Tesserae: Mosaics, Materials & Cluster Design

• Tesserae are discrete elements that serve as fundamental tiling units in mosaics, soft-matter lattices, and GPU scheduling, each defined by application-specific properties.
• In mosaic analysis, fine-tuned segmentation models like SAM 2 achieve an average IoU of 91% and recall of 95%, enabling high-fidelity digital preservation.
• In materials science and computing, tesserae underpin programmable microgel patterning and resource allocation via generalized Hertzian models and graph-matching algorithms.

Tesserae are discrete elements foundational to multiple scientific and technical domains, ranging from the fabrication and analysis of ancient mosaics to applications in self-assembled materials and high-performance computing system design. In all contexts, a tessera (plural: tesserae) denotes a fundamental unit participating in an overall tiling or decomposition, with properties and constraints varying by application. This article surveys the experimental, computational, and algorithmic roles of tesserae, as formalized in selected state-of-the-art research across cultural heritage preservation, soft-matter physics, and cluster resource management.

1. Tesserae in Mosaic Art and Digital Preservation

Tesserae are the constitutive tiles—typically stone, glass, or ceramic—used in the assembly of mosaic artworks, a practice with millennia of historical continuity. The physical vulnerability of mosaics (to environmental stress, material fatigue, or anthropic damage) motivates the development of computational methods for digital preservation, notably accurate segmentation of tesserae in images. Historically, manual annotation provides maximal precision but is infeasible at scale, necessitating automated segmentation models capable of high-fidelity tessera extraction (Kapelonis et al., 20 Dec 2025).

Recent advances implement foundation models such as Segment Anything Model 2 (SAM 2) fine-tuned for the tesserae segmentation task. The architecture employs an image encoder (kept frozen), alongside a prompt encoder and mask decoder trained on domain-specific datasets. The segmentation process leverages Dice loss and a calibration loss that aligns the model's confidence score with Intersection over Union (IoU):

$$\mathcal{L}_{\mathrm{Dice}} = 1 - \frac{2 (P \cdot G)}{\lVert P\rVert_1 + \lVert G\rVert_1}$$

where $P$ is the model output and $G$ the ground truth; a scoring loss further reduces confidence-score discrepancies with actual IoU (Kapelonis et al., 20 Dec 2025). Augmentation strategies (cropping, geometric transforms) and a curated dataset of annotated mosaics yield statistically significant improvements over previous approaches. On test splits, SAM 2 _ft achieves an IoU of 91.02%, recall of 95.89%, and a tessera count error of 0.02—marked improvements versus baseline models. These results establish a direction for near-real-time, artifact-agnostic segmentation and quantitative mosaic analysis.

2. Tesserae in Self-Assembled Soft-Matter Lattices

In soft condensed matter, tesserae generalize to discrete, microscale units (e.g., soft microgels) assembling into two-dimensional tessellations with programmable order. The work of performing sequential depositions of poly(N-isopropylacrylamide) (PNIPAM) microgels at a liquid–liquid interface demonstrates the capacity for a single microgel species to organize into a wide variety of structures—ranging from hexagonal and rectangular tilings (Archimedean) to more exotic rhomboidal, herringbone, and multi-scale superlattices—by varying the respective packing fractions ($\varphi_1, \varphi_2$) of two monolayers (Grillo et al., 2019).

The interparticle interaction is modeled as a “generalized Hertzian” potential,

$$U(r) = \epsilon (1 - r/\sigma)^\alpha \Theta(1 - r/\sigma)$$

with the exponent $\alpha \approx 1.9$ extracted from force-distance measurements, supporting a bounded, monotonic short-range repulsion. Experimental protocols involve transfer of a first fixed monolayer at prescribed packing, followed by a rotated deposition of a second, mobile monolayer. At equilibrium, the combined packing fraction and the template–mobile mismatch define the symmetry and coordination of the resulting tessellation. Imaging (AFM) and bond-orientational analysis reveal stability of rectangular (4-fold), honeycomb (3-fold), rhomboidal, and 12-fold local motifs depending on these tunable parameters. Phase diagrams are constructed by analytically and numerically minimizing the per-particle energy for candidate lattices as functions of $(\varphi_1, \varphi_2)$ (Grillo et al., 2019).

3. Tesserae as Graph-Matching Units in GPU Cluster Scheduling

In distributed deep learning (DL) resource management, "tesserae" analogously serve as mapping units for complex placement and colocation tasks within DL clusters. The Tesserae framework formalizes resource placement, job migration, and packing constraints as instances of (bi)partite graph matching problems (Bian et al., 7 Aug 2025).

