Power Spiral Map: Geometry & Analysis

• Power Spiral Map is a concept combining recursive Euclidean constructions driven by a seed angle with analytic spiral-stretch maps that exhibit power-law scaling and logarithmic twists.
• It provides explicit formulations such as the Angular Seed Power Map with area-preserving partitions and a variational minimizer for finite distortion between annuli.
• Its descriptive usage extends to astronomy and IFU studies, where spiral structure is quantitatively mapped via tracer densities, contrast metrics, and scaling laws.

In the cited arXiv literature, the expression Power Spiral Map is not used for a single universally fixed object. In one explicit sense, it denotes the Angular Seed Power Map, a geometric construction in which a seed angle $\theta$ controls both an internal area partition of a unit square and an external recursive scaling of squares that unfold as a spiral-like lattice (Dijksman, 24 Jun 2026). In another, closely related sense, the extremal spiral-stretch map between annuli is described as being “in spirit a ‘power spiral map’,” because it combines a power-law radial scaling with a logarithmic spiral twist, namely

$$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$

and serves as the unique minimizer of a mean distortion functional in a prescribed boundary class (Balogh et al., 6 Nov 2025). A broader descriptive usage also appears in several astronomy-oriented mappings of spiral structure, where the phrase denotes maps that make the geometry and relative strength of spiral arms explicit (Vallee, 2017).

1. Distinct meanings and terminological scope

The recent literature supports two explicit mathematical usages and a broader descriptive one. The first is synthetic-geometric and recursive; the second is analytic and variational; the third is interpretive and cartographic.

Usage	Governing parameter(s)	Canonical form
Angular Seed Power Map	$\theta$	recursive scaling by $\sec\theta$ and $\cos\theta$
Spiral-stretch or power spiral map	$q,k,\theta$	$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}$
Descriptive spiral-structure map	tracer-dependent	overdensity, intensity, or age-resolved spiral maps

The Angular Seed Power Map is introduced as a “continuous angular evolution of the linear coordinate grid” in which a single angle produces an infinite family of expanding and contracting squares (Dijksman, 24 Jun 2026). The spiral-stretch map arises in the planar theory of finite distortion mappings between annuli and has the characteristic power-spiral decomposition into radial scaling and logarithmic twisting (Balogh et al., 6 Nov 2025).

This non-uniform usage matters technically. In the geometric construction, the “map” is a recursive Euclidean configuration. In the annulus problem, it is an explicit homeomorphism between domains in $\mathbb C$. A plausible implication is that the shared phrase refers less to a single formal definition than to a common structural motif: a parameter-driven coupling of power-law scaling and spiral organization.

2. Angular Seed Power Map as a Euclidean construction

The geometric version begins with a unit diameter circle $C_u$ whose endpoints are $O(0,0)$ and $$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$0, together with the unit square $$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$1 built on the segment $$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$2 (Dijksman, 24 Jun 2026). For a seed angle $$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$3, the ray from $$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$4 at angle $$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$5 meets the circle at a point $$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$6. Because the circle is the Thales circle over $$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$7, the triangle $$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$8 is right-angled at $$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$9: $\theta$0

The construction assigns the basic trigonometric lengths

$\theta$1

If $\theta$2 is the vertical projection of $\theta$3 onto the base segment $\theta$4, then the partition of the base is

$\theta$5

Since the reference square has height $\theta$6, these segment lengths are simultaneously rectangle areas, so the square is decomposed into two parts with areas $\theta$7 and $\theta$8. The same values reappear as the areas of the squares erected on the legs $\theta$9 and $\sec\theta$0. This is the paper’s internal area-preserving partition of unity: $\sec\theta$1

The external part of the construction comes from extending the seed ray until it meets the vertical line $\sec\theta$2 at

$\sec\theta$3

Using $\sec\theta$4 as a side produces the expanding square $\sec\theta$5, with

$\sec\theta$6

At the same time, the line through $\sec\theta$7 and $\sec\theta$8 cuts this larger square so that one of the resulting rectangles has invariant area

$\sec\theta$9

This links the internal unit-area partition to the external scaling regime.

