Articles

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

10 articles · Mathematics

Mathematics

Featured

5–7 min

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 →

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 →

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 →

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 →