What agrees
Energy: −2.25.
Angular momentum: 3.00.
Both orbits satisfy Newton’s equations.
A GALLERY OF MATHEMATICAL IDEAS / 001
Three bodies.
Centuries of
imagination.
An equation keeps an insight available to another person. Meet the people, shapes and questions behind three bodies in motion, then follow their ideas into the mathematics.
Working paper v0.7 · Website edition 016
Explore exact classical families and an infinite local series. This exhibition does not establish a general finite closed-form solution.
01 / A SHARED CENTERθ = 15°
Five encounters / a selected history
The things we
learned to see.
Give an idea room. Name the person and the work. Stay with the question long enough for a relationship to become visible.
This art is important. Let’s put it in an important building.
George Lucas, quoted on the Lucas Museum’s campus page.
That care for narrative art inspires this Ergentics gallery. The museum was founded by George Lucas and Mellody Hobson; its account of art as a shared human encounter informs our approach. The mathematical diagrams and narration here are original modern presentations. Read the inspiration trace.
A small gallery of people, relationships and the patient work of understanding. Follow the labels into the original texts; follow the diagrams into the mathematics.
-
Ergentics · modern schematic · mutual attraction 01 / Isaac Newton / 1687
A language
for attraction.What if motion in the heavens could be described through the same mathematical relationships we use on Earth? In the Principia, mutual attraction belongs to a common account of matter and motion.
Look for the relationship: a greater separation weakens the pair force.
Read the force law in §2.1 ↗Primary text · Latin transcription
Principia · Laws I–III; Book III, Proposition VIITranscribed by Jonathan Ingram, Keith Edkins and the Online Distributed Proofreading Team. Their work makes this text newly reachable.
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Ergentics · modern exact reference · special initial data 02 / Leonhard Euler / 1763 → 1767
The possibility
of a line.Three bodies need not lose every simple shape. Euler studied their mutual attraction on a straight line. Our workbench separately presents a symmetric rotating member of the classical collinear family.
Look for what stays fixed while the direction changes.
Inspect the exact families ↓Archive catalogue · original text access pending
Euler Archive E327 · written 1763, published 1767The catalogue establishes the work and dates. Its original PDF was inaccessible during this trace; the rotating reference is checked in our own derivation.
-
A shared shape · Ergentics, 2026 · AI-generated interpretation 03 / Joseph-Louis Lagrange / 1772
A triangle
held together.Begin with what lies between the bodies. Lagrange introduces his method through their three mutual distances—the triangle they form at each instant. A relationship becomes a way of seeing the whole.
A shared shape can hold unequal lives in balance.
In the circular equilateral family below, unequal masses move at different radii while every side stays the same length.
Change the view of this triangle ↓Primary text · collected-edition transcription
Essai sur le Problème des trois Corps · Avertissement, pp. 229–230Wikisource revision 13975608, Œuvres, tome VI. The companion portrait is a contemporary interpretation, not an authenticated likeness or historical painting.
The mathematical referenceExact circular Lagrange family · phase 15°.
The painted triangle is an artistic suggestion. -
Looking again · Ergentics, 2026 · AI-generated interpretation 04 / Henri Poincaré / 1890
The value of
looking again.In the introduction to his revised memoir, Poincaré thanks Edvard Phragmén: a careful question helped him discover and correct an important error. The history of mathematics includes the people who help us see what we missed.
What does your evidence establish—and what can it miss?
Challenge an apparent match ↓Primary text · author's acknowledgment
Introduction · Acta Mathematica 13, pp. 5–7 · acknowledgment p. 5The portrait and returning paths are contemporary interpretations, not an authenticated likeness or a mathematical proof.
A separate teaching exampleOur phase comparison tests an apparent match.
It is not Poincaré's original diagram. -
Ergentics · local Taylor bound · distinct from Sundman’s theorem 05 / Karl F. Sundman / printed 1912 · catalogued 1913
An answer
can be infinite.Sundman asks us to take an infinite representation seriously. His closing theorem changes the time variable and states a nonzero-angular-momentum condition, together with a convention for continuing through collisions.
