RADIAL Method: Quantitative Order Analysis

• RADIAL Method is a quantitative technique that projects visible points onto a circle to generate a normalized one-dimensional spacing distribution.
• It translates complex two-dimensional structural information into statistical measures, revealing distinct order signatures in lattices, quasicrystals, and random sets.
• By combining visibility tests, angular normalization, and algebraic criteria, the method efficiently analyzes intrinsic order in locally finite planar point sets.

The RADIAL method, as introduced in "Radial spacing distributions from planar point sets" (Baake et al., 2014), is a quantitative technique for extracting and comparing the intrinsic order in locally finite planar point sets such as periodic lattices, aperiodic tiling vertices, or atomic positions. By compressing complex two-dimensional structural information into a sequence of angular spacings via a radial projection, the method enables the rigorous comparison of patterns with widely varying geometric, algebraic, or probabilistic origins. The procedure consists of visibility determination, angular projection, normalization, and the extraction of a spacing distribution, thus translating spatial order into a one-dimensional gap measure whose statistical properties can be analyzed with precision.

1. Radial Projection Method: Formulation and Workflow

The workflow of the radial projection method involves three main stages:

1. Visibility Determination: For a point set $S \subset \mathbb{R}^2$ and a distinguished reference point $x_0$ (often chosen for symmetry, e.g., the origin or the center of an aperiodic patch), a point $p \in S$ is deemed "visible" from $x_0$ if no other point $p_0 \in S$ lies strictly between $x_0$ and $p$. Formally,

$$p_0 = x_0 + t (p - x_0),\quad \text{for some}\ t \in (0,1).$$

The set of visible points is denoted $V$.

1. Projection onto the Circle: For a fixed radius $r > 0$, consider visible points inside the disk $B_r(x_0)$ to create $V(r) = V \cap B_r(x_0)$. Every $v \in V(r)$ has a polar representation $v = s e^{i\theta}$, $0 \leq s \leq r$. Each $v$ is "projected" onto the boundary circle $C_r$ by recording its angular coordinate $\theta$.
2. Normalizing Angles and Constructing the Spacing Distribution: After collecting and sorting the set of angles $\Theta(r) = \{\theta_1, ..., \theta_n\}$ in increasing order, the (normalized) spacings $d_i = \theta_{i+1} - \theta_i$ are computed, and the empirical spacing measure

$$\nu_r = \frac{1}{n-1} \sum_{i=1}^{n-1} \delta_{d_i}$$

is formed. By scaling the angles so that the mean spacing is unity, $\nu_r$ becomes a probability measure for statistical analysis. The method then inquires whether $\nu_r$ converges weakly to a limiting measure $\nu$ as $r \to \infty$, which would represent the "signature" of order for $S$.

2. Quantitative Order Signatures in Planar Point Sets

The core aim is to translate the geometric or combinatorial order of $S$ into a quantitative statistical signature via the limiting distribution $\nu$.

• Perfectly Ordered Case ($\mathbb{Z}^2$ lattice): Visible points from $x_0=0$ are lattice points with coprime coordinates—an arithmetic-geometric condition related to Farey sequences. The limiting spacing distribution exhibits a "gap" (a zero-density interval at small $t$), a structured bulk, and a power-law tail, all encoded in a piecewise density $g(t)$.
• Poisson (Disordered) Case: For $S$ given by a homogeneous Poisson point process, the spacing distribution is exponentially distributed:

$f(t) = e^{-t},\quad t \geq 0.$

This reflects complete randomness and absence of spatial correlation.

• Aperiodic and Quasicrystalline Tilings: The method is applied to vertex sets of tilings generated by inflation, substitution, or cut-and-project schemes (e.g., Ammann–Beenker, Tübingen triangle, Gähler Shield, Lançon–Billard). The resulting gap distributions interpolate between the lattice and Poisson extremes—a marked gap for Ammann–Beenker and Tübingen triangle (indicating robust aperiodic order), versus the absence of a gap and exponential-like decay for Lançon–Billard (a non-Pisot inflation tiling).

