Articles

Scientific explorations of spirals and convergence phenomena in mathematics, physics, biology, astronomy, computer science and complex systems.

59 articles

Physics

22–30 min

From Filament to Point: Quantum Vortices Across 3D, 2D, and Topological Singularity

What becomes of a quantum vortex when moving from a three-dimensional world, where it is a filament or ring, to a quasi-2D system, where it appears as a simple point? This reduction reveals the essential topological structure of the phenomenon and opens onto the collective physics of BKT pairs, clusters, and 2D quantum turbulence.

Jun 27, 2026Read →

Biology

20–27 min

Cardiac Spiral Waves: How Electrical Reentry Sustains Arrhythmia

Every serious arrhythmia conceals an electrical spiral rotating indefinitely in the myocardium. This article explores the physics of reentrant waves, from the FitzHugh-Nagumo model to realistic ionic models, and the strategies for breaking these lethal spirals.

Two AIs, One Web Project: The Generalist, the Specialist, and the Human in Control

A single AI does not simultaneously possess strategic vision, project knowledge, and control of the target environment. This article explores a real proof of concept: making a generalist and a specialist cooperate under the direction of a human who retains intention, authority, and final responsibility.

Why Do Spirals Appear Everywhere?

A shell, a cyclone and a galaxy can look alike without sharing the same cause. This article proposes a method to distinguish geometric spirals, helices, vortices, converging trajectories and simple visual analogies.

Jun 22, 2026Read →

Mathematics

14–19 min

Mathematical Spirals: One Family, Several Growth Laws

Archimedean, logarithmic, Fermat and hyperbolic spirals look alike, but their radius does not evolve the same way. Their equations reveal radically different growth mechanisms.

Jun 20, 2026Read →

Biology

16–22 min

Phyllotaxis: How a Plant Organizes Space

The spirals of a sunflower are not drawn by a central plan. They emerge from the successive position of new organs around a meristem, under the effect of local growth and space constraints.

Jun 18, 2026Read →

Physics

18–24 min

Vortices and Cyclones: When a Fluid Starts to Spin

A vortex is not a spiral drawn in air or water, but a rotating flow field. The visible trajectories depend on pressure, viscosity, conservation of angular momentum and the scale of the phenomenon.

Jun 15, 2026Read →

Astronomy

20–27 min

Spiral Galaxies: A Living Form at Cosmic Scale

Spiral galaxies are disks of stars, gas, dust and dark matter. Their arms are not simple material ribbons: they form and are maintained by several dynamic mechanisms.

Jun 10, 2026Read →

Computer Science

14–19 min

Spirals in Computer Science: Represent, Traverse and Compute

In computer science, the spiral can be a traversal structure, a visualization or the result of a generative algorithm. It does not automatically follow from recursion.

Jun 5, 2026Read →

Complex Systems

30–40 min

Convergence in Complex Systems: Resemblance, Constraint or Kinship?

Two systems can converge toward the same form because they share a constraint, without sharing a history or mechanism. An analytical grid separates analogy, homology and functional convergence.

Jun 1, 2026Read →

Perception

15–20 min

Illusory Spirals: When the Brain Curves What Is Straight

Some images seem to contain spirals or continuous rotation while they are made of circles, segments or static contrasts. These illusions reveal the shortcuts of the visual system.

Jun 28, 2026Read →

Mathematics

18–24 min

The Golden Ratio: What the Evidence Actually Shows

The golden ratio has real mathematical properties and appears in some growth models. However, it is often added after the fact to works, bodies and monuments that do not precisely follow it.

Jun 25, 2026Read →

Biology

17–23 min

DNA: Why the Molecule Forms a Double Helix

DNA is a helix, not a flat spiral. Its shape results from the geometry of nucleotides, base stacking, hydrogen bonds and the aqueous environment.

Jun 23, 2026Read →

Astronomy

21–28 min

Accretion Disks: How Matter Loses Its Orbit

Matter around a black hole initially retains too much angular momentum to fall directly. It forms a disk, exchanges energy and angular momentum, then drifts inward.

