Shadow–Microstructure Diagrams

Updated 5 July 2026

Shadow–microstructure diagrams are reduced representations that project complex microstructures onto observable charts while preserving phase, morphology, or topology.
They span multiple disciplines by mapping optical, stereological, thermodynamic, and combinatorial features into lower-dimensional, interpretable formats.
Applications include material gradient visualization, image mosaic alignment, black-hole phase analysis, and the combinatorial classification of topological structures.

Searching arXiv for the cited papers to ground the article in current records. Shadow–microstructure diagrams are diagrammatic constructs that relate a reduced or observable “shadow” to underlying microstructure. The term is explicit in geometrothermodynamic studies of black holes, where an SM diagram is a $2$-D or $2.5$-D representation of a macroscopic parameter, the shadow radius $R_{sh}$ , and a curvature-based microstructural indicator; in other literatures the same idea appears as an interpretive umbrella for spatially resolved chord-length maps, shadow-XMCD-PEEM contrasts, bridge and shadow diagrams in topology, shadow-controlled empirical distributions in random complexes, and parametric diagram fits of grain maps (Ladino et al., 2 Apr 2026, Whitman et al., 2024).

1. Cross-disciplinary scope and defining features

The literature does not support a single field-independent formalism. Instead, it supports a family of related constructions in which a lower-dimensional representation encodes local structure, interactions, or topology. In some settings the “shadow” is optical or geometric; in others it is combinatorial, stereological, thermodynamic, or probabilistic.

Domain	Diagram variables	Encoded microstructure
Heterogeneous materials	spatial coordinate, chord length, probability $P_i^{(k)}$ (Whitman et al., 2024)	constituent size, gradients, anisotropy
Shadow-XMCD-PEEM	image position, XMCD contrast, transmitted intensity (Jamet et al., 2015)	3D magnetization texture
Thin-film growth	processing/composition axes and generated SEM morphology (Banko et al., 2019)	structure zones under shadowing and mobility
Low-dimensional topology	arc/shadow data on a spine or trisection surface (Brand et al., 2022, Zupan, 2022)	link or surface isotopy class
Random complexes	scale $x$ , shadow density $s(x)$ , density $(1-s(x))/(d+1)$ (Fraiman et al., 2020)	MSA-weight and persistence microstructure
Black-hole thermodynamics	$Q$ or $J$ , $R_{sh}$ , GTD curvature (Ladino et al., 2 Apr 2026)	phase and interaction type
Grain-map surrogates	polynomial minimisation parameters and fitted cells (Bourne et al., 20 May 2026)	compressed grain-boundary geometry

A plausible synthesis is that a shadow–microstructure diagram is any representation that projects high-dimensional structure onto a reduced observable chart while preserving phase, morphology, topology, or interaction signatures. The reduction may be statistical, optical, combinatorial, or thermodynamic, but the recurrent purpose is the same: to make latent structure legible.

2. Materials-science formulations

In heterogeneous materials, the most direct realization is the spatially-resolved chord length distribution (SR-CLD). A chord is a contiguous run of pixels belonging to a constituent along a prescribed direction, and for each row or column $2.5$0 a local chord length distribution is formed and stored as a probability vector $2.5$1. The paper defines

$2.5$2

so that each line index $2.5$3 carries a normalized distribution over chord-length bins. The resulting $2.5$4 array is visualized as a heat map whose axes are spatial position and chord length, with color equal to probability. In this sense the SR-CLD heat map is a directional shadow diagram of local length scales. The reported case studies show that SR-CLD distinguishes a uniform polycrystal from a gradient polycrystal even when their overall CLDs are nearly identical, and that it resolves fine-to-coarse transitions and coarse bands in additively manufactured Ti alloys. The same formalism is also used for image alignment via a scaled Euclidean distance between overlapping SR-CLDs, enabling automated merging of large microstructure mosaics (Whitman et al., 2024).

