between 0.3 and 0.8 in bovid horns and tusks enabling age and sex determination from growth trajectories.^[37] Recent studies have confirmed spiral patterns in hurricanes through fractal analysis of cloud boundaries and logarithmic fitting of rainband signatures, revealing self-similar structures in tropical cyclone inflows with crossing angles correlating to wind speeds up to 75 m/s.^[38]^[39] The self-similarity of logarithmic spirals under scaling transformations underpins their recurrence in these natural growth and dynamic processes.^[33]

Uses in Engineering and Technology

Logarithmic spirals are employed in antenna design, particularly in log-periodic and spiral antennas, to achieve broadband performance and frequency invariance. These antennas maintain consistent radiation patterns across a wide range of frequencies due to the self-similar geometry of the spiral, enabling applications in ultra-wideband direction finding and high-power UHF systems. For instance, a dual-arm conical logarithmic spiral antenna has been developed for operation from 500 MHz to 1000 MHz, providing wideband reception suitable for radar and communication systems. Similarly, miniaturized multi-arm logarithmic spiral antennas offer compact, low-profile designs with improved bandwidth for modern wireless devices. In mechanical engineering, logarithmic spirals inform the design of helical components such as springs and gears, promoting uniform stress distribution and efficient energy transfer. Spiral power springs, modeled as logarithmic spirals, utilize wide-plate theory to ensure consistent torque output in applications like measuring tapes and clockwork mechanisms.^[40] Logarithmic spiral gears and sprag one-way clutches leverage the curve's equiangular property to minimize wear and enable smooth meshing under varying loads, as seen in high-performance transmissions and turbines.^[41]^[42] In turbines, logarithmic spiral profiles optimize fluid flow paths, enhancing aerodynamic efficiency compared to Archimedean alternatives by maintaining proportional expansion for better energy capture.^[43] Optical and acoustic technologies benefit from logarithmic spirals for wavefront manipulation and sound propagation. Spiral phase plates, fabricated with helical thickness profiles based on logarithmic spirals, generate optical vortex beams for laser applications, enabling precise beam shaping and phase singularities in structured light systems. In acoustics, exponential horn loudspeakers incorporate logarithmic spiral flares to achieve impedance matching and uniform directivity, improving efficiency in audio reproduction by exponentially expanding the wavefront.^[44] Recent advancements in the 2020s integrate logarithmic spirals into robotics and additive manufacturing for enhanced functionality. In AI-driven path planning, logarithmic spiral trajectories guide underactuated robots and vehicles, iteratively connecting waypoints to optimize navigation in constrained environments with limited sensor fields. Soft robotic grippers shaped as logarithmic spirals enable versatile grasping across scales, mimicking scalable expansion for delicate or oversized objects via 3D printing and cable actuation.^[45] The constant angle of logarithmic spirals facilitates scalable manufacturing, allowing designs to be efficiently prototyped and produced at varying sizes without altering performance ratios, as their growth properties ensure proportional stress handling in engineered components.^[46]

