Proposal of the “Logarithmic Spiral Coordinate System”（English version）

Proposal of the “Logarithmic Spiral Coordinate System”

I have conceived a “Logarithmic Spiral Coordinate System” (with an appearance similar to the Fibonacci spiral), akin to an “oblique coordinate system,” but with the oblique angle unfolding according to the logarithmic spiral pattern. The X-axis becomes a background without markings, used to define the spiral angle, and the independent variable represents the trajectory of the spiral, with its unit based on the differential form of the natural base e; the dependent variable represents the rotational extent of the spiral angle, with its unit based on the differential form of the circular constant π. In this way, information about rotational periodicity and the non-closure of the curve is fully incorporated. Moreover, the unit scaling on the coordinate axes is no longer the natural number 1, but e and π respectively—this will undoubtedly have significant applications in the future.

In this coordinate system, a point P is uniquely determined by a pair of coordinates (s, φ), where:

• Independent variable s: the radial scaling parameter, defined as s = ln(r / r0), where r is the radial distance in polar coordinates, and r0 is a reference scaling constant (usually taken as r0 = 1 for simplification).
The unit of s is based on the differential form of the natural logarithm base e: ds corresponds to a multiplicative radial growth of e^{ds}, which inherently supports exponential compound growth rather than linear addition.

• Dependent variable φ: the normalized angular parameter, defined as φ = θ / π, where θ is the standard polar coordinate angle (in radians).
The unit of φ is based on the differential form of π: dφ corresponds to a differential rotation of π radians, which avoids the 2π periodic closure problem in traditional polar coordinates and ensures that rotational information remains continuous and infinite (the curve never self-intersects or closes).

Trajectory conversion equations in polar coordinates:

r(s) = r0 * exp(s)
θ(φ) = π * φ

Complete parametric equations in Cartesian coordinates (from logarithmic spiral coordinates to standard x-y coordinates):

x(s, φ) = r(s) * cos(θ(φ)) = r0 * exp(s) * cos(π φ)
y(s, φ) = r(s) * sin(θ(φ)) = r0 * exp(s) * sin(π φ)

Properties of the coordinate grid:

• Constant-s lines (fixed s, varying φ): form a family of logarithmic spirals with fixed radial scale r0 * exp(s), while the angle rotates uniformly in units of π, embodying the exponential growth of the Fibonacci golden spiral.
• Constant-φ lines (fixed φ, varying s): form rays emanating from the origin at a fixed angle of π φ, ensuring non-periodicity (no 2π modulo operation) and preserving complete information about infinite rotation.

The value of this coordinate system lies in its ability to capture the compound growth trends in nature through the differential unit of e, and to avoid angular periodic closure through the differential unit of π, thereby providing the most natural description of common natural phenomena involving self-similarity, perpetual spirals, and non-closed dynamic processes (such as phyllotaxis in plants, galactic arms, and vortex formation).

In the logarithmic spiral coordinate system, a logarithmic spiral with a growth rate based on the golden ratio (and all constants set to 1) can be taken as the basic reference curve, analogous to the coordinate axes in a Cartesian system. However, this is equivalent to transforming the X-axis in the Cartesian system into a scaleless background axis used solely to define angles, while the Y-axis becomes the golden logarithmic spiral with specific values. This aligns with the polar coordinate formula for the “coordinate axis,” which can be written as r(θ) = c · e^(dθ), where r(θ) is the radial distance, θ is the angle, and the curvature κ = b / r. Setting d/c = (5² - 1)/2 + 1 and taking c as the natural number 1, the formula becomes r(θ) = e^｛［（5^2-1）/2+1］ θ｝. This is the most “natural” logarithmic spiral with the highest degree of self-similarity. Straight lines, other curves, additional logarithmic spirals, and the positions and coordinate values of points defined in the logarithmic spiral coordinate system can all be compared against this spiral reference axis, thereby revealing deviations.

In polar coordinates, the equation of the golden spiral can be simplified as r(θ) = e^{(2/π) • ln[（5^2/1+1）/2] • θ}. Thus, in the logarithmic spiral coordinate system where a point P is represented as [s = ln r, φ = θ / π] (with r being the polar radial distance and θ the standard polar angle in radians), its coordinate expression can be transformed to s = 2 ln [（5^2/1+1）/2] • φ, φ = φ. The golden logarithmic spiral no longer manifests as a curve equation, but transforms into a linear baseline (while retaining its helical geometric form). Let the point on the golden spiral be P’ (s’, φ’). Then, any point P(s, φ) in the logarithmic spiral coordinate system can be expressed relative to the golden spiral baseline, with P relative to P’ written in the form P → s = s’ + Δs, φ = φ’ + Δφ, where Δφ = Δ(θ/π), and Δs = ln(r) - ln(r’), with r being the polar radial distance of arbitrary point P and r’ the polar radial distance of point P’ on the golden spiral.

In Cartesian coordinate system, basis vectors exist. Similarly, in this logarithmic spiral coordinate system, basis vectors can also be defined as [e, (1/π)°]. If the logarithmic spiral is to be based on a different logarithm base, simply replace e with the corresponding real number serving as the base (this real number must not be 0), the scale units for the corresponding helical length are also changed to that non-zero real number as the unit, this transformation is analogous to altering the angle between the two coordinate axes in a rectangular coordinate system.

When this logarithmic spiral coordinate system is applied to the complex plane, the baseline spiral curve in the logarithmic spiral coordinate system can also be transformed into a scaleless X-axis used to define the magnitude of angles. This axis serves qualitatively as a scaleless background dedicated to representing the imaginary unit i. The primary logarithmic spiral curve, however, retains scales—similar to polar coordinates—where the independent variable (radial) and dependent variable (angular) are measured in units of e and π, respectively. Thus, the length unit is e, and the angle unit is π. The presence or absence of an imaginary part is determined by the value in π units, when the angle is a specific numerical value, there is no imaginary part(when the angle is exactly an integer multiple of π).

【First proposed on December 25, 2025. Citations are welcome with attribution to the source.】