Jun 20, 2026Read →

Computer Science

16–22 min

Cellular Automata: From Local Rules to Global Forms

A grid, a few states and a neighborhood rule can produce stability, propagation, chaos or rotating patterns. Spirals only appear in certain families of automata.

Jun 17, 2026Read →

Complex Systems

19–26 min

Reaction-Diffusion: How Uniformity Becomes Pattern

Two substances that react and diffuse at different speeds can destabilize a uniform state. Depending on parameters and geometry, spots, bands, fronts or spirals appear.

Jun 12, 2026Read →

Mathematics

14–19 min

Fibonacci: Why Ratios Converge Toward the Golden Ratio

The convergence of Fibonacci ratios toward φ follows from the characteristic equation of the recurrence. This exact result does not transform every biological pattern into proof of the golden ratio.

Jun 8, 2026Read →

Biology

16–22 min

Mollusk Shells: Growing Without Changing Shape

A shell grows by adding material at its opening. When proportions remain stable, its shape can be modeled by a logarithmic spiral, but each species has its own parameters.

Jun 6, 2026Read →

Physics

17–23 min

Circular Polarization: A Field Rotation in a Wave

In a circularly polarized wave, the electric field rotates as the wave propagates. The tip of the vector traces a circle at a fixed point and a helix when space is represented at a given instant.

Jun 4, 2026Read →

Astronomy

18–24 min

Pulsars and Magnetars: Rotation, Beams and Extreme Fields

Pulsars are rotating neutron stars whose radiation sweeps space. Their physics is primarily about rotation and the magnetosphere, not a single geometric spiral.

Jun 2, 2026Read →

Computer Science

15–20 min

AVL Rotations and Space-Filling Curves: Geometry of Algorithms

An AVL rotation is a local reorganization of a tree, not a spiral motion. Hilbert curves and other recursive traversals are about continuity and space-filling rather than spiraling.

May 30, 2026Read →

Complex Systems

20–27 min

Deterministic Chaos: The Lorenz Attractor Beyond the Butterfly

The Lorenz attractor has two lobes around which a trajectory winds irregularly. It is not two stable spirals, but a deterministic dynamic sensitive to initial conditions.

May 28, 2026Read →

Perception

16–22 min

Music, Circle of Fifths and Cochlea: Three Distinct Geometries

The circle of fifths organizes keys, harmonics follow frequency ratios and the cochlea is coiled in the inner ear. These three structures are linked to music, but not by a single spiral mechanism.

May 25, 2026Read →

Mathematics

16–22 min

Fractal Dimension: Measuring What Fills Space Differently

A fractal can be too irregular to be described as an ordinary line without being a surface. The Hausdorff dimension formalizes this scale complexity.

May 22, 2026Read →

Mathematics

18–24 min

Fourier Transform: Complex Rotations, Not Automatic Spirals

In the complex plane, a Fourier exponential rotates on a circle of constant radius. A true spiral only appears if the amplitude varies at the same time as the phase.

May 18, 2026Read →

Mathematics

13–18 min

Complex Numbers: Rotations, Dilations and Logarithmic Spirals

Multiplying by e^(iθ) performs a rotation. Simultaneously multiplying by a factor with modulus different from one produces a sequence of points on a logarithmic spiral.

May 14, 2026Read →

Mathematics

15–20 min

Möbius Strip and Klein Bottle: Torsion Is Not a Spiral

The Möbius strip has a single face and a single edge. Its property comes from a topological identification after a half-twist, not from a spiral trajectory.

May 10, 2026Read →

Mathematics

17–23 min

Continued Fractions: The Best Rational Approximations

A continued fraction transforms a real number into a sequence of integers. Its convergents provide exceptionally efficient rational approximations, without necessarily drawing a spiral.

May 6, 2026Read →

Mathematics

14–19 min

Parametric Curves: Describing a Spiral Through Motion

A parameterization gives the position of a point as a function of a parameter. It allows studying speed, curvature and length of spirals, including when polar coordinates are insufficient.