The construction also generates higher powers inside the contracting square $\cos\theta$0. A line through the projection point $\cos\theta$1 parallel to $\cos\theta$2 yields a sub-rectangle with area $\cos\theta$3 and a complementary region of area $\cos\theta$4. Thus the geometry realizes not only $\cos\theta$5 and $\cos\theta$6, but also higher polynomial combinations, directly as Euclidean areas (Dijksman, 24 Jun 2026).

3. Recursive scaling, self-similarity, and algebraic identities

The recursive mechanism is encoded by

$\cos\theta$7

The outer generations of squares have side lengths and areas

$\cos\theta$8

while the inner generations satisfy

$\cos\theta$9

The unit reference square is $q,k,\theta$0, with $q,k,\theta$1 and $q,k,\theta$2.

The recursion is linear in the side lengths: $q,k,\theta$3 The paper also gives the Global Scaling and Partition Laws

$q,k,\theta$4

$q,k,\theta$5

for all integers $q,k,\theta$6 (Dijksman, 24 Jun 2026). The left linear partition is therefore always a factor of $q,k,\theta$7 of the total base, and the same partition law persists at every scale.

The paper does not classify the resulting spiral as strictly logarithmic or Archimedean. Instead, it states that the structure behaves like a logarithmic scaling lattice: side lengths grow geometrically as $q,k,\theta$8, and successive orientations are tied to the seed angle and orthogonal square directions. The object is self-similar in the sense that each generation is a scaled copy of the reference square, with the same partition ratio at every level (Dijksman, 24 Jun 2026).

A central feature is the appearance of algebraic identities from discrete alignments. The key condition is

$q,k,\theta$9

equivalently

$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}$0

After squaring and substituting $g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}$1, the paper derives the polynomial family

$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}$2

For $g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}$3, this becomes

$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}$4

whose positive root is the golden ratio. For $g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}$5, the polynomial

$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}$6

factors as

$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}$7

and the positive real root of $g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}$8 is the plastic ratio $g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}$9 (Dijksman, 24 Jun 2026). The paper characterizes these as arising through purely planar intersections, meaning that the constants emerge from circles, rays, squares, and parallel lines before any analytic reformulation.

4. Power spiral maps as spiral-stretch maps between annuli

In the analytic setting, the relevant domains are the concentric annuli

$\mathbb C$0

with parameters $\mathbb C$1 and $\mathbb C$2 (Balogh et al., 6 Nov 2025). The extremal spiral-stretch map is

$\mathbb C$3

where $\mathbb C$4 is prescribed. Writing $\mathbb C$5 gives the polar form

$\mathbb C$6

This has the characteristic power spiral structure. The radial part is the power law

$\mathbb C$7

while the angular part is

$\mathbb C$8

Geometrically, radial segments are sent to logarithmic spirals: radius obeys a power law, angle varies linearly in $\mathbb C$9 (Balogh et al., 6 Nov 2025).

The boundary data are explicit. On the outer boundary,

$C_u$0

so the outer circle is fixed pointwise. On the inner boundary,

$C_u$1

so the inner circle is sent to the inner target circle by a radial stretch and a rotation by angle $C_u$2. The parameters $C_u$3 and $C_u$4 are thus determined by the source and target radii together with the prescribed boundary rotation (Balogh et al., 6 Nov 2025).

The map is studied in the class of finite distortion homeomorphisms. For an orientation preserving homeomorphism $C_u$5, finite distortion means, in particular, that $C_u$6 and that there exists a measurable $C_u$7 with $C_u$8 a.e. such that

$C_u$9

where

$O(0,0)$0

The linear distortion function is

$O(0,0)$1

The variational functional is the weighted mean distortion

$O(0,0)$2

where $O(0,0)$3 is increasing, strictly convex, and satisfies $O(0,0)$4 (Balogh et al., 6 Nov 2025). In this sense, the power spiral map is not only explicit but variationally distinguished.

5. Extremality and quantitative stability

The spiral-stretch map is the unique minimizer of the mean distortion functional among orientation preserving finite distortion maps in $O(0,0)$5 with the same boundary values. The recalled Feng–Hu–Shen theorem states that

$O(0,0)$6

with equality if and only if $O(0,0)$7 (Balogh et al., 6 Nov 2025).

