Topographs: Spatial & Graph Representations

• Topographs are structured representations of spatial, geometric, or relational forms whose meanings vary by discipline and are used to support analysis, prediction, and inference.
• They integrate methodologies like coordinated 3D terrain visualizations, planar trees in arithmetic, and graph kernels in machine learning to extract actionable insights.
• In both scientific and mathematical contexts, topographs serve as computational frameworks for tasks such as weather downscaling, coastal monitoring, graph-based segmentation, and reduction theories.

Topographs are structured representations of spatial, geometric, or relational form whose meaning depends strongly on disciplinary context. In contemporary arXiv literature, the term ranges from coordinated 3D terrain visualizations for meteorological downscaling, beach elevation profiles, and image-space maps of surface micro-structure, to Conway’s planar trees for binary quadratic forms, graph frameworks for topology-preserving segmentation, and physics-informed graphs for decay reconstruction. The common thread is not a single ontology of “topography,” but a recurrent methodological move: local geometric or adjacency relations are made explicit in a representation that supports analysis, reduction, inference, or prediction (Höhlein et al., 2024, Zeppelzauer et al., 2015, O'Sullivan, 2024, Lux et al., 2024, Ehrke et al., 2023).

1. Terminological scope

The term “topograph” is therefore polysemous rather than field-invariant. In some domains it refers to a depiction of a physical surface; in others it denotes a graph encoding of algebraic or topological structure; in scanning probe microscopy it denotes an image produced by the instrument; and in recent machine learning work it can be the name of a specific architecture.

Domain	Meaning of “topograph”	Structural form
Meteorological downscaling	Coordinated visualization of terrain, atmosphere, and elevation variability	Terrain map, atmosphere layer, elevation summary plots
Coastal monitoring	Surveyed beach cross-section	Elevation-versus-chainage profile
Surface analysis	2D representation of 3D micro-structure	Topography map or Enhanced Topography Map
Arithmetic	Conway topograph	Planar trivalent tree with labeled regions and edges
Image segmentation	Topograph framework	Combined component graph of prediction and ground truth
Particle reconstruction	Topograph framework	Sparse, topology-aware graph with intermediate-particle nodes

In meteorology, the representation is explicitly a visual analytics workflow: a terrain map, an atmosphere layer with volume rendering and vertical slices, and elevation summary plots over a user-defined neighborhood radius (Höhlein et al., 2024). In coastal monitoring, a topographic beach profile is a 2D profile of elevation versus chainage, typically surveyed from the back of the beach toward tidal datums such as MLWN and MLWS (Munian et al., 11 Jan 2025). In surface metrology, a topography map $\bar{T}$ is a 2D compensated depth image obtained from a high-resolution 3D reconstruction after removing global curvature (Zeppelzauer et al., 2015). In arithmetic, Conway’s topograph is a planar trivalent tree whose regions are labeled by values of an integral binary quadratic form or by rationals linked by Farey adjacency (O'Sullivan, 2024). In STM, “topographs” are constant-height images whose apparent contrast may continue to reflect molecular structure even when the tunneling energy lies inside the transport gap (Grewal et al., 2023). In segmentation and event reconstruction, “Topograph” designates graph-based frameworks in which topology is built directly into the computational object (Lux et al., 2024, Ehrke et al., 2023).

A recurrent misconception is that topographs are necessarily maps of terrain height. The literature does not support that restriction. The mathematical and graph-theoretic usages are fully established, and even within experimental physics the word may denote an image of electronic structure rather than a literal height field (O'Sullivan, 2024, Grewal et al., 2023).

