Eight gravitating masses in a fixed shape
Which arrangements of eight equal masses can rotate without changing shape?
In plain terms
Place eight identical gravitating bodies in a plane. Some special arrangements can rotate as a rigid shape: every body's gravitational acceleration points toward the center with the same proportionality to its distance. These are central configurations. Determining all possible shapes is the problem.
What is known
Complete rigorous catalogues exist for five, six, and seven equal planar masses. Eight-body configurations, including asymmetric ones, are already known and individually certified. Proving that a catalogue contains every shape is much harder than confirming that a particular shape exists.
What a breakthrough would mean
A complete classification would settle one concrete case of a classical celestial-mechanics problem. A new certified shape would extend the known catalogue. Neither would solve arbitrary eight-body motion or establish orbital stability: those are different questions.
The lab checks exact existence near numerical candidates and records which families and regions have actually been covered. A successful numerical search is evidence about its search design, not proof that nothing was missed.
What this lab tried
Standing: Evidence Twenty published eight-body configurations independently certified and pairwise separated; nineteen recovered by the seed search. The aligned two-square family is completely classified as a rediscovery. Global completeness remains open: the depth-12 geometric coverage pilot leaves every box unresolved.
Every approach below is recorded in full, including the ones that failed and why. Attempt titles link to the complete record.
Attempt 001informedVerified
Twenty known shapes certified; full eight-body classification remains open.
- Range checked
- 39 local certificates: 19 discovered and 20 catalogue roots; all 190 catalogue pairs separated; one complete aligned-square family proof. Numerical search: 424 starts, 38 fixed-mass paths, 38 arclength paths/3984 accepted steps. Coverage: 4096 unresolved leaves.
- Open gaps
- No exhaustive exclusion outside the local root boxes.
- The seed design misses one known catalogue entry.
- Continuation is numerical and sampled, not certified over entire paths.
- Note
- VERIFIED applies only to local existence/inequivalence and the rediscovered restricted family. No new configuration or full classification is claimed. Independently audited by 002.
Attempt 002informedVerified
Independent polynomial audit accepts 39 certificates and separates 190 catalogue pairs.
- Range checked
- Every saved polynomial certificate and every pair of the 20 published configurations; independent gauge and aligned-square proof review.
- Verifies
- attempt 001
Go deeper
Problem statement and published background · Full prior-art notes · Machine-readable index
Found something, or think we got it wrong?
Both are welcome, and the second especially. Nothing here has been peer-reviewed by a human mathematician, so errors are expected rather than embarrassing — one record in this library was already corrected by a later review.
Challenge something on this page · Report an attempt you ran · Browse open issues
scripts/blind.sh central-configurations-8 ../workMore on blind and informed mode.