The stronger statement is quantitative stability. The paper defines the spiral-stretch deficit

$O(0,0)$8

If $O(0,0)$9, then $$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$00; the stability theorem shows that almost equality forces quantitative closeness. Under the assumptions that $$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$01 is increasing, strictly convex, $$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$02, and $$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$03 a.e. on $$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$04, there exist $$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$05 and $$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$06 such that

$$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$07

Moreover, the exponent $$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$08 is sharp (Balogh et al., 6 Nov 2025).

The proof proceeds by reducing the annulus problem to a rectangle via logarithmic and exponential coordinates. In the rectangular model, the extremal is the linear stretch

$$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$09

whose Beltrami coefficient and distortion are constant. A second-order Taylor inequality for convex $$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$10, structural inequalities comparing $$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$11, $$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$12, and $$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$13, and chain-rule estimates for the conjugated map $$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$14 yield near-conformality in the form

$$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$15

The passage from near-conformality to $$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$16-closeness uses the Cauchy–Pompeiu formula, and the annulus result is recovered through the conformal change of variables

$$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$17

together with a logarithmic map on the target side (Balogh et al., 6 Nov 2025).

Examples in the rectangle and annulus settings show that the square-root rate is optimal. A plausible implication is that the power-spiral structure is not only extremal but rigid in a quantitatively sharp sense: small defect in weighted mean distortion controls the full map in $$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$18.

6. Descriptive extensions in spiral-structure mapping

A broader descriptive usage of Power Spiral Map appears in several astronomy-oriented studies. This suggests a looser meaning in which the phrase designates a map that couples spiral geometry to a measure of relative strength, tracer concentration, or temporal contrast, rather than a single formally defined transformation.

For the Milky Way, the phrase is used for an overview in which the spiral pattern is represented as an approximately symmetric four-armed spiral with logarithmic arm geometry, arm tangents, pitch angles, onion-like tracer offsets, bar connections, and density-wave shock signatures (Vallee, 2017). In that context, the map is “drawn in cold gas, hot dust, young stars, and magnetic fields,” while the “power behind it is gravity,” organized into a long-lived spiral mode.

For M51, the expression is used for a spatially resolved CO $$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$19 map in which the “power” of the spiral structure is identified with dense, excited molecular gas, arm–inter-arm contrast, and radius-dependent line-ratio behavior (Vlahakis et al., 2013). The paper defines the arm–inter-arm contrast

$$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$20

and shows that for CO$$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$21 this contrast decreases strongly with radius, while CO line ratios and PAH correlations provide an excitation-weighted view of spiral structure.

In time-sliced IFU studies of barred spirals, the phrase is used for maps that encode the strength and geometry of bar and spiral structure as a function of stellar age (Peterken et al., 2019). There the key quantity is the spiral-arm contrast

$$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$22

with an age-dependent summary $$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$23 obtained by taking the median over radius. The resulting “time slices” show that old stars can be nearly axisymmetric, intermediate-age stars trace an underlying density wave, and the youngest populations and H$$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$24 display the highest arm contrast.

For Gaia EDR3 maps of the Galactic disk, an analogous descriptive use appears in the construction of overdensity and wavelet maps of young stellar tracers (Poggio et al., 2021). The basic overdensity field is

$$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$25

so the map highlights spiral structure as relative enhancement over a smoothed background. In that setting, the Local Arm, Sagittarius–Carina, Scutum, and a large-pitch Perseus geometry emerge as coherent overdensity arches.

Taken together, these usages show that Power Spiral Map functions in current arXiv literature as both a precise mathematical designation and a transferable structural metaphor. In its strictest forms, it denotes either a recursive Euclidean construction driven by $$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$26 or an extremal annulus map of the form $$g^\ast(re^{i\phi})=r^k e^{i(\phi+\beta\log r)}, \qquad \beta=\frac{\theta}{\log q},$$27 (Dijksman, 24 Jun 2026, Balogh et al., 6 Nov 2025). In broader descriptive settings, it denotes a map whose spiral organization is made quantitatively explicit by an intensity, distortion, overdensity, or age-resolved strength field (Vallee, 2017, Vlahakis et al., 2013, Peterken et al., 2019, Poggio et al., 2021).