3. Local Visibility Criteria for Aperiodic and Model Sets

In complex aperiodic settings, direct visibility tests (ray-casting) are computationally prohibitive. The method exploits algebraic structure to derive local visibility criteria:

• Ammann–Beenker (AB) Tiling: Vertices $$T_{\mathrm{AB}} = \{x \in \mathbb{Z}[\zeta_8]: x^{\star} \in W_8\}$$ (with star-map conjugation and window $W_8$ an octagon) are visible if their star-images, after scaling by the silver mean inflation $A_\mathrm{sm}$, remain inside a certain belt in $W_8$ and if the coordinates are coprime in the underlying $\mathbb{Z}$-module.
• Similar number-theoretic (gcd-like) local conditions are constructed for Tübingen triangle and Gähler’s shield tilings, utilizing module decompositions of points and algebraic representation of vertices.

These local algebraic tests obviate global geometric visibility checks, ensuring robust and efficient extraction of the visible set $V$ for large and complicated tilings.

4. Statistical Measures: Formulas and Scaling

The central mathematical formulas are:

• Visibility Condition:

$$p_0 = x_0 + t (p - x_0)\ \text{for some } t \in (0,1).$$

• Spacing Distribution:

$$d_i = \theta_{i+1} - \theta_i,\quad \nu_r = \frac{1}{n-1} \sum_{i=1}^{n-1} \delta_{d_i}$$

with normalization such that $\frac{1}{n-1} \sum_{i=1}^{n-1} d_i = 1$.

• Limiting Behavior: Investigation of weak convergence $\nu_r \rightharpoonup \nu$ as $r \to \infty$. The limit $\nu$ encodes a fundamental order–disorder signature of the original point set, independent of local patch idiosyncrasies.

5. Numerical Experiments and Observed Phenomena

Empirical computation for various $S$ shows:

Point Set	Gap in Distribution	Decay of Tail	Order Pattern
$\mathbb{Z}^2$ lattice	Yes	Power-law	Perfect order
Poisson (random)	No	Exponential	Total disorder
Ammann-Beenker, Tübingen triangle	Yes	Structured, heavy	Quasiperiodic
Lançon–Billard (non-Pisot inflation)	No	Exponential	Weak order

These distinctions are visible in histogram plots of the normalized angular spacings: aperiodic tilings with Pisot inflation and strong algebraic order show a clear gap and robust structure analogous to the lattice, while non-Pisot and random sets are gapless and appear exponential.

6. Significance, Applications, and Methodological Considerations

The radial method bridges geometric, algebraic, and probabilistic analysis by converting a high-dimensional spatial problem into a tractable one-dimensional statistical analysis of spacings. It provides:

• A rigorous framework for quantitative comparison of order among highly disparate structures: perfect lattices, quasicrystals, and disordered systems.
• Tools for distinguishing order signatures in aperiodic and quasicrystalline systems, even when visual inspection alone is inconclusive.
• Efficient realizations for large domains, especially when algebraic visibility conditions can be used instead of geometric ray-tracing.
• Scalability, as the method delivers robust statistics in the large-$r$ limit.

Methodological choices—such as the selection of $x_0$ (to maximize symmetry or represent typical patches), normalization conventions, and windowing in cut-and-project sets—can affect empirical convergence rates but do not alter the fundamental discriminative power of the method.

7. Conclusion and Context within Order Theory

The RADIAL method, via radial projection and angular spacing statistics, provides an effective order parameter for locally finite planar point sets, resolving a longstanding challenge in comparing the degree of order in atomic, tiling, and lattice point structures. The technique is especially valuable when structural features do not map cleanly to traditional correlation functions or when intermediate forms of order (quasiperiodic, aperiodic) require classification beyond visual or local arithmetic inspection. Its construction allows direct translation of geometric order into a measurable statistical quantity, supporting both theoretical investigations of aperiodicity and practical assessments in crystallography, materials science, and mathematical physics.