PACKING PREVIEW60° / 1.50:1
filled by trianglesempty space
Loading the selected packing evidence…
PACKING COMPASS
What are you trying to discover?
Choose one goal. The Atlas will show only the decisions needed for that research task, while the complete technical interface remains available.
Checking the verified Atlas release before answering a research question…
YOUR RESEARCH QUESTION
The guided question controls will appear after the integrity-checked release is ready.
PACKING WORKSHOP
Test a packing without overstating it.
Start from integrity-checked coordinates, adjust one triangle at a time, and run the Atlas geometry verifier in this browser. Nothing here changes the published record.
Verified baseline search is waiting for release data.
No verified baseline loaded
The Workshop will preserve the exact container, triangle shape, and published incumbent identity.
- Published fill
- —
- Pieces
- —
- Evidence
- —
Select or drag a triangle. Arrow keys move the selected triangle by 0.01; hold Shift for 0.1. Press R to rotate.
No candidate packing is available.
QUICK EDIT
Selected triangle
Choose a verified baseline to begin.
No local validation yet. Published evidence remains authoritative.
NEXT SAFE STEP
Waiting for verified baseline
A verified baseline must load before the Workshop can suggest a next step.
Recovery is waiting for a verified baseline.
Choose a verified baseline to inspect one triangle.
Unvalidated candidateRun local validation before drawing any conclusion.
COORDINATE EDITOR
Move one triangle precisely.
Select a placement, enter finite coordinates and angle, then apply. Reflection stays locked when the verified problem forbids it.
Triangle navigation is waiting for a verified baseline.
Choose a verified baseline to begin.
LOCAL VALIDATION
Check constraints and compare.
The same geometry engine checks bounds, overlap, spacing, fingerprint, and the current incumbent. Local success is still not proof or publication.
No local validation yet
Published evidence remains authoritative.
- Published fill
- —
- Candidate fill
- —
- Difference
- —
Validation findings
- Run local validation to inspect geometry and submission-readiness findings.
SAVE · RECOVER · CONTRIBUTE
Keep the work reproducible.
Browser saves and exported bundles are checksummed and bound to this verified release. They are working files, not scientific evidence.
Web-to-Git handoff
Run local validation to unlock an exact, reviewable contribution plan.
- Nothing has been submitted from this browser.
No draft has been saved or exported.
- Export the candidate and exact verified baseline identity.
- Run the full CLI verifier and retain its report.
- Open a reviewable GitHub contribution only when local checks support it.
- Wait for independent verification and maintainer review before claiming improvement or proof.
OPEN TRIANGLE-PACKING DATA
See how triangles
fit in rectangles.
Choose a triangle shape and rectangle size. Explore verified packings, compare results, and find problems that still need better solutions.
— verified records
04 family slices
— open problems
Try the packing map ↓
INTERACTIVE PACKING MAP
Change the shape. See what works best.
Adjust the triangle angle and rectangle ratio. The map shows the best verified pattern in our sampled data. It does not claim that every result is optimal.
Checking verified research data…The map remains a modeled preview until release integrity checks finish.
Showing a representative overview. Sliders and evidence use the complete verified map.
TRIANGLE APEX →
Use the arrow keys to move through sampled triangle angles and rectangle ratios. Press Enter or Space to choose a result.
RECTANGLE ASPECT RATIO → · loading computed release…
BROWSE BY TRIANGLE
Choose a triangle family.
Right triangles
Includes exact packings that fill the whole rectangle.
SOME RESULTS PROVEN BESTEquilateral triangles
Shows how much empty space remains near rectangle edges.
VERIFIED RESULTSIsosceles triangles
Covers many shapes; most cases can still be improved.
MOSTLY OPENFEATURED PACKINGS
Open a result to inspect it.
PORTABLE RESEARCH SESSION
Move this investigation safely.
Save the current map, filters, comparison, and shortlist in one integrity-checked file. A session stores research context—not new scientific evidence.
Waiting for an integrity-checked Atlas release.
ALL VERIFIED RESULTS
Search the full dataset.
Filter by triangle family or evidence level. Open any result to see its coordinates, verification details, known bounds, and reproduction command.
EVIDENCE LADDER
How strong is the claim?
Each step includes the checks below it. Only the final step says that no better packing exists for the exact problem.
- CandidateProposed; not independently verified
- Verified constructionCoordinates fit under published rules
- Best knownStrongest verified result in this release
- Proven optimalReaches a rigorous upper bound
Machine-checkable proof scopeFor researchers auditing finite-domain claims
Triangle and rectanglePacking patternWhat we knowFilled / possible gain
WHERE PATTERNS CHANGE
Changes seen between neighboring test cases.
COMPARE TWO RESULTS
See which packing uses more space.
Choose two packings to compare how much of the rectangle they fill, how much remains empty, and which pattern they use.
versus
GUIDED COMPARISONS
Start with a research question.
Preparing verified suggestions…
SAVED WORKSPACE
Keep a research shortlist.
Save up to six verified records. The shortlist stays in this browser and is checked again whenever the Atlas release changes.
- Waiting for verified records.
The shortlist will be restored after release integrity checks finish.
Comparison will be available after release integrity checks finish.
Preparing verified comparison records…
OPEN CHALLENGES
Help improve a packing.
Each challenge starts with a verified result. Reproduce it, fit more triangles, or prove a tighter limit.
HIGHEST FILL RATES
Results that use the most space.
Ranked by independently checked fill percentage. Every result includes its author and the data needed to reproduce it.
CONTRIBUTE A RESULT
Submit a packing
we can verify.
- 01 Submit JSON
- 02 Verify geometry
- 03 Compare record
- 04 Review evidence
- 05 Publish
Submit the triangle coordinates as JSON. Automated checks confirm that pieces stay inside the rectangle, do not overlap, and improve on the current result. A maintainer reviews proofs and citations.
- Review pipeline
- Loading current status…
PRIVATE BROWSER PREFLIGHT
Check readiness before you submit.
Your file stays in this browser. Preflight checks structure and provenance; the full local verifier must still check geometry and improvement.
Waiting for an integrity-checked release.
From baseline to review
- Select a verified baseline to prepare a safe starter.
The starter repeats the current verified result and is not an improvement. GitHub submission and maintainer review remain separate steps.
HOW THE DATA STAYS TRUSTWORTHY
Every result can be checked again.
Traceable results
Each result has a stable ID, coordinates, source, history, and citations.
Run it again
Saved commands, seeds, limits, and fingerprints make results reproducible.
Visible uncertainty
We show the best result found, the proven limit, and the gap between them.
Clear next tasks
The work queue lists specific cases where contributors can improve the atlas.
RELATED RESEARCH
Sources and how we use them.
Each paper is labeled by its relationship to this project. A related method or topic is not treated as proof of an atlas result.
DOWNLOAD ATLAS V2
Verified data you can reuse.
Atlas v2 contains 304 verified packing results. Download the coordinates as JSON or CSV, rerun the checks, and use the open work queue to choose a research problem.
- Release status
- loading…
Browser support
Checking this browser’s verification and interaction support…
- FirefoxCurrent release target
- SafariCurrent release target
- ChromiumCurrent release target
Live workflow diagnostics will run after verified data loads.