2. Earth-system and geomorphometric usages

In geoscience, topographs are closely tied to the representation of topographic surfaces, elevation profiles, and terrain-dependent physical processes. A prominent recent example is near-surface temperature downscaling in complex terrain. Numerical weather prediction models often resolve terrain too coarsely, so operational postprocessing applies a vertical temperature correction using a lapse rate, often the ICAO standard atmosphere value of $-6.5\ \mathrm{K/km}$. The cited work shows that true lapse rates can vary from a temperature drop of $1.2\ \mathrm{K}$ per $100\ \mathrm{m}$ of ascent to more than $10\ \mathrm{K}$ of temperature rise over the same distance, with sunny conditions producing around $10$–$12\ \mathrm{K/km}$ of cooling with height and stable nocturnal inversions reaching around $100\ \mathrm{K/km}$ across shallow layers. The corresponding topographs expose when a fixed lapse rate is wrong in both magnitude and sign (Höhlein et al., 2024).

The meteorological workflow combines three coordinated views. The terrain map can overlay coarse model terrain and finer $1\ \mathrm{km}$ terrain, display stations as spheres, and encode height differences directly by color. The atmosphere layer adds 3D model levels, volume rendering, and movable vertical slices through temperature and temperature-gradient fields. The elevation summary plots compute local terrain statistics, including quantiles and an interquartile-range-based whisker construction,

$$z_\text{upper} = z(75\%) + f \cdot \text{IQR}, \qquad z_\text{lower} = z(25\%) - f \cdot \text{IQR}.$$

These views revealed that summer-afternoon isotherms are often approximately parallel to terrain contours, whereas winter-morning cases show strongly bent isotherms, local gradients exceeding $-6.5\ \mathrm{K/km}$0 near the surface, and stacked valley layers with alternating gradient signs. The resulting local lapse-rate model estimates

$-6.5\ \mathrm{K/km}$1

with $-6.5\ \mathrm{K/km}$2 obtained from a weighted local regression over neighborhoods usually $-6.5\ \mathrm{K/km}$3–$-6.5\ \mathrm{K/km}$4 in radius, Gaussian distance weighting, and $-6.5\ \mathrm{K/km}$5-dependent clamping. Validation against about $-6.5\ \mathrm{K/km}$6 million valid observations at about $-6.5\ \mathrm{K/km}$7 sites yielded RMSE improvements of about $-6.5\ \mathrm{K/km}$8–$-6.5\ \mathrm{K/km}$9 for both valley and mountain stations (Höhlein et al., 2024).

A second geoscientific usage concerns systematic geomorphometric mapping. The proposed geomorphometric atlas of ice-free Antarctic areas treats topography as a quantitative surface described by a coordinated map system. For each of 194 areas, the atlas is planned to include a hypsometric map, eleven morphometric variables, and reference texts and tables. The variables are slope, aspect, horizontal curvature, vertical curvature, minimal curvature, maximal curvature, catchment area, topographic wetness index, stream power index, total insolation, and wind exposition index (Florinsky, 4 Aug 2025).

The atlas is organized explicitly around the theory of the topographic surface and general geomorphometry. REMA versions 1.1 and 2, with grid spacings of $1.2\ \mathrm{K}$0, $1.2\ \mathrm{K}$1, and $1.2\ \mathrm{K}$2 for local analysis and $1.2\ \mathrm{K}$3 and $1.2\ \mathrm{K}$4 products for broader overviews, provide the elevation basis. Local variables are computed using the finite-difference method of Evans; catchment area uses the maximum-gradient-based multiple flow direction algorithm of Qin et al.; total insolation and wind exposition use methods by Böhner (Florinsky, 4 Aug 2025). This usage of topographic representation is not merely descriptive. It links geometry to drainage, erosion, wind exposure, radiative forcing, soil moisture, and habitat structure.

Coastal monitoring adds a profile-based variant of the same logic. A topographic beach profile is the surveyed cross-section

$1.2\ \mathrm{K}$5

and the operational problem is often to extrapolate the part between MLWN and MLWS when field surveys do not reach the lower datum. This makes the topograph a one-dimensional but process-relevant representation of morphology, used for erosion monitoring, flood-risk assessment, habitat mapping, sediment transport analysis, and coastal planning (Munian et al., 11 Jan 2025).