Manifestations in Other Fields

In art, M.C. Escher incorporated logarithmic spirals into several prints to achieve aesthetic self-similarity and visual depth, as seen in his 1958 woodcut Sphere Surface, where loxodromic spirals—spherical analogs of planar logarithmic spirals—arrange fish motifs along curved paths on a globe.^[47] Similarly, in Path of Life III (1962), Escher employed projective geometry with logarithmic spirals to create interlocking patterns that evoke infinite progression and ambiguous perspective.^[48] These designs highlight the spiral's equiangular property, allowing seamless scaling without distortion, which enhances the perceptual illusion of continuity. Antoni Gaudí drew inspiration from natural logarithmic spirals in his architectural works, particularly in the Sagrada Família basilica in Barcelona, where helical towers and staircases approximate Fibonacci spirals—a discrete approximation of the golden logarithmic spiral with growth factor φ ≈ 1.618.^[49] Gaudí's use of these forms, derived from shell and plant geometries, promotes structural harmony and organic flow, as the spirals distribute loads evenly while mimicking self-similar expansion.^[50] Fibonacci tiles, consisting of squares with side lengths following the Fibonacci sequence (1, 1, 2, 3, 5, ...), serve as a polygonal approximation to the golden spiral by inscribing quarter-circles at each square's corners, converging to a true logarithmic spiral as the sequence progresses.^[51] This method, rooted in the limit ratio of consecutive Fibonacci numbers approaching φ, has been applied in artistic tilings to simulate continuous spiral growth with rectilinear elements.^[52] In physics, hydrodynamic quantum analogs demonstrate logarithmic spirals through walking droplets on vibrating fluid surfaces, which mimic quantum wave functions and exhibit spiral trajectories around obstacles, replicating bound states like those in the hydrogen atom.^[53] These experiments reveal self-similar spiral patterns in the droplet's path, analogous to radial decay in quantum orbitals, where the logarithmic form arises from balanced centripetal and viscous forces.^[54] Logarithmic spirals also appear in models of black hole accretion disks, where spiral density waves propagate through the disk, forming tightly wound patterns due to differential rotation and gravitational instabilities. In simulations of pre-merger binary black holes, these spirals sculpt the disk's outer edge, channeling material inward while preserving angular momentum conservation. In economics, the continuous compounding interest model, given by A = P e^{rt}, traces a logarithmic spiral in polar coordinates when radius r corresponds to the amount and angle θ to time (with θ = kt), as the exponential growth aligns with the spiral's form r = a e^{bθ}. This representation visualizes sustained proportional growth, common in long-term investment projections. Population growth models in ecology, such as the exponential phase of the logistic equation dN/dt = rN, similarly map to logarithmic spirals in polar plots, illustrating unbounded expansion before density-dependent limits, as used in analyzing invasive species dynamics.^[55] Post-2010 applications in computer graphics leverage logarithmic spirals for procedural generation, such as log-polar transforms to create scalable textures and terrains in video games, enabling efficient rendering of infinite landscapes like those in No Man's Sky (2016), where spiral patterns generate planetary flora and structures.^[56] In data visualization for big data, logarithmic spiral timelines facilitate analysis of temporal trends in large datasets; for instance, a 2019 IEEE method uses spiral maps to evolve literature review themes, placing events at angular intervals for uniform scaling across scales. Similarly, two-dimensional logarithmic spirals serve as timelines in medical big data classifiers, mapping patient records radially to detect regional patterns in electronic health records.^[57] These approaches handle volume by compressing outer loops logarithmically, improving readability for datasets exceeding millions of entries.^[58]