In shadow-XMCD-PEEM, the shadow is literal. The transmitted photon intensity beneath the geometric shadow of a three-dimensional magnetic object depends on a path integral of magnetization-dependent absorption along the X-ray trajectory through the object. The paper formulates the attenuation through helicity-dependent coefficients $2.5$5 and $2.5$6, so that each shadow pixel records an integral of $2.5$7 through the volume. This makes the shadow a microstructure map of three-dimensional magnetization texture rather than merely a silhouette. The reported textbook cases are uniform transverse magnetization, which yields a single-sign shadow XMCD, and orthoradial curling, which yields a dipolar shadow XMCD with a central node. That distinction is then used to identify Bloch-point walls and curling end domains in cylindrical permalloy wires. The same work also emphasizes that photon energy, extraction voltage, focus plane, background level, electrostatic distortion, Fresnel diffraction, and scattering all affect how faithfully the shadow represents the internal texture (Jamet et al., 2015).

In thin-film growth, the classical structure zone diagram is already a shadow–microstructure diagram in a process–morphology sense. The paper describes SZDs as low-dimensional, abstracted, graphical representations of possible polycrystalline thin-film microstructures as functions of processing parameters, with the underlying competition between atomic shadowing and adatom mobility. It then generalizes that idea by replacing hand-drawn zones with learned image synthesis. The reported conditional variables are deposition temperature $2.5$8, average ion energy $2.5$9, degree of ionization $R_{sh}$ 0, deposition pressure $R_{sh}$ 1, Al concentration, and O concentration. A variational autoencoder organizes microstructure patches into a latent manifold, and a conditional generative adversarial network generates SEM-like patches over chosen 2D slices of the 6D condition space, producing generative structure zone diagrams. The reported dataset contains 123 thin-film samples and more than 10,000 image patches derived from 128 random 128×128 patches per SEM image, downsampled to 64×64. The authors use these gSZDs for interpolation within the sampled domain and for process-window identification under application constraints (Banko et al., 2019).

3. Topological shadow diagrams and combinatorial microstructure

In low-dimensional topology, the term “shadow diagram” has a distinct but closely related meaning: it is a combinatorial encoding of how arcs or surfaces sit relative to a lower-dimensional skeleton. For links in a 3-manifold with trivalent spine $R_{sh}$ 2, a generic diagram is a projection $R_{sh}$ 3 with over/under data, while a crossingless projection is an arc diagram. The paper proves that any link admits an arc diagram on a trivalent spine and that isotopic links are related by a complete move system consisting of Clearing, Finger, Vertex, Reidemeister, and spinal isotopy moves; for crossingless diagrams the calculus collapses to Finger, Exchange, Vertex, and spinal isotopy. It then connects these diagrams to Turaev’s shadow theory through shadow cones $R_{sh}$ 4, showing that crossingless projections of isotopic framed links on a fixed spine give shadow-equivalent cones. Here the microstructure is the local combinatorics of how arcs run through faces, edges, and vertices of the spine (Brand et al., 2022).

For surfaces in $R_{sh}$ 5, shadow diagrams arise from bridge trisections. A surface $R_{sh}$ 6 in $R_{sh}$ 7-bridge position determines three shadow sets $R_{sh}$ 8 of arcs on the central torus $R_{sh}$ 9, and a hexagonal lattice diagram is a shadow diagram in which arcs of the same color do not intersect in their interiors, intersections of differently colored arcs occur only at triple bridge points, and the union $P_i^{(k)}$ 0 tiles $P_i^{(k)}$ 1 by hexagons. The paper proves that if a positive-genus surface minimizes genus in its homology class, then it is isotopic to a complex curve $P_i^{(k)}$ 2 if and only if it admits a hexagonal lattice diagram. It also gives formulas

$P_i^{(k)}$ 3

and

$P_i^{(k)}$ 4

for genus and self-intersection in terms of the diagram. Two explicit algebraic families,

$P_i^{(k)}$ 5

and

$P_i^{(k)}$ 6

are shown to realize the efficient hexagonal families $P_i^{(k)}$ 7 and $P_i^{(k)}$ 8 with respect to the standard Stein trisection. In this setting, the diagram is not a summary of geometry but a complete combinatorial encoding of it (Zupan, 2022).