3. Surface micro-structure, imaging, and molecular topographs

At smaller spatial scales, topographs represent local surface deviation or structural organization rather than regional terrain. In 3D surface analysis, surface topography is defined as the geometric micro-structure of a surface, including roughness, waviness, peaks, valleys, crannies, and lay. The efficient image-space extraction method begins with a high-resolution point cloud, estimates a support plane, uses orthographic projection to create a depth map $1.2\ \mathrm{K}$6, smooths it to obtain $1.2\ \mathrm{K}$7, and defines the topography map

$1.2\ \mathrm{K}$8

The representation then separates valleys and peaks, smooths them again, and applies logarithmic enhancement to produce the Enhanced Topography Map. The overall complexity is effectively linear in both point count and image size, and on archaeological rock-surface data the best ETM feature combination, GHS + SF, achieved $1.2\ \mathrm{K}$9, outperforming PFH, depth-map, and visual-texture baselines (Zeppelzauer et al., 2015).

A distinct imaging usage appears in STM. For metal phthalocyanines on NaCl/Au(111), STM topographs recorded inside the transport gap remain clearly molecular. For PtPc on trilayer NaCl, the gap images show a four-lobe pattern on the isoindole units; for MgPc the same pattern rotates with the molecular orientation. Theory and experiment agree that the dominant determinant is the molecule’s absolute rotational orientation relative to the NaCl lattice. The wave functions inside the gap are well approximated by linear combinations of bound molecular orbitals, expanded over all 182 molecular eigenstates, and the weight is not dominated by frontier orbitals alone: the HOMO contributes only about $100\ \mathrm{m}$0, the LUMO about $100\ \mathrm{m}$1, while lower-lying orbitals provide much of the remaining weight (Grewal et al., 2023). The topograph here is not a simple height image. It is a structured probe of molecular symmetry and substrate-mediated tunneling states.

In OCR, “topographic features” denote structural descriptors of character strokes as seen from different viewing directions on a 2D plane. The method for Bengali and Hindi character images extracts closed regions, convexities of strokes, and straight-line strokes from a thinned skeleton, viewing the character from North, South, East, and West. Closed regions are detected via connected-component analysis; straight lines require at least $100\ \mathrm{m}$2 horizontally 8-connected neighbor pixels; convexities are matched against a database after directional scanning. The extracted features are encoded in an undirected shape-based graph $100\ \mathrm{m}$3 whose vertices are placed at component centroids (Bag et al., 2011). Here, “topography” refers to structural stroke organization rather than elevation.

These examples indicate that topographs often privilege geometry that is stable under nuisance variation. In surface analysis that nuisance is global curvature; in STM it is the misleading expectation that in-gap images are featureless; in OCR it is font or handwriting variability. The representational objective is therefore comparative structure rather than raw appearance (Zeppelzauer et al., 2015, Grewal et al., 2023, Bag et al., 2011).

4. Learned representations and predictive models for topographic data

Recent machine learning work treats topographs not only as objects to visualize but also as signals from which generic embeddings or predictive extrapolations can be learned. Topo2vec is a self-supervised method for topographic images based on the “fractal-effect,” the assumption that structures such as rivers, peaks, and saddles appear across scales. The encoder $100\ \mathrm{m}$4 maps a topographic image $100\ \mathrm{m}$5 into a latent representation, and the decoder $100\ \mathrm{m}$6 reconstructs a version at another scale $100\ \mathrm{m}$7, with $100\ \mathrm{m}$8. Training minimizes an $100\ \mathrm{m}$9 reconstruction distance, with an adversarial variant Topo2vec-adv adding a conditional GAN term. The architecture is a U-Net-like encoder-decoder without skip connections, precisely so that the latent representation must hold the information required for decoding (Kavitzky et al., 2021).

The model is pretrained on 100,000 images derived from ALOS DTM data over Europe and evaluated by training an SVM on embeddings. On OSM-labeled topographic classes, the reported accuracies include $10\ \mathrm{K}$0 for peaks with Topo2vec-1, $10\ \mathrm{K}$1 for rivers with Topo2vec-adv, $10\ \mathrm{K}$2 for saddles with Topo2vec-4, and $10\ \mathrm{K}$3 for cliffs with Topo2vec-adv. On topography-correlated classes, Topo2vec again outperforms baselines, although waterfalls are identified as a class that does not exhibit the same fractal properties (Kavitzky et al., 2021). This suggests that the learned representation is explicitly scale-aware rather than scale-invariant in the usual computer-vision sense.