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We may make the scaled curve congruent with the original curve by scaling the curve with different factors in the tangent and normal directions (affine ...
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Logarithmic spirals and continue triangles - ScienceDirect.com
Logarithmic spirals and continue triangles. Author links open overlay panel ... paper he called it continue spiral). In the figure it is represented by ...
[30]
[PDF] Approximating Logarithmic Spirals by Quarter Circles
The approximation of a golden logarithmic spiral by quarter circles is well known. Starting from this, we show that any logarithmic spiral can be approximated ...
[31]
Nautilus Spirals and the Meta-Golden Ratio Chi
Dec 12, 2018 · The Nautilus shell is the popular iconic image for a logarithmic spiral. It is also frequently cited as an example of a golden ratio logarithmic spiral in ...
[32]
Phyllotaxis. II. A Description in Terms of Intersecting Logarithmic ...
Abstract. The descriptive theory of phyllotaxis developed by Richards is re-examined. It is shown that, given a set of primordia characterized by a plastoc.
[33]
Phyllotaxis, or the properties of spiral lattices. - Journal de Physique
In this paper, we devote our attention to plane lattices of points. Through a conformal transformation, one gets a lattice of points aligned along a logarithmic ...Missing: scientific | Show results with:scientific
[34]
The observed spiral structure of the Milky Way
We found that the models of three-arm and four-arm logarithmic spirals are able to connect most spiral tracers.
[35]
A universal power law for modelling the growth and form of teeth ...
Mar 30, 2021 · One such model is the logarithmic spiral, which has been purported to explain the growth of biological structures such as teeth, claws, horns, ...
[36]
Estimating the Intensity of Tropical Cyclones from Spiral Signatures ...
May 15, 2024 · Additionally, the edges of the spiral signatures were partially fitted using a logarithmic spiral to improve the reliability of the HLS ...
[37]
A validation of the fractal dimension of cloud boundaries
Feb 6, 2007 · In this paper the problem of the fractal dimension of the perimeter of self-organized vortices is investigated. Results from two numerical ...
[38]
[PDF] DNA Spiral Supercoiling and Intramolecular Topological Interlink
Oct 22, 2011 · In the present study, we show that pBR322 DNA isolated from cells can be transformed from plectonemical negative supercoiling to spiral.
[39]
Spiral Power Springs. Part 1—Theory | J. Manuf. Sci. Eng.
A key assumption in the model is that the uncaged spring has the shape of a logarithmic spiral, and wide plate theory instead of narrow beam theory is applied ...
[40]
Study on spatial curve meshing and its application for logarithmic ...
The goal of the paper is to develop the theory of curving meshing and apply it in high performance gear transmission. With this aim, this study investigates ...Missing: turbines | Show results with:turbines
[41]
Design and analysis of logarithmic spiral type sprag one-way clutch
Dec 16, 2015 · A complete mathematical model for logarithmic spiral type sprag one-way clutch design and analysis is given.
[42]
A comparative examination of the aerodynamic performance of ...
Two types of turbine spiral profiles (logarithmic and Archimedean) are investigated with changing the turbine opening angle (θ). Utilizing the turbulence model ...
[43]
Magnificent Spiral Speakers Shaped Like Nautilus | Tech Trends
Jun 25, 2021 · Audio engineers adopted logarithmic spiral shape to construct a sound system that not only looks great but also sounds better.
[44]
Logarithmic spiral-shaped robots for versatile grasping across scales
Apr 18, 2025 · This paper introduces a new class of soft robots that replicate a pattern observed in nature: the logarithmic spiral.
[45]
Constant Force Spring System With a Spiral: Accuracy Assessment
Jan 4, 2024 · Measuring tapes and springs in traditional watches belong to this group. ... logarithmic spiral, although some comparisons with these two most ...
[46]
Loxodromic Spirals in M. C. Escher's Sphere Surface
M.C. Escher's 1958 woodcut Sphere Surface is an image of black and white fish arranged along eight spiral paths on the surface of a sphere. By connecting the ...
[47]
[PDF] Logarithmic Spirals and Projective Geometry in M.C. Escher's "Path ...
Jan 1, 2012 · angles but need not preserve length, area, or curvature. Indeed ... This equation describes a logarithmic spiral with pitch e1/m. Since ...
[48]
Designing with Nature's Math - Mathematical Association of America
Jan 23, 2025 · Gaudí leveraged anthropomorphism and flowing organic surfaces while employing some of the strongest mathematical forms in nature. His famous ...
[49]
Antoni Gaudi Architecture: The Spiritual Geometry Behind Barcelona ...
The spiral staircases ascend at calculated angles that minimize climbing effort while maximizing structural strength. The perforation patterns in tower ...
[50]
[PDF] Fibonacci Numbers and Chebyshev Polynomials
Dec 2, 2015 · is often called the Golden Spiral, which can be observed in the arms of the spiral galaxies as illustrated in Fig. 1, a shape nautilus shell and ...
[51]
On the Construction of 3D Fibonacci Spirals - Academia.edu
It is to be mentioned that this 2D classical Fibonacci spiral is an approximation of the golden spiral, a logarithmic spiral with a growth ratio equal to the ...
[52]
[PDF] Hydrodynamic quantum analogs - MIT
Dec 21, 2020 · logarithmic spiral. The color bar indicates the wave amplitude in microns. (e) A droplet spirals about a pillar with radius 2.5 mm. As γ/γF.
[53]
[PDF] Chapter 2 The State of Play in Hydrodynamic Quantum Analogs - MIT
The interaction of a walking droplet and a submerged pillar: From scattering to the logarithmic spiral. Chaos 28: 096106. Hasan, M.A., L. Calderin, T. Lata ...
[54]
[PDF] mathematical models in population dynamics - CORE
2.1 Malthusian Exponential Growth Model ... continued exponential growth of human populations against Earth's limited resources,.
[55]
Speeding up the log-polar transform with inexpensive parallel ...
Aug 7, 2025 · Log-polar mapping has been proven useful in many areas like computer vision, robotics, computer graphics, and image processing because of its ...
[56]
[PDF] Developing a Regional Classifier to Track Patient Needs in Medical ...
In this study, we use two-dimensional logarithmic spirals as spiral timelines. A logarithmic spiral can be described as θ b ea. ⋅= r using polar coordinates ...Missing: post- | Show results with:post-
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[PDF] iMap: A stable layout for navigating large image collections with ...
The variation of image size in the spiral layout makes the problem more complicated, since those large images will greatly limit the possible moves we can take.