A GALLERY OF MATHEMATICAL IDEAS / 001

Three bodies.
Centuries of
imagination.

An equation keeps an insight available to another person. This is an invitation to meet the people, shapes and questions behind three bodies in motion—and to look closely enough to see something new.

Enter the gallery

Working paper v0.7 · Website edition 013

Classical exact families and an infinite local series. The general finite closed-form question remains open here.

Exact Lagrange triangle at phase 15 degrees. Bodies 1, 2 and 3 have masses 1, 2 and 3 and rotate on different radii about their common center of mass.
01 / A SHARED CENTERθ = 15°
Different masses. One rotating triangle.The circles trace each body’s path. The cross marks the center of mass. Side length = 1; G = 1.

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.

Frozen Lagrange triangle, orthographic camera tilt 0 degreesErgentics, LLC; edition 013. Phase 15 degrees; masses 1,2,3; a=1; G=1. Physical z=0. Camera X=x, Y=y cos(alpha), Z=y sin(alpha). Symbolic marker sizes; projected lengths are not physical distances. 123
A frozen exact state, seen from a changing angle.Phase 15° · masses 1, 2, 3 · a = 1 · G = 1. Numbered markers have symbolic sizes.

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

Unchanged plan view of the same triangle, for comparison.
Keep the plan beside you.
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
Frozen phase 15°. Physical (x, y, z = 0); camera (X = x, Y, Z). Values in reference units. The camera changes only the projection.
BodyPhysical xPhysical yCamera YCamera Z
1-0.45138-0.56924-0.569240.00000
20.51454-0.31042-0.310420.00000
3-0.192570.396690.396690.00000

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 a convincing match, then examine an infinite local series. It does 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.

CHAPTER 01 / SHAPE

A shape holds.
A world moves.

These positions come directly from a formula. You are changing its phase and parameters, not watching a numerical integration.

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.

Lagrange triangle at phase 0 degrees, with three numbered bodies and their circular paths.
Begin with the shape.
Lagrange triangle at phase 60 degrees, with three numbered bodies and their circular paths.
60°Turn around the shared center.
Lagrange triangle at phase 120 degrees, with three numbered bodies and their circular paths.
120°Keep the same three distances.
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.

Exact Euler solution at phase 30 degrees; the outer bodies are opposite and the middle body is at the center.
2.0

Outer masses = 1. Adjacent gap = 1. G = 1.

30°

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 symmetric Euler family is collision-free for all real time and linearly unstable. An exact orbit need not be stable.

Start with a still view, or take one quarter-turn. Playback targets one cycle per 14 display seconds; a quarter-turn targets 3.5 seconds and pauses. Slow or interrupted rendering can lengthen these durations. Time is measured from phase zero in reference units and returns to zero each cycle. It is not elapsed viewing time. With reduced motion, quarter-turn shows its endpoint immediately.

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. Exact coordinates remain in the table below.

  1. Body 1Teal · mass 1
  2. Body 2Ochre · mass 2
  3. Body 3Violet · mass 1
Inspect positions and velocities
Current exact state; coordinates in the reference plane, z = 0. G = 1.
BodyMassxyvₓvᵧ
11.0−0.86603−0.500000.75000−1.29904
22.00.000000.000000.000000.00000
31.00.866030.50000−0.750001.29904

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.

Read the equation in three parts

The minus and plus signs place the outer bodies on opposite sides of the center. The distance a stays fixed.

r1=au r2=0 r3=au
u=(cosθ,sinθ,0) θ=ωt
The rhythm of the rotationω2=G(m0+m4)a3
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.

CHAPTER 02 / EVIDENCE

Look beyond
matching numbers.

A shifted circular orbit has the same energy and angular momentum—and still misses the requested initial state. Move the shift to see what invariant checks leave out.

What agrees

Energy: −2.25.
Angular momentum: 3.00.
Both orbits satisfy Newton’s equations.

What still needs checking

Do the positions and velocities match the requested state at the requested time?

Solid reference bodies and outlined bodies shifted by a quarter period have different positions, despite matching energy and angular momentum.
90°
Energy, both orbits
−2.25
Angular momentum, both
3.00
Maximum position difference / a
1.4142

Filled bodies: requested reference.
Outlined bodies: shifted comparison.
Dashed segments: position differences.

Both satisfy Newton’s equations. A check of the intended initial positions and velocities detects the mismatch.

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. The initial states differ. Energy and angular momentum are unchanged.

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 time fraction, more terms make this upper bound smaller. Near the edge of the guaranteed interval, the bound grows.

bN(η)=ηN+12(1η)

bN is the position bound divided by δ.
η = |t| / T < 1 · N is the truncation degree.

The analytical truncation bound rises as time approaches the guaranteed local radius. This is a bound, not measured error.
0.50
8
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 excludes floating-point roundoff and does not represent a measured error or a collision time.

Read sample bounds as a table
Horizontal coordinate η = |t| / T; vertical value = bound / δ. Values update with degree N. These are analytical upper bounds, not measured errors.
Time fraction ηPosition bound / δ
0.252.543e-06
0.501.953e-03
0.751.502e-01
0.956.302e+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.

  1. 1687

    Newton

    The Principia joins laws of motion and universal gravitation into a mathematical account of celestial motion.

  2. 1767

    Euler

    Euler’s published straight-line study is a landmark in the search for special three-body configurations.

  3. 1772

    Lagrange

    Equilateral configurations offer another remarkable family, including unequal masses.

  4. LATE 1800s

    Poincaré

    Qualitative dynamics brings new ways to understand complexity beyond a single explicit formula.

  5. 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.

  6. 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 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.

All-time special-family motion is distinct from a general local series. A display check is distinct from a mathematical examination.

About this website edition

Website edition 013 is a visual companion to manuscript v0.7. 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.

How this page is checked · edition 013

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

  1. 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.
  2. 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.
  3. 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.
  4. 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.
  5. 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.
  6. 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.