TopoFormer addresses a more applied forecasting problem in coastal morphology. Its target is the lower part of a beach profile, especially the segment from MLWN to MLWS that many surveys do not observe. Each profile is resampled to 100 elevation-chainage pairs and represented as an input array of size $10\ \mathrm{K}$4, where the first 80 elevation samples up to MLWN plus all 100 chainage samples are the inputs and the remaining 20 elevation samples are the target. The architecture uses six transformer blocks, each with multi-head attention and four ConvLSTM layers, followed by a two-layer MLP and a dense output layer (Munian et al., 11 Jan 2025).

On WCMC survey data, TopoFormer is evaluated against DenseNet, LSTM, biLSTM, ConvLSTM, 1D-CNN, and 2D-CNN. The reported in-distribution performance is MAE $10\ \mathrm{K}$5, RMSE $10\ \mathrm{K}$6, with 761K trainable parameters; the paper interprets this as an error as low as 2 cm. It also reports superior out-of-distribution generalization when only the MLWN portion of a profile is observed (Munian et al., 11 Jan 2025). In this usage, the topograph is not only a representation of terrain but the direct input-output object of a sequential prediction problem.

5. Conway topographs and arithmetic structure

In mathematics, the topograph has a precise and classical meaning due to Conway. For integral binary quadratic forms, it is a planar trivalent tree whose regions are labeled by integers and satisfy an arithmetic progression rule. If the edge labels directed away from a vertex are $10\ \mathrm{K}$7, $10\ \mathrm{K}$8, and $10\ \mathrm{K}$9, then the discriminant is

$10$0

Equivalently, if nearby region labels are $10$1, then the local relation is

$10$2

A fundamental theorem identifies the topograph with the values of a quadratic form $10$3 on coprime integer pairs (O'Sullivan, 2024).

This geometry supports a uniform reduction theory across all discriminant types. For $10$4, topographs of definite forms have wells. For $10$5 non-square, they have a unique infinite periodic river separating positive and negative regions. For square discriminants there are lakes connected by a finite river. The 2024 treatment develops a novel continued fraction for complex numbers and uses it to obtain explicit reduction to canonical representatives for negative, positive, square, and non-square discriminants within a single framework (O'Sullivan, 2024).

A related but distinct arithmetic development concerns Markov fractions and Cohn matrices. In that setting, the ordinary Farey mediant

$10$6

is replaced by Springborn’s mediant

$10$7

This defines a Markov fraction topograph $10$8 parallel to the rational Conway topograph. The principal theorem states that the Markov fraction associated to $10$9 coincides with the index of the corresponding Cohn matrix at parameter $12\ \mathrm{K/km}$0,

$12\ \mathrm{K/km}$1

One consequence is a simple concatenation rule for the corresponding continued fractions,

$12\ \mathrm{K/km}$2

which makes the common trivalent-tree structure explicit across rationals, Markov fractions, Cohn matrices, and continued fractions (Veselov, 19 Apr 2026).

The arithmetic literature therefore uses “topograph” in a strong formal sense: it is not merely a visualization aid but an object that organizes equivalence classes, reduction, continued fractions, class numbers, and matrix recursions (O'Sullivan, 2024, Veselov, 19 Apr 2026).

6. Algorithmic, combinatorial, and telescoping extensions

The topograph perspective has been generalized into explicit algorithms. One line of work uses Conway’s theory to decide whether two integral binary quadratic forms in two variables are isomorphic over $12\ \mathrm{K/km}$3. The construction starts from primitive vectors, bases, and superbases in $12\ \mathrm{K/km}$4, passes to lax vectors and lax superbases, and obtains an infinite 3-regular tree. The sign structure of a form then classifies the topograph into well, river, lake, lake-pair, or weir types, and the resulting invariants can be compared algorithmically (Fehér, 2023).

