A Dyson swarm built as a toroidal-shell construction: rings of square solar collectors at strategically chosen inclinations, stacked so their union approximates a thin spherical shell. Each tiny square is a "captor" — a platform converting sunlight into useful work. The blue traces are their orbits.

Ring inclinations follow either the greedy (shared-node meridian) or seashell (Pokrovsky) construction. Try both and watch the Coverage and Overlap numbers in the Data tab.

In 1960, physicist Freeman Dyson proposed that an advanced civilization might surround its star with collectors to capture a significant fraction of its energy output.

The Sun outputs about 3.8×10²⁶ watts continuously. Even capturing a small percentage of this would vastly exceed current human energy production.

Astronomer Nikolai Kardashev proposed classifying civilizations by energy use:

Type I — Energy equivalent to what reaches a planet (~10¹⁶ W)
Type II — Full output of a star (~10²⁶ W)
Type III — Output of an entire galaxy

Earth currently sits at about K = 0.73. The simulation shows how a swarm configuration maps to this scale.

Rings: True ring count, 1 up to the full swarm (N★ = πr/L, typically millions). Slider left half hand-picks 1–50; right half sweeps log-scaled to 100%. Type an exact count in the box. Rendering samples the true set (up to 866 panel-cloud rings + 4,000 orbit guides); physics always uses the true count.

L²: Collector area (km², log scale 1 → 10¹² km², i.e. L from 1 km to 10⁶ km — the top end flagged ILLUSTRATIVE). Each ring packs N = fill·2πr/L squares. The Feasibility pill flips TOO FLIMSY when σ_max drops below the 10 g/m² reference.

Radius: Mean orbital distance from the Sun (0.01–1.0 AU). Inverse-square sets the flux; cube law sets l_min.

Scheme: Ring placement. Greedy = shared-node meridian set (default). Seashell = Pokrovsky spiral.

ΔR: Radial stagger between rings, % of mean r. 0% = single shell; >0 = thin toroidal shell, breaks line-of-sight occlusion.

α_seash: Ascending-node step for seashell scheme. Only active when Scheme = Seashell.

Fill: Per-ring occupancy, 1–100% (linear). 100% = collectors tile the orbit with minimal gaps; 1% = sparse. Since Maxwell–Tisserand depends on N³, fewer collectors per ring relaxes the bound: σ_max scales as 1/fill³.

Where did the σ slider go? Areal density is now an output: the Data tab shows σ_max — the heaviest a panel of side L can be per unit area while its ring stays Maxwell-stable — compared against a 10 g/m² thin-film reference design.

Click the "Orbit" button to ride along with one of the captors.

You'll see the swarm from the perspective of a single collector as it orbits the Sun. Drag to look around, scroll to adjust distance. Click "Orbit" again to return to the overview.

Sun-to-orbit ratio: Accurate.

Captors: Rendered at TRUE physical size. Sub-pixel panels keep their exact light as pixel-coverage alpha (nothing is enlarged); the fidelity note names any active aggregation.

Orbital arrangement: Illustrative. See the Physics tab for details.

Cockpit variants: theme gallery · classic original.

Data tab: Live statistics as you adjust the configuration.

Physics tab: Equations and assumptions behind the calculations.

View source on GitHub

This simulation follows the toroidal-shell construction of Grand Futures §8.7. Rings of square collectors at strategically chosen inclinations approximate a sphere; the union forms a thin toroidal shell of width ΔR. Ring placement uses the greedy (shared-node meridian) or seashell (Pokrovsky) scheme.

Place collectors along a circular orbit of radius r until they form a thin band of width l. Stack rings at successive inclinations; their union approximates a spherical surface. Each ring is geometrically a torus of minor radius ∼l/2 and major radius r — stacked at different inclinations, they tile out the sphere.

Because every shared-node ring passes through the same two points, the construction is "pages of a book" around the node axis: N rings = 2N half-meridians, widest gaps on the axis equator with spacing πr/N. Full coverage therefore needs N★ = πr/L — exactly half the captors-per-ring count 2πr/L, and a factor π/2 more than the naive area bound 2r/L (that surplus is the navel self-shadowing of Fig 8.4). For 10 km collectors at 0.03 AU, N★ ≈ 1.4×10⁶ rings.