Read the boundary of a guarantee as carefully as the formula.
Our local Taylor series is a separate construction, with its own limited domain.
Explore the local bound ↓Primary text · article scan
Mémoire sur le problème des trois corps · §34, pp. 178–179The opening leaf says “Imprimé le 8 juillet 1912.” The publisher catalogues the work as 1913. Both pieces of evidence belong in its history.
And around every named work: teachers, translators, librarians, makers of instruments, careful readers. This gallery is one path through a much wider human inheritance.
The viewing room / a study in perception
A different view.
The same relationship.
The triangle is still. Tilt the view and its silhouette narrows. Its three sides remain one unit long in the mathematical plane. A picture and the thing it describes are related, but they are not interchangeable.
You choose the viewpoint
Arrow keys change one degree. Home returns to the plane; End selects 60°. These controls change the view, not the orbit’s time.
Plan view. The physical state is fixed.
Each physical side1.00000 unit
The reference shape stays here.
How the depth is made. Foreshortening, a reference plane and depth ordering supply visual cues on this flat screen. This is an orthographic projection, not stereoscopic output. We have not measured a perceptual improvement.
With camera tilt α, X = x, Y = y cos α, Z = y sin α. Physical z remains zero; camera Z measures depth in the view. Screen distances generally shrink. Trace the design foundations.
Inspect the physical and camera coordinates
| Body | Physical x | Physical y | Camera Y | Camera Z |
|---|---|---|---|---|
| 1 | -0.45138 | -0.56924 | -0.56924 | 0.00000 |
| 2 | 0.51454 | -0.31042 | -0.31042 | 0.00000 |
| 3 | -0.19257 | 0.39669 | 0.39669 | 0.00000 |
Still editions: plan · 30° · 60°. The comparison is available without animation or JavaScript.
The shape of the inquiry
Three ways into
the same question.
What does it mean to solve a motion? Begin with an exact shape, challenge an apparent match, then examine the limits of an infinite local series. These studies do not establish a general finite closed-form solution to the three-body problem.
Read with context. The mathematical derivations are coordinator self-review. The three-model AI study is exploratory and does not establish a reliable model ranking. Read the claims and limitations. How these inquiry examples were checked.
CHAPTER 01 / SHAPE
A shape holds.
A world moves.
The formula gives each state directly. Choose a family and phase to inspect its motion; this display does not numerically integrate a trajectory.
One triangle, three moments.
These are three exact states of the same circular Lagrange orbit: masses (1, 2, 3), side length 1. Every side stays the same length as the triangle turns.
THE LIVE INSTRUMENT
Select a family, then move its phase. The formulas below follow your choice.
The saved figures and explanations are available below. Interactive controls become available when JavaScript finishes loading.
Outer masses = 1. Adjacent gap = 1. G = 1.
Keyboard: arrows move 1°; Shift + arrows move 0.1°; Page Up/Down move 15°; Home/End select 0°/360°.
- Time within this cycle
- 0.3491
- Angular speed
- 1.5000
- Period
- 4.1888
The circular symmetric Euler family is collision-free for all real time and linearly unstable. An exact orbit need not be stable.
Playback targets one cycle per 14 display seconds; a quarter-turn targets 3.5 seconds and then pauses. Slow or interrupted rendering can lengthen both. With reduced motion, quarter-turn shows its endpoint immediately.
The time readout measures physical time from phase zero in reference units and returns to zero each cycle. It is not elapsed viewing time.
Motion paused.
The horizontal axis is x; the vertical axis is y; O marks the center of mass. Numbered markers identify bodies and are enlarged on narrow screens; they do not show physical sizes. The table below gives coordinates and velocities rounded to five decimal places.
- Body 1Teal · mass 1
- Body 2Ochre · mass 2
- Body 3Violet · mass 1
Inspect positions and velocities
| Body | Mass | x | y | vₓ | vᵧ |
|---|---|---|---|---|---|
| 1 | 1.0 | −0.86603 | −0.50000 | 0.75000 | −1.29904 |
| 2 | 2.0 | 0.00000 | 0.00000 | 0.00000 | 0.00000 |
| 3 | 1.0 | 0.86603 | 0.50000 | −0.75000 | 1.29904 |
Mathematics, written in motion
A line becomes an orbit.
