Splatonic: Packings & 3DGS-SLAM Advances

• Splatonic is a dual research area comprising studies on random packings of Platonic solids and a co-designed 3D Gaussian Splatting SLAM technology.
• It examines experimental protocols, packing fraction metrics, and shape dependence to reveal material properties and stability in granular matter.
• Its real-time SLAM pipeline uses adaptive sparse sampling and hardware acceleration to achieve significant speedup and energy savings on mobile platforms.

Splatonic encompasses two distinct lines of research in contemporary computational science: (1) measurement and analysis of random packings of Platonic solids, with implications for granular materials and metamaterials, and (2) “Splatonic” as a co-design architecture for sparse, real-time 3D Gaussian Splatting SLAM (3DGS-SLAM) on mobile platforms, characterized by algorithmic, pipeline, and hardware advances for efficient dense reconstruction. This article presents a detailed survey of both domains, including definitions, key methodologies, results, and ongoing research directions.

1. Definitions and Conceptual Overview

The term “Splatonic” first refers to systematic studies of random packings of the five Platonic solids—tetrahedron, cube, octahedron, dodecahedron, and icosahedron—which serve as an archetype for analyzing the influence of particle geometry and friction on maximum and minimum stable packings (Baker et al., 2010). It also designates a recent co-designed algorithm-hardware system for efficient 3D Gaussian Splatting SLAM, which overcomes computational bottlenecks on mobile hardware by exploiting adaptive sparsity and fine-grained pipeline parallelism (Huang et al., 24 Nov 2025).

2. Random Packing of Platonic Solids: Experimental Methodology

The Splatonic framework in granular matter research defines the global packing fraction as

$$\phi = \frac{N\,V_\mathrm{solid}}{V_\mathrm{container}}$$

where $N$ is the number of particles, $V_\mathrm{solid}$ the mean solid volume (measured by water displacement), and $V_\mathrm{container}$ the container volume (Baker et al., 2010). Experiments involve four preparation protocols:

1. Sequential Addition (Unshaken, U): Incremental random placement without agitation.
2. Hand Shaken (H): Periodic gentle shaking to allow rearrangement.
3. Mechanical Vibration (M, “Random Close Packing”): Electromagnetic agitation at $f\approx50$ Hz, $\Gamma=5g$ accelerations.
4. Fluidization (F, “Random Loose Packing”): Hydraulic suspension, followed by gentle sedimentation.

Plastic dice ($\rho=1.16$ g/cm$^{3}$, $\mu_{\rm plastic}\approx0.375$) and ceramic tetrahedra ($\rho=1.63$ g/cm$^{3}$, $\mu_{\rm ceramic}\approx0.48$) were used, with all Platonic forms represented.

3. Packing Fraction Results and Shape Dependence

Measured packing fractions for the five Platonic solids (plastic, mechanically vibrated and fluidized protocols) are summarized below:

Shape	$\phi_{\rm rlp} \pm \sigma$	$\phi_{\rm rcp} \pm \sigma$
Tetrahedron	$0.51\pm0.01$	$0.64\pm0.01$
Cube	$0.54\pm0.01$	$0.67\pm0.02$
Octahedron	$0.52\pm0.01$	$0.64\pm0.01$
Dodecahedron	$0.51\pm0.01$	$0.63\pm0.01$
Icosahedron	$0.50\pm0.01$	$0.59\pm0.01$

Both $\phi_{\rm rlp}$ (loose) and $\phi_{\rm rcp}$ (dense) packings peak at the cube ($F=6$), which uniquely tessellates space, then decrease monotonically with increasing face count ($F=4\rightarrow 20$). Increased friction (ceramic vs. plastic) reduces $\phi$ by up to 9%. In all cases, as $F\to\infty$, values approach those for spheres, but remain systematically lower than frictionless theoretical maxima or empirical sphere values in the loose limit (Baker et al., 2010).

4. Elastic Platonic Shells: Buckling and Morphogenesis

Another “Splatonic” context concerns buckling transitions of elastic spherical shells with frozen defect topology, yielding faceted shapes with Platonic polyhedral symmetry (Yong et al., 2013). The configuration space is determined by Euler’s theorem,

$\sum_{z} (6-z)\,N_z = 12$

and symmetry constraints:

• Tetrahedron: 4 three-fold disclinations.
• Cube/Octahedron: 6 four-fold disclinations.
• Icosahedron/Dodecahedron: 12 five-fold disclinations.

Upon quasi-static deflation, the shell undergoes a first-order, hysteretic transition from spherical to Platonic-faceted morphologies. The Landau free energy, constructed in spherical-harmonic order parameters, includes quadratic and cubic invariants, with the latter ensuring first-order (hysteretic) behavior. Group-theoretic analysis links symmetry with the spherical harmonic $\ell$ number:

• Tetrahedral: $\ell=3$;
• Cubic/Octahedral: $\ell=4$;
• Icosahedral/Dodecahedral: $\ell=6$.

Critical buckling pressures and hysteresis widths are measured (e.g., $p_b^u \simeq 210$–230 for Platonic cases), with partial shape memory upon unloading. The $(Q_{\ell}, W_{\ell})$ invariants serve as order parameters for distinguishing symmetries (Yong et al., 2013).

5. Splatonic in 3D Gaussian Splatting SLAM

The “Splatonic” framework for real-time 3DGS-SLAM centers on scene representation by a sparse set of 3D Gaussians $\{G_i\}$, each parameterized by spatial mean $\mu_i$, covariance $\Sigma_i$, color $c_i$, and opacity $\alpha_i$ (Huang et al., 24 Nov 2025). Rendering projects these Gaussians and composites via alpha blending,

$$C(p) = \sum_{i} \Gamma_i\,\alpha_i\,c_i,\quad \Gamma_i=\prod_{j< i}(1-\alpha_j)$$

Algorithmic contributions include:

• Adaptive Sparse Pixel Sampling: Reduces tracking and mapping pixel counts by $256\times$ (tile size $w_t=16$) with $<1$ cm degradation in ATE. Mapping samples “unseen” and texture-rich pixels using gradient-based weighting.
• Preemptive Alpha-Checking: Gaussians are culled early if per-pixel $\alpha_i(p)<10^{-4}$, eliminating >90% of candidates.
• Gaussian-Parallel Rendering: Assigns one warp per pixel, partitions Gaussians across threads for efficient forward and backward (reverse rasterization) passes.

A pipelined hardware accelerator implements five functional units: projection, hierarchical sorting, rasterization, reverse rasterization, and aggregation, with resource allocation tuned to input sparsity.

6. Architectural Performance and Evaluation

Splatonic’s co-design achieves:

• GPU evaluation: $14.6\times$ speedup and $86.1\%$ energy savings (vs. dense 3DGS-SLAM).
• Hardware accelerator: $274.9\times$ tracking speedup, $4738.5\times$ energy savings over a mobile GPU.
• Area and power: $1.07$ mm$^2$ (16 nm), $220$ mW active at $500$ MHz, per-frame energy $\sim 2$ mJ.
• Accuracy: Tracking ATE matches dense baseline (Replica dataset: $0.46$ cm), and reconstruction PSNR improves ($+1$ dB) with sparse adaptive mapping.

Limitations include diminished tile-vs-pixel efficiency crossover at fine spatial sampling and increased complexity in DRAM scheduling under irregular per-pixel accesses. Mapping requires $4\times$ more pixels than tracking, resulting in relatively lower speedups for that phase (Huang et al., 24 Nov 2025).

7. Implications and Ongoing Extensions

Splatonic’s empirical findings in particle packing illuminate how geometry and friction dictate the attainable spectrum between random loose and close packings, extending the paradigm beyond spheres to faceted particles (Baker et al., 2010). The elasticity-driven route to Platonic morphologies in shells enables control of shape and memory effects in soft matter and provides model systems for morphological transitions (Yong et al., 2013). In 3DGS-SLAM, adaptive sparsity and pixel-based rendering pipelines allow SLAM workloads to scale efficiently on resource-constrained devices, with direct applicability to real-time robotics, AR/VR, and city-scale reconstruction (Huang et al., 24 Nov 2025).

Extensions include dynamic, scene-adaptive sampling, integration of learned importance maps for pixel selection, multi-view/foveated rendering, and scaling to hierarchical scene representations (e.g., Gaussian octrees). A plausible implication is that the Splatonic approach could generalize to other differentiable rendering and registration tasks across computer vision and robotics.