Another extension appears in crystallography. Powder auto-indexing must reconstruct a reciprocal lattice from observed squared lengths despite systematic absences. The 2012 work generalizes Conway’s rank-2 topographs to higher dimensions using Voronoi’s second reduction theory, defining a graph $12\ \mathrm{K/km}$5 of primitive $12\ \mathrm{K/km}$6-type domains. In rank 2, Conway’s four associated lengths,

$12\ \mathrm{K/km}$7

remain linked by the parallelogram law

$12\ \mathrm{K/km}$8

The algorithm exploits distribution rules for systematic absences on the topograph, is implemented in Conograph, and reduces computation time to between $12\ \mathrm{K/km}$9 and $100\ \mathrm{K/km}$0 on the reported diffraction patterns (Oishi-Tomiyasu, 2012).

A third development studies sums indexed by topograph vertices. For a topograph associated to a binary quadratic form, with region labels $100\ \mathrm{K/km}$1 and edge labels $100\ \mathrm{K/km}$2, the local identity

$100\ \mathrm{K/km}$3

turns global series into telescoping sums over edges. Cutting the topograph along a root edge produces an admissible half-topograph, and the sum over all vertices of one side collapses to an explicit boundary expression depending only on the root labels and the discriminant. This framework yields arithmetic proofs of modular graph function identities, alternative derivations of Hurwitz-style class number formulas, and a unified treatment of Mordell–Tornheim series and Hata’s series for $100\ \mathrm{K/km}$4 (Kalinin, 2 Oct 2025).

These developments show that the mathematical topograph is both a geometric picture and a computational device. Its local rules are sufficiently rigid that global algebraic quantities become recoverable from finite or boundary data (Fehér, 2023, Oishi-Tomiyasu, 2012, Kalinin, 2 Oct 2025).

7. Topograph as a graph framework in modern machine learning and physics

In recent machine learning literature, “Topograph” is used for graph frameworks whose aim is topology preservation or topology-aware inference rather than terrain representation. In image segmentation, the 2024 framework constructs a combined component graph $100\ \mathrm{K/km}$5 from prediction and ground truth. The four induced regions are true positives, true negatives, false negatives, and false positives, and connected components of these regions become graph vertices. A critical set $100\ \mathrm{K/km}$6 of topologically consequential error components is identified, and the loss

$100\ \mathrm{K/km}$7

is applied only to those regions. The paper also introduces the DIU metric based on the inclusion of the intersection into the union of thickened foregrounds, with strict topological error

$100\ \mathrm{K/km}$8

It proves that $100\ \mathrm{K/km}$9 implies the relevant inclusions are homotopy equivalences, and reports average loss computation times of 95.94 ms for Topograph compared with 327.64 ms for Betti Matching and 602.99 ms for HuTopo (Lux et al., 2024).

In particle physics, the Topograph is a modular graph built from particle blocks representing intermediate particles and candidate final-state daughters. For all-hadronic $1\ \mathrm{km}$0 reconstruction, each top-quark block contains a nested $1\ \mathrm{km}$1 block, message passing occurs between candidate jets and injected intermediate-particle nodes, and the network predicts both assignment scores and intermediate-particle kinematics. The approach is sparse and topology-aware: jets connect only to possible mother particles rather than forming a fully connected event graph. In the benchmark with 20 million generated events, the event-level reconstruction efficiency for fully reconstructable events with at least 6 jets and exactly 2 $1\ \mathrm{km}$2-jets is $1\ \mathrm{km}$3 for Topograph, $1\ \mathrm{km}$4 for SPA-Net, and $1\ \mathrm{km}$5 for the $1\ \mathrm{km}$6 baseline (Ehrke et al., 2023).

These usages are terminologically related to the older mathematical sense only at a high level. In both cases, a graph is designed so that permitted adjacencies encode the structure one wants to preserve or reconstruct. A plausible implication is that “Topograph” has become a productive label whenever topology is shifted from an implicit property of the data to an explicit property of the representation (Lux et al., 2024, Ehrke et al., 2023).