For the symmetric Euler family, three positions share one rotating direction. The outer bodies mirror each other; the middle body rests at their center.
The minus and plus signs place the outer bodies on opposite sides of the center. The distance a stays fixed.
The rhythm of the rotation
- a
- Gap from the center to either outer body
- m
- Mass of each outer body
- m₀
- Mass of the middle body
- G
- Gravitational constant
Here θ is the phase, ω is the angular speed, and t is physical time. Written in the center-of-mass frame, with positive masses and a > 0. Phase is zero at t = 0; the slider selects a state on that orbit.
Matching initial velocities are required. This symmetry describes a special family, not arbitrary initial conditions.
The rhythm of the rotation
- a
- Side length of the equilateral triangle
- M
- Total mass: m₁ + m₂ + m₃
- c
- Center of mass: (m₁s₁ + m₂s₂ + m₃s₃) / M
- R
- Counterclockwise rotation by phase θ
The initial vertices are s₁ = (0, 0), s₂ = (a, 0), s₃ = (a/2, √3 a/2). Subtracting c centers the triangle. R(θ) sends (x, y) to (x cos θ − y sin θ, x sin θ + y cos θ).
Here G = 1, a = 1, and masses are (1, 2, 3), so ω = √6. The motion lies in z = 0. Matching initial velocities are required: vᵢ = ω(−yᵢ, xᵢ, 0). This is the circular Lagrange family; arbitrary initial data need not preserve this triangle.
CHAPTER 02 / EVIDENCE
Look beyond
matching numbers.
A phase-shifted circular orbit keeps the same energy and angular momentum, yet can miss the requested initial state. Change the shift to compare positions and velocities.
What still needs checking
Do the positions and velocities match the requested state at the requested time?
- Energy, both orbits
- −2.25
- Angular momentum, both
- 3.00
- Maximum position difference / a
- 1.4142
- Maximum velocity difference / (aω)
- 1.4142
Positions and velocities differ.
Both gaps are dimensionless: position is scaled by a, velocity by aω. Compare corresponding body identities. At 180°, the two outer positions exchange places; the labeled initial state still differs.
Filled bodies: requested reference.
Outlined bodies: shifted comparison.
Dashed segments: position differences.
Check both gaps to decide whether the labeled initial states match.
G = 1; masses (1, 2, 1); a = 1. This is a constructed analytical control, not a measured software failure.
Shift 90°. Maximum position difference / a: 1.4142. Maximum velocity difference / (aω): 1.4142. The initial states differ. Energy and angular momentum are unchanged.
An interlude from Ergentics Music Theory · Recital 025
The art of
returning.
A musician counts the space between entrances. A mathematician asks what returns—and what stays unchanged. Follow two voices through one phrase, and see how a small change of phase transforms their meeting.
The score · one phrase, six seconds
II · Three against two
A strikes at 0, ⅓ and ⅔ of the phrase. B strikes at 0 and ½. Both return to their starting phase at 1.
A
B
Opening · 0½Return · 1
The score is readable without playback. Sound is optional.
The connection
Phase is a useful shared language.
In this score, each hand records a repeating rhythm. In the circular Lagrange and Euler examples above, one rotation carries the whole configuration: every body has the same angular speed. Their radii may differ.
Return to the exact circular families →The distinction
A common language is a place to begin.
Three against two is a musical example, not a frequency ratio assigned to the three gravitating bodies. These clicks are an authored listening aid, not sound from an orbit. The connection invites a question; it does not establish a physical equivalence.
Open the score, equations and source notes
Let u be the fraction of a phrase. The two phases, in turns, are pA = a·u mod 1 and pB = (b·u + δ) mod 1. A strike occurs at an integer turn. The three movements use (a,b,δ) = (1,1,0), (3,2,0), (3,2,½). Display time is six seconds per phrase; it is not physical orbital time.