4. Statistical and reduced-order constructions

In random topology, the shadow is a probabilistic-combinatorial object controlling the bulk distribution of critical values. For a $P_i^{(k)}$ 9-dimensional simplicial complex with complete $x$ 0-skeleton, the $x$ 1-shadow is the set of missing $x$ 2-faces whose insertion creates a new $x$ 3-cycle. The paper studies the empirical distribution of rescaled minimum spanning acycle weights and of persistence-diagram death times, and proves that both converge to a deterministic probability measure $x$ 4 whose density is

$x$ 5

where $x$ 6 is the limiting shadow density of the Linial–Meshulam complex $x$ 7. Thus the “microstructure” of MST/MSA weights and persistence deaths is read directly from the shadow process. The same paper explicitly interprets the pair $x$ 8 and $x$ 9 as a shadow-based microstructure curve, which is closely aligned with the general notion of a shadow–microstructure diagram (Fraiman et al., 2020).

A different reduced-order construction appears in polynomial diagrams for grain-map modelling. A minimisation diagram is defined by cells

$s(x)$ 0

and the linear parametrised form $s(x)$ 1 yields

$s(x)$ 2

Power diagrams and anisotropic power diagrams appear as degree-1 and degree-2 cases, while higher-degree polynomial diagrams allow algebraic boundaries of prescribed degree. The fitting objective is a multinomial-logistic-regression likelihood over a segmented grain map, optimized in a GPU-accelerated framework using Legendre polynomials. Reported EBSD applications include a small dataset with $s(x)$ 3 grains and $s(x)$ 4 pixels and a large dataset with $s(x)$ 5 grains and $s(x)$ 6 pixels. Reported compression ratios for the large dataset are $s(x)$ 7 at $s(x)$ 8, $s(x)$ 9 at $(1-s(x))/(d+1)$ 0, $(1-s(x))/(d+1)$ 1 at $(1-s(x))/(d+1)$ 2, and $(1-s(x))/(d+1)$ 3 at $(1-s(x))/(d+1)$ 4. This suggests a shadow–microstructure interpretation in which a fitted polynomial partition is a compact parametric shadow of a much larger pixel-level microstructure (Bourne et al., 20 May 2026).

5. Black-hole shadows, thermodynamics, and microstructure

The most explicit use of the term occurs in black-hole thermodynamics. In that setting, a Shadow–Microstructure diagram is a $(1-s(x))/(d+1)$ 5-D or $(1-s(x))/(d+1)$ 6-D plot whose axes and color coding are chosen so that one axis is a macroscopic parameter, such as $(1-s(x))/(d+1)$ 7 for Reissner–Nordström or $(1-s(x))/(d+1)$ 8 for Kerr, the second axis is the shadow radius $(1-s(x))/(d+1)$ 9, and the color or region label encodes the sign and magnitude of a geometrothermodynamic curvature scalar $Q$ 0 or $Q$ 1, together with thermodynamic phase information. The same work introduces microscopic thermodynamic phase labels AS, RS, NS, AL, and RL, corresponding to attractive, repulsive, or non-interacting small or large black-hole phases. It further argues that $Q$ 2 encodes the same phase information as entropy and applies the construction to Sagittarius A* by overlaying observationally allowed $Q$ 3 bands from EHT “Images” and “mG-Rings” analyses onto the SM diagrams (Ladino et al., 2 Apr 2026).

Related AdS analyses use shadow observables as phase probes even when the term SM diagram is not yet foregrounded. For the RN–AdS black hole, the paper “Ruppeiner Geometry of the RN-AdS Black Hole Using Shadow Formalism” introduces the normalized curvature scalar $Q$ 4, shows that the shadow radius $Q$ 5 is monotonic with the horizon radius in the physical domain, and uses color-coded shadow silhouettes to represent phase structure and microstructure. In that construction, Van der Waals-like SBH/LBH transitions, coexistence radii, and the critical point are all localized in shadow space, while along the Hawking–Page line the normalized curvature takes the universal constant value $Q$ 6 (Wang et al., 2021).