May 2, 2026Read →

Biology

15–20 min

Meristems: The Cellular Workshop of Plant Growth

Apical meristems are the growth zones of plants. The way these cells divide and organize produces spiral arrangements, governed by the same equations as phyllotaxis.

Apr 28, 2026Read →

Biology

17–23 min

Alpha Helix: How a Protein Chain Stabilizes Itself

The alpha helix is one of the two fundamental secondary structures of proteins. It forms because hydrogen bonds between amino acids spaced four residues apart create a helical geometry that minimizes free energy.

Apr 24, 2026Read →

Biology

14–19 min

Horns and Tusks: The Geometry of Asymmetric Growth

The ram's horns, the mammoth's tusks, the narwhal's tusk grow in a spiral. The reason is biomechanical: helical growth distributes mechanical stresses optimally.

Apr 20, 2026Read →

Biology

13–18 min

Migratory Raptors: Climb in Circles, Travel in Lines

Migratory raptors use rising columns of warm air to gain altitude without expending energy. They spiral upward in these columns, then glide in a straight line to the next thermal.

Apr 16, 2026Read →

Biology

16–22 min

Bacterial Flagella: Turning Rotation into Propulsion

The bacterial flagellum is a rigid helix driven by a rotary molecular motor. The helical geometry converts rotation into translation with remarkable efficiency.

Apr 12, 2026Read →

Physics

20–27 min

Quantum Spin: A Rotation Without a Spinning Ball

Spin is the intrinsic angular momentum of quantum particles. A spin-1/2 electron must complete two full rotations to return to its initial state — a property with no classical analogue.

Apr 8, 2026Read →

Physics

18–24 min

Quantum Vortices: When Circulation Becomes Discrete

In type-II superconductors, the magnetic field penetrates as quantum vortices — flux tubes surrounded by circular currents. Each vortex carries exactly one flux quantum.

Apr 4, 2026Read →

Physics

15–20 min

Gyroscopes and Precession: From Cones to Dissipative Spirals

A spinning gyroscope resists changes in orientation. Under gravity, its axis describes a conical precession. With dissipation, this precession becomes a converging spiral.

Mar 30, 2026Read →

Physics

16–22 min

Tornadoes: Concentrating Rotation in a Storm

A tornado is a rapidly rotating column of air. Conservation of angular momentum, pressure gradient and the Coriolis force combine to concentrate rotation in a narrow column.

Mar 26, 2026Read →

Physics

17–23 min

Molecular Chirality: When the Mirror Changes the Chemistry

Chirality is the property of an object not being superimposable on its mirror image. Natural amino acids are almost all left-handed. This asymmetry manifests in the helical structure of DNA and in the efficacy of chiral drugs.

Mar 22, 2026Read →

Astronomy

15–20 min

Planetary Nebulae: Sculpting a Dying Star's Envelope

When an intermediate-mass star exhausts its fuel, it ejects its outer layers. The star's rotation and interactions with a binary companion create spectacular spiral structures.

Mar 18, 2026Read →

Astronomy

18–24 min

Galactic Arms: Density Waves, Instabilities and Interactions

If spiral arms were rigid material structures, differential rotation would wind them up. Density wave theory explains why they persist, but other mechanisms also contribute.

Mar 14, 2026Read →

Astronomy

19–26 min

Orbits and Precession: Why a Rosette Is Not a Spiral

In Newtonian mechanics, planetary orbits are closed ellipses. Perihelion precession produces a rosette — a trajectory that rotates without continuously approaching the center.

Mar 10, 2026Read →

Astronomy

16–22 min

From Cloud to Planetary System: The Role of the Protoplanetary Disk

Conservation of angular momentum flattens a collapsing cloud into a rotating disk. Gravitational instabilities in this disk create spiral structures that lead to planet formation.

Mar 6, 2026Read →

Astronomy

38–51 min

Compact Object Merger: The Spiral That Becomes a Chirp

Two compact objects lose energy as gravitational waves. Their orbit winds inward as a dissipative spiral, frequency rises, and the signal produces a chirp — from GW150914 to GW250114.