For movement II, the cycle lengths are two and three cells on a six-cell grid. Their common period is lcm(2,3) = 6. These cycle lengths are distinct from the counts of strikes per phrase. In movement III, B strikes at ¼ and ¾; A still strikes at 0, ⅓ and ⅔. No strike times coincide, although the pair of phases repeats after one phrase.
Adapted from Ergentics Music Theory’s phase-shift and common-period primitives, with reference clicks from its Euclidean-rhythm telescope. This is an original teaching score, not a historical composition or an instrument simulation. Read the source trace and validation boundaries. Save the three-movement score (SVG).
CHAPTER 03 / BOUNDARY
Every guarantee
has a domain.
The paper’s infinite local series covers general distinct-position initial data. Its truncation bound is explicit, but guaranteed only inside a conservative local radius.
READ THE BOUND
Two parts of a guarantee.
At a fixed nonzero time fraction, a higher degree makes this upper bound smaller. Near the edge of the guaranteed interval, the bound grows.
bN is the position bound divided by δ.
η = |t| / T < 1 · N is the truncation degree.
Choose a guarantee
A question the bound can answer.
At η = 0.50, degree 13 is the least degree from 1 to 24 meeting this target. Bound / δ: 6.104e-5.
A sufficient bound under the paper’s assumptions; no measured accuracy or floating-point error guarantee. An unavailable target leaves the selected degree unchanged.
Inside / 0 ≤ η < 1
A finite bound.
For example, η = 0.50 and N = 8 give bound / δ = 1/512. The chart lets you inspect other interior values.
At the edge / η = 1
Outside this guarantee.
The denominator vanishes. No finite value is assigned here. This does not imply that the physical motion collides or that every series diverges there.
Position bound / δ1.953 × 10⁻³
Bound / δ = ηN+1 / [2(1 − η)], where η = |t|/T < 1 and δ is the initial minimum separation divided by 16. Vertical values below 10⁻¹⁵ are clipped on the plot; the readout retains them.
This bound excludes floating-point roundoff. It is not a measured error or a collision time.
Read sample bounds as a table
| Time fraction η | Position bound / δ |
|---|---|
| 0.25 | 2.543e-06 |
| 0.50 | 1.953e-03 |
| 0.75 | 1.502e-01 |
| 0.95 | 6.302e+00 |
In gratitude
Every equation carries
a human history.
To the observers who kept records, the mathematicians who found patterns, the teachers who made difficult ideas approachable, and the builders who gave others new ways to see.
This work stands within that human history. Some contributors are celebrated; many are unnamed. Their care made these questions ours to inherit. We offer this study with respect for what came before, and excitement for what people may discover next.
An original dedication for this project—not a quotation from a historical source.
A few landmarks in a much wider history
An unfinished conversation.
- 1687
Newton
The Principia joins laws of motion and universal gravitation into a mathematical account of celestial motion.
- 1767
Euler
Euler’s published straight-line study is a landmark in the search for special three-body configurations.
- 1772
Lagrange
Equilateral configurations offer another remarkable family, including unequal masses.
- LATE 1800s
Poincaré
Qualitative dynamics brings new ways to understand complexity beyond a single explicit formula.
- PRINTED 1912 · CATALOGUED 1913
Sundman
Convergent-series work expands what an exact representation can mean. The original article carries a 1912 printing notice; the publisher catalogues it as 1913.
- AN OPEN FUTURE
All of us
Better explanations, careful computation, independent review, and new questions carry the work forward.
Selected landmarks, not a complete history or claim of priority. See the reading notes. The classical families belong to the history of mathematics; this project contributes its documented exploration and reference resources.
Read the research
Follow the idea
to its workings.
What the mathematics establishes
Explicit classical circular and nonradial parabolic families, radial collapse and expansion with collision excluded, the fixed-planar-shape boundary, linear instability of the circular symmetric Euler family, and an infinite local series with a convergence radius and truncation bound.