Greedy (default). All rings share their ascending node — pages of a book around the node axis — at uniform dihedral spacing. This is the completed version of the greedy build order (ring 1 at 90°, then 45°, 135°, … each into the largest gap): the build sequence converges to uniform spacing, and only the uniform set closes the sky exactly at N★ = πr/L. The "navels" at the shared nodes (Grand Futures Fig 8.4) concentrate overlap.

Seashell (Pokrovsky 1973). Inclinations sweep uniformly while the ascending node rotates by a constant small step α. The pattern spirals around the sphere like a nautilus shell — no navels, slightly smoother shadow distribution at the cost of less aggressive gap-filling per ring.

Why not the golden angle? Plants use 137.5° in phyllotaxy to minimise leaf shadowing on a 2D surface. Dyson rings are 1D great circles on a sphere — the geometry doesn't carry over. Grand Futures (p.436) shows golden-angle is worse than both greedy and seashell for coverage.

Does Tisserand give spacing? No — it bounds mass per element given N. Spacing comes from your packing rule. If you assume gapless, N = 2πr/L ⇒ spacing = L. Sparser rings (smaller N) relax the bound as fill³.

Density as output. Inverting the bound: σ_max = 2.3 M_⊙L / (8π³r³fill³) — the stability ceiling on areal density for panels of side L. Anchor: at 1 AU, L = 272 m, fill = 100% → σ_max = 1.5 g/m² (Grand Futures p.435). The Data tab compares σ_max to the 10 g/m² thin-film reference: FEASIBLE means a real panel that size can fly.

Anchor values. At 1 AU and σ = 1.5 g/m² (light-pressure threshold), l_min = 272 m. At σ = 10 kg/m² (heavy PV), l_min = 1815 km. The lighter the collector, the smaller it can be: statite-density designs allow free-flying meter-scale units; heavy designs require continent-sized panels or active station-keeping. The live Stability pill in the Data tab lights red when L < l_min.

Light pressure cancels gravity at the statite limit: σ = (1+R)L_⊙/(4πcGM_⊙) ≈ 1.5 g/m² for R=1. Below it, v² ≤ 0 — the collector hovers instead of orbits (statite regime). Above it, v approaches Kepler's value asymptotically.

Practical PV (~10 kg/m²) sits squarely in the gravity-dominated regime; thin-film mirrors (~1 g/m²) cross into statite. The sim's orbital periods use the 10 g/m² reference design: light pressure cancels σ_statite/σ_ref = 15% of gravity, slowing orbits ~8% below Kepler — independent of r, since both forces fall as 1/r².

Sphere coverage. Fraction of the sky with ≥1 panel in the line of sight (fill-aware), by Monte Carlo over up to 1024 scheme-true ring normals (band-widened stride sample beyond that) on 4 000 sphere points, recomputed on every geometry change.

Overlap fraction. Fraction of covered solid angle hit by ≥2 rings — a proxy for self-shadowing. At ΔR = 0 every overlap is a wasted collector. ΔR > 0 stages rings into a thin toroidal shell, breaking line-of-sight occlusion at the cost of radial spread.

Reproduces the metric behind Grand Futures Fig 8.5: as ring count grows or scheme changes, coverage rises and overlap depends on scheme + α.

At 0.03 AU: 1361 / 0.03² = 1,512 kW/m² (~1100× Earth)
At 0.05 AU: 1361 / 0.05² = 544 kW/m² (~400× Earth)
At 1.0 AU: 1.36 kW/m² (Earth's orbit)

At 0.03 AU: T ≈ 1,911 K (1638°C) — optimal for thermionic
At 0.05 AU: T ≈ 1,480 K (1207°C) — still viable
Tungsten melts at 3695 K; sublimates significantly above ~2800 K

Thermionic emission: At 1200–1700°C, electrons "boil off" a hot cathode and are collected at a cooler anode, generating current directly. No moving parts.

Efficiency: ~15–25% demonstrated (we use 20%). Soviet TOPAZ reactors flew thermionic converters in 1987–1988.

Why not photovoltaic? Silicon PV degrades above ~200°C. At 