Mar 2, 2026Read →

Computer Science

35–47 min

Neural Networks: How Optimization Actually Converges

How do SGD, momentum, AdamW and Muon find solutions in enormous spaces? Explore loss landscapes, Hessians, noise, edge of stability and generalization interactively.

Feb 26, 2026Read →

Computer Science

17–23 min

Elliptic Curves: A Geometry in Service of Cryptography

Elliptic curve cryptography secures the majority of modern encrypted communications. Elliptic curves are cubic equations whose points form an algebraic group.

Feb 22, 2026Read →

Computer Science

30–40 min

Fractal Compression: Reconstructing an Image Through Transformations

Fractal compression encodes an image as a set of contracting geometric transformations. The image is the fixed point of this system: by applying the transformations infinitely, one converges toward the original image.

Feb 18, 2026Read →

Computer Science

16–22 min

Small-World Networks: Local Proximity, Global Reach

Small-world networks combine a high local clustering coefficient with short global distances. This topology emerges from simple connection rules and characterizes social networks, the brain and the Internet.

Feb 14, 2026Read →

Computer Science

14–19 min

Genetic Algorithms: Exploration, Diversity and Convergence

Genetic algorithms mimic Darwinian evolution to solve complex optimization problems. The population evolves through selection, crossover and mutation, progressively converging toward the optimum.

Feb 10, 2026Read →

Complex Systems

24–32 min

Emergence: When the Whole Is Not Visible in the Parts

Emergence refers to the appearance of global properties that cannot be deduced from the properties of individual components. Starling murmurations, ant colonies and financial markets are examples.

Feb 6, 2026Read →

Complex Systems

22–30 min

Long Economic Cycles: The Kondratiev Hypothesis

Kondratiev identified economic cycles of about 50 years in historical data. These waves of expansion and contraction correspond to major technological revolutions, but their existence remains debated.

Feb 2, 2026Read →

Complex Systems

23–31 min

Epidemics and the SIR Model: Threshold, Peak and Extinction

The SIR model describes epidemic spread through three coupled differential equations. In phase space, the trajectory traces a curve converging toward an equilibrium point.

Jan 28, 2026Read →

Complex Systems

22–30 min

Food Webs: How a Disturbance Travels Through an Ecosystem

A food web is a graph of predator-prey relationships. The disappearance of a keystone species can trigger a trophic cascade that propagates through the entire network.

Jan 24, 2026Read →

Complex Systems

20–27 min

Feedback Loops: Stability, Runaway and Oscillation

Positive feedback amplifies perturbations; negative feedback dampens them. The dynamics of complex systems are governed by the interaction of these two types of loops.

Jan 20, 2026Read →

Complex Systems

20–27 min

Stigmergy: Coordinating Without a Central Plan

Stigmergy is an indirect coordination mechanism where agents modify their environment. Ants build nests with complex structures without central coordination, each individual following simple local rules.

Jan 16, 2026Read →

Perception

18–24 min

Curvilinear Perspective: Convergence Without Mandatory Spiral

Linear perspective relies on the convergence of parallel lines toward a vanishing point. In curvilinear perspective, these lines curve. The brain uses these cues to reconstruct three-dimensional depth.

Jan 12, 2026Read →

Perception

20–27 min

Turning Dance: Body, Balance and Cultural Meaning

Rotation is present in the ritual practices of all human cultures. Whirling dervishes, shamanic dances and trance rituals use prolonged rotation to induce altered states of consciousness.

Jan 8, 2026Read →

Perception

20–27 min

Prehistoric Spirals: A Common Pattern, Uncertain Meanings

Spirals carved in rock appear on every continent since the Neolithic. Is it a universal symbol rooted in human cognition, or an independent convergence toward a naturally salient form?

Jan 4, 2026Read →

Perception

22–30 min

Vertigo and Nystagmus: How the Vestibule Encodes Rotation

The semicircular canals of the inner ear detect angular rotations of the head. Vertigo occurs when this system is disrupted, and the nystagmus that accompanies it itself traces a spiral trajectory.

Dec 30, 2025Read →