Download the manuscript · Markdown ↓Download the worked reference card ↓
Read the circular and parabolic equations ↓
Read the radial collision-boundary equations ↓
Try the worked collision-boundary check ↓
What the AI study observes
Three original responses under a shared prompt and profile, followed by an unblinded synthesis. A reported intake deviation is retained. The study supports inspection of these outputs, not a model ranking.
The full local record preserves those responses and review history.
What remains open
No general finite closed form, independent mathematical certification, protected examination, numerical trajectory study, or measured learning or deployment benefit is established.
Each special family has its own time domain; radial collision is excluded. The general series is local and infinite. A display check is distinct from a mathematical examination.
About this website edition
Website edition 016 is a visual companion to manuscript v0.7 and its equation supplements 018–019. The included Markdown edition carries its scientific main text, declarations, and reading notes, with internal file links and local operational records removed from the deployable files. The full research record retains the original model responses, configurations, and review history. This page explains the results; the manuscript provides their assumptions and limitations.
Excitement for what comes next
More people able to
see, question & contribute.
The next step can be modest and meaningful: help someone catch a wrong calculation, understand a difficult distinction, or follow an idea all the way back to its evidence.
01
Build trust by checking
Compare an independent implementation with exact reference cases, including examples designed to expose misleading agreement.
02
Make understanding shareable
Pair motion with equations, words, and inspectable states. Then ask whether people learn something they can use elsewhere.
03
Keep the invitation open
Credit inherited knowledge, preserve uncertainty, and give future contributors a clear place to begin.
These are directions for future evaluation. Their educational, accessibility, efficiency, and societal effects have not been measured.
The reading room / trace & acknowledgment
The ideas have a history.
Follow the thread.
Mathematical works
- Newton, I. Philosophiæ Naturalis Principia Mathematica (1687). Latin transcription, Project Gutenberg 28233: Laws I–III; Book III, Proposition VII. Modern transcription inspected; original scan not compared.
- Euler, L. De motu rectilineo trium corporum se mutuo attrahentium. Written 1763; published 1767, pp. 144–151. Euler Archive E327 catalogue. Original PDF access failed; no primary-text verification claimed.
- Lagrange, J.-L. Essai sur le Problème des trois Corps (1772). Œuvres, tome VI, pp. 229–331; Avertissement, pp. 229–230. Inspected transcription, revision 13975608.
- Poincaré, H. “Introduction.” Acta Mathematica 13 (1890), pp. 5–7. Primary PDF. His acknowledgment of Phragmén appears on p. 5. This source is the introduction, not the complete memoir.
- Sundman, K. F. “Mémoire sur le problème des trois corps.” Acta Mathematica 36, 105–179. Primary article: p. 105 printing notice, 8 July 1912; §34, pp. 178–179, final theorem. Publisher catalogue: 1913.
- Marchal, C. The Three-Body Problem. Elsevier (1990). Retained background reference; full text not newly inspected.
Narrative art & human encounter
The Lucas Museum of Narrative Art, founded by George Lucas and Mellody Hobson, is the user’s stated inspiration. Its campus account credits Ma Yansong/MAD and Mia Lehrer/Studio-MLA. Its art and artists page keeps the works and their makers visible. We transfer that care into original mathematical narration and diagrams; no affiliation is implied.
Seeing depth on a screen
Torralba, Isola and Freeman’s Foundations of Computer Vision: Single View Metrology and Stereo Cues. Web rendering context: W3C CSS Transforms Level 2. Motion control: W3C animation-from-interactions guidance.
Closer to the work. This edition reads selected passages in four primary texts, building on the fifteen contextual records traced in edition 012. Euler remains catalogue-only. A source supports the claim beside it; it does not independently certify our derivations. Read the primary-text ledger and precise access limits. Earlier contextual trace.
The companion art. A shared shape and Looking again are original AI-generated interpretations produced with image_gen under Ergentics art direction in 2026. They are not authenticated likenesses or historical paintings. The mathematical diagrams are separate, modern reference figures. Read the art notes and save the originals.
Ergentics, LLC directs this project. Codex supplied mathematical exposition, coding, editorial and image-generation assistance. Classical mathematical contributions retain their historical attribution; the librarians, transcribers and careful readers who make them available are part of this story.