For GPU cluster scheduling, each job–GPU assignment or migration is representable as an edge in a weighted bipartite graph:

$G = (V_1, V_2, E, W)$

where $V_1, V_2$ are sets of jobs and resources (or consecutive placement plans), $E$ edges indicating feasible assignments, and $P$0 edge weights or migration costs. The Hungarian algorithm solves maximum or minimum weighted matching instances in polynomial ($P$1) time. For migration minimization, node-level costs are defined by:

$P$2

summed over the symmetric difference between jobs on GPU $P$3 (previous round) and $P$4 (current round), amortizing move-in/-out events.

The placement architecture integrates these policies in a modular layer beneath existing scheduling methods (e.g., FIFO, LAS/Tiresias), enabling both compatibility and extensibility. Empirical evaluation on clusters up to 256 GPUs and workloads including ResNet-50 and GPT-3 derivatives demonstrates improvements of 1.62× in average job completion time and 1.15× in makespan over representative baselines, with robust scaling and near-linear matching time per round (<1.6 s with 2,048 jobs) (Bian et al., 7 Aug 2025).

Domain	Formal Role of Tesserae	Key Techniques/Models
Mosaic art/preservation	Discrete tiles in mosaics	SAM 2 segmentation, polygonal annotation
Soft matter self-assembly	Microgel tiling units	Generalized Hertzian model, phase diagram
DL cluster scheduling	Graph-matching resources	Bipartite matching, Hungarian algorithm

4. Evaluation Methodologies and Quantitative Results

The referenced works employ rigorous, domain-specific approaches for the performance evaluation of tesserae-based systems:

• Segmentation tasks (mosaic analysis): Pixelwise IoU, recall, precision, F-measure, and tessera count error quantify segmentation fidelity. The fine-tuned SAM 2 achieves average IoU of 91.02%, recall of 95.89%, and a tessera count error reduction from 0.20 (U-Net3 baseline) to 0.02 (Kapelonis et al., 20 Dec 2025).
• Soft-matter tessellation: Analytical lattice energy and bond-orientational order parameters ($P$5) distinguish equilibrium phases; AFM and 2D FFT detect local/global motif symmetry. Theoretical and simulation results predict the stable phase for each (packing fraction) parameter pair (Grillo et al., 2019).
• Cluster scheduling frameworks: Metrics include average job completion time (JCT), makespan, migration count, and fairness ratio (FTF), computed on both physical and simulated cluster traces spanning diverse DL model workloads. Tesserae demonstrates up to 1.62× lower average JCT and 1.15× reduction in makespan compared to heuristic and optimization-based baselines, with matching solutions computed in practical time (Bian et al., 7 Aug 2025).

5. Algorithmic and Mechanistic Insights

Across domains, tesserae-centric systems benefit from algorithmic frameworks exploiting locality, modularity, and optimization:

• In digital mosaic segmentation, leveraging the frozen backbone of a universal image encoder combined with domain-adapted decoders enables transfer-efficient, annotation-driven fine-tuning.
• For programmable soft-matter tessellations, the template–mobile interplay (annealing over a fixed substrate) and the use of bounded concave repulsion enable access to nontrivial tilings and superlattices through control of packing fractions. Frustration and commensurability effects underpin the stability and richness of emergent patterns (Grillo et al., 2019).
• GPU scheduling with Tesserae capitalizes on the reduction of placement constraints (migration minimization, colocation optimization) to matching problems, enabling tractable solution space exploration via classical combinatorial optimization algorithms. This abstraction layer allows independent innovation in scheduling policies and placement policies, supporting scalability and adaptability to hardware and workload variance (Bian et al., 7 Aug 2025).

6. Implications, Datasets, and Prospects

Research into tesserae as foundational units yields implications and actionable resources across fields:

• The release of richly annotated mosaic datasets fosters reproducibility and comparative benchmarking in digital cultural heritage, providing data for diverse segmentation models (Kapelonis et al., 20 Dec 2025).
• Robust control over soft-matter tessellation processes can advance the fabrication of metasurfaces, with implications for optoelectronics and responsive interfaces. The accessibility of varied tessellation phases via a single, isotropic interaction model (generalized Hertzian) points to broad applicability (Grillo et al., 2019).
• Tesserae’s formalization of cluster resource matching supports integration with arbitrary DL scheduling policies and facilitates further exploration of placement constraints—such as device topology, energy, and latency objectives—by translating them into graph-matching terms. Empirical robustness to profiling noise and support for parallelism strategy tuning solidify its utility in production systems (Bian et al., 7 Aug 2025).

A plausible implication is that these unified perspectives on tesserae—across computational, material, and architectural levels—enable systematic design and analysis of large-scale, heterogeneous systems in science and engineering.