For Kerr–AdS, geometrothermodynamics is combined with finite-distance shadow optics. The reported thermodynamic critical point in the canonical ensemble occurs at

$Q$ 7

and the corresponding critical photon region gives

$Q$ 8

The paper then constructs $Q$ 9, $J$ 0, and $J$ 1, showing that the singularities of GTD curvature align with divergences in thermodynamic response functions directly in shadow space. It also uses asymmetric shadow profiles, rather than only a scalar shadow radius, to encode rotating phase structure (Ladino et al., 5 May 2025).

A distinct gravitational meaning of shadow–microstructure appears in fuzzball geometries. There, the microstructure is not thermodynamic but geometric and string-theoretic. The paper reports that four-dimensional horizonless fuzzballs trap light rays on long-lived, highly redshifted chaotic orbits near the would-be horizon, producing a black-hole-like shadow in sufficiently deep scaling solutions. Time-delay, redshift, curvature, and four-color lensing maps then function as microstructure-sensitive shadow diagnostics. Away from the scaling limit, residual glow and shadow-size deviations can distinguish fuzzballs from black holes and from other exotic compact objects (Bacchini et al., 2021).

6. Common logic, applications, and limitations

Across these disparate settings, the recurring structure is a mapping from hidden organization to a reduced diagram whose coordinates are chosen for observability or algorithmic tractability. In SR-CLD the coordinates are position and chord length; in shadow-XMCD-PEEM they are image-plane position and helicity-dependent transmitted intensity; in random complexes they are filtration scale and shadow density; in black-hole GTD they are $J$ 2 or $J$ 3, $J$ 4, and a curvature scalar; in trisection topology they are arc systems on a low-dimensional carrier surface (Whitman et al., 2024, Fraiman et al., 2020, Ladino et al., 2 Apr 2026, Zupan, 2022).

This common logic also explains the range of applications. Reported uses include visualization of spatial gradients and anisotropy, automated alignment of large microstructure image sets, quantitative discrimination of three-dimensional magnetic states, classification of links and surfaces up to isotopy, construction of efficient bridge and Stein trisections, prediction of unseen thin-film morphologies within an interpolative regime, compact modelling of EBSD grain maps, and observational inference of black-hole phase structure or microscopic interaction type (Jamet et al., 2015, Brand et al., 2022, Banko et al., 2019, Bourne et al., 20 May 2026).

A common misconception is that “shadow” always denotes a literal optical shadow. The literature does not support that restriction. In some papers it is optical and radiative, in others it is a combinatorial shadow on a spine or trisection surface, a stereological projection of constituent lengths, or a shadow set of cycle-creating simplices. Another misconception is that these diagrams are necessarily invertible. The XMCD work explicitly states that the mapping from 3D magnetization to 2D shadow pattern is not a bijection; the thin-film work warns that the models interpolate well within the data manifold but should not be trusted for extrapolation; the SR-CLD work notes sensitivity to segmentation quality, directional sampling, and the distinction between 2D sections and 3D microstructure; the black-hole GTD constructions depend on thermodynamic representation, observer setup, and metric choice; and the polynomial-diagram framework depends on basis choice, degree, and gauge structure (Jamet et al., 2015, Banko et al., 2019, Whitman et al., 2024, Ladino et al., 5 May 2025, Bourne et al., 20 May 2026).

A plausible implication is that shadow–microstructure diagrams are best understood not as a single specialized formalism but as a cross-disciplinary strategy for encoding latent structure in reduced coordinates. What changes from field to field is the nature of the shadow—optical, combinatorial, statistical, or thermodynamic. What persists is the attempt to render microstructure legible without carrying the full complexity of the underlying object.