Symmetry Discovery Questionnaire
Workflow
Copy this checklist and track your progress:
Symmetry Discovery Progress:
- [ ] Step 1: Classify your domain and data type
- [ ] Step 2: Analyze coordinate system choices
- [ ] Step 3: Test candidate transformations
- [ ] Step 4: Analyze physical constraints
- [ ] Step 5: Determine output behavior under transformations
- [ ] Step 6: Document symmetry candidatesStep 1: Classify your domain and data type
Ask user what their primary data type is. Use this table to identify likely symmetries and guide further questions. Images (2D grids) → likely translation, rotation, reflection. 3D data (point clouds, meshes) → likely SE(3), E(3). Molecules → E(3) + permutation + point groups. Graphs/Networks → permutation. Sets → permutation. Time series → time-translation, periodicity. Tabular → rarely symmetric. Physical systems → conservation laws imply symmetries. For detailed worked examples by domain, consult Domain Examples.
Step 2: Analyze coordinate system choices
Guide user through coordinate analysis questions: Is there a preferred origin? (NO → translation invariance). Is there a preferred orientation? (NO → rotation invariance). Is there a preferred handedness? (NO → reflection invariance). Is there a preferred scale? (NO → scale invariance). Is element ordering meaningful? (NO → permutation invariance). Document each answer with reasoning.
Step 3: Test candidate transformations
For each candidate transformation T, ask: "If I transform my input by T, should my output change?" If NO → invariance to T. If YES predictably → equivariance to T. If YES unpredictably → no symmetry. Use domain-specific checklists from Domain Transformation Tests. Test all relevant transformations systematically. For the detailed methodology behind this testing approach, see Methodology.
Step 4: Analyze physical constraints
Ask about conservation laws and physical symmetries. Noether's theorem: every conservation law implies a symmetry. Energy conserved → time-translation symmetry. Momentum conserved → space-translation symmetry. Angular momentum conserved → rotation symmetry. Ask: Are there physical conservation laws? Is system isolated from external reference frames? Are there gauge freedoms?
Step 5: Determine output behavior under transformations
Critical question: When input transforms, how should output transform? Classification labels → stay same (invariance). Bounding boxes → move with object (equivariance). Force vectors → rotate with system (equivariance). Scalar properties → stay same (invariance). Segmentation masks → transform with image (equivariance). This determines whether you need invariant or equivariant architecture.
Step 6: Document symmetry candidates
Create summary using Output Template. List identified symmetries with confidence levels. Note uncertain cases that need empirical validation. Identify non-symmetries (transformations that DO matter). Recommend next steps for validation and formalization. Quality criteria for this output are defined in Quality Rubric.
Domain Transformation Tests
Image Symmetries
| Transformation | Test Question | If NO → |
|---|---|---|
| Translation | Does object position matter for label? | Translation invariance |
| Rotation (90°) | Would rotated image have same label? | C4 symmetry |
| Rotation (any) | Would any rotation preserve label? | SO(2) symmetry |
| Horizontal flip | Would mirror image have same label? | Reflection |
| Scale | Would zoomed image have same label? | Scale invariance |
3D Data Symmetries
| Transformation | Test Question | If NO → |
|---|---|---|
| 3D Translation | Does absolute position matter? | Translation invariance |
| 3D Rotation | Does orientation matter? | SO(3) or SE(3) |
| Reflection | Does handedness matter? | O(3) or E(3) |
| Point permutation | Does point ordering matter? | Permutation invariance |
Graph Symmetries
| Transformation | Test Question | If NO → |
|---|---|---|
| Node relabeling | Does node ID matter, or just connectivity? | Permutation invariance |
Molecular Symmetries
| Transformation | Test Question | If NO → |
|---|---|---|
| Rotation | Is property independent of orientation? | SO(3) |
| Translation | Is property independent of position? | Translation |
| Reflection | Are both enantiomers equivalent? | Include reflections |
| Atom permutation | Do identical atoms behave identically? | Permutation |
Temporal Symmetries
| Transformation | Test Question | If NO → |
|---|---|---|
| Time shift | Can pattern occur at any time? | Time-translation |
| Time reversal | Is forward same as backward? | Time-reversal |
| Periodicity | Do patterns repeat with period T? | Cyclic symmetry |
Quick Reference
The 5 Key Questions:
- Is there a preferred coordinate system? (origin, orientation, scale)
- Does element ordering matter?
- What transformations leave the label unchanged?
- What physical constraints apply?
- How should outputs transform when inputs transform?
Common Symmetry → Group Mapping:
- Rotation (2D, discrete) → Cyclic group Cₙ
- Rotation + reflection (2D) → Dihedral group Dₙ
- Rotation (2D, continuous) → SO(2)
- Rotation (3D) → SO(3)
- Rotation + translation (3D) → SE(3)
- Full Euclidean (3D) → E(3)
- Permutation → Symmetric group Sₙ
Output Template
SYMMETRY CANDIDATE SUMMARY
==========================
Domain: [Data type]
Task: [Classification/Regression/Detection/etc.]
IDENTIFIED SYMMETRIES:
1. [Transformation]: [Invariance/Equivariance]
- Evidence: [Why you believe this]
- Confidence: [High/Medium/Low]
2. [Transformation]: [Invariance/Equivariance]
- Evidence: [Why you believe this]
- Confidence: [High/Medium/Low]
UNCERTAIN SYMMETRIES (need validation):
- [Transformation]: [Reason for uncertainty]
NON-SYMMETRIES (transformations that DO matter):
- [Transformation]: [Why it matters]
NEXT STEPS:
- Empirically validate uncertain symmetry candidates
- Map confirmed symmetries to mathematical groups
- Design architecture based on validated group structure