Elastic tensors are 3×3 symmetric positive-definite matrices on GL+(3)/SO(3). Proprietary geometric descriptors classify crystal symmetry, interpolate alloy properties, and detect phase transitions — all respecting the manifold constraint.
Elasticity tensors describe how materials respond to stress. The Voigt-averaged 3×3 stiffness matrix is symmetric positive-definite, but standard analysis treats it as a flat vector of 6 numbers.
This creates real problems: Euclidean interpolation between alloys can produce non-physical negative eigenvalues. Classification by crystal symmetry (cubic, hexagonal, tetragonal, etc.) fails under arbitrary coordinate rotations. Standard distance metrics can’t handle the 3+ orders of magnitude eigenvalue spread across material classes.
On the Riemannian symmetric space GL+(3)/SO(3), the affine-invariant metric naturally respects positive-definiteness. Geodesic interpolation between materials stays physical at every intermediate point. The V+/V− decomposition separates shear anisotropy (5D) from bulk modulus (1D).
Proprietary geometric descriptors derived from the manifold structure achieve 96.8% crystal symmetry classification on 1,181 real materials — with complete rotation invariance guaranteed by construction.
For 3×3 Voigt stiffness matrices, the tangent space decomposes into:
Crystal symmetry is determined by the pattern of anisotropy (V+), not the overall stiffness (V−). The geometric framework naturally focuses on this distinction.
Validated on 1,181 real materials from the Materials Project database.
7 symmetry classes (cubic, hexagonal, tetragonal, orthorhombic, monoclinic, triclinic, trigonal)
Published elastic constants from Materials Project (Jain et al. 2013)
Improvement over Euclidean methods under random SO(3) coordinate rotations
Patent-pending geometric fingerprint: anisotropy index, Zener ratio, geodesic spread, curvature signature
Crystal symmetry classification on 1,181 materials. Riemannian methods dramatically outperform Euclidean approaches.
Crystal symmetry classification must be independent of the coordinate frame. Euclidean methods fail catastrophically under rotation; Riemannian methods are invariant by construction.
Treat the 6 Voigt components as a flat vector. A simple coordinate rotation scrambles the components completely. Classification accuracy drops from 71.7% to 0.005% under random rotations — worse than random guessing.
After random rotation
Operate on the full 3×3 SPD matrix. The affine-invariant metric satisfies d(RART, RBRT) = d(A, B) for all rotations R. Classification accuracy is identical before and after rotation — guaranteed by the math.
Unchanged under any rotation
Geometric materials analysis enables capabilities impossible with standard methods.
Classify build quality from measured elastic tensors. Detect porosity, grain misalignment, and annealing defects by their geometric signature on the SPD manifold. Few-shot: characterize a new alloy from 5-10 measured samples.
Geodesic interpolation between known alloys produces physically valid intermediate stiffness tensors at every composition. Euclidean interpolation violates positive-definiteness — the geometric approach guarantees physical validity.
Monitor geodesic distance between rolling elasticity measurements to detect structural phase transitions in real time. The V+/V− decomposition separates symmetry-breaking (V+ shift) from thermal softening (V− shift).
Design metamaterials with target anisotropy profiles by optimizing on the SPD manifold. The 8-feature StiffnessDescriptor provides a compact fingerprint for rapid similarity search across material databases.
import requests
# Classify crystal symmetry from elastic tensor
response = requests.post(
"https://api.omnisciences.io/materials/classify",
json={
"stiffness_matrix": C.tolist(), # 3x3 Voigt-averaged SPD
}
)
result = response.json()
print(f"Symmetry: {result['prediction']}")
# Symmetry: cubic
print(f"Confidence: {result['confidence']:.1%}")
# Confidence: 97.0%
# Geodesic interpolation between two alloys
interp = requests.post(
"https://api.omnisciences.io/materials/interpolate",
json={
"tensor_a": C_steel.tolist(),
"tensor_b": C_titanium.tolist(),
"t": 0.5, # midpoint on geodesic
}
)
# Result is guaranteed SPD (physically valid)Find all elastic tensors matching target stiffness, weight, and symmetry constraints. Inverse design gives the complete solution set, not one optimizer guess.
Design lattice parameters for target effective elastic properties. Map the full feasible design space via parameter homotopy.
Validate elastic tensor databases at scale. We flagged 15.2% of Materials Project entries as non-SPD (thermodynamically impossible). Catch bad data before it trains your ML models.
Database validation consulting: $20K–$50K per database audit. Learn more