symmetry-discovery-questionnaire

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 candidates

Step 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:

1. Is there a preferred coordinate system? (origin, orientation, scale)
2. Does element ordering matter?
3. What transformations leave the label unchanged?
4. What physical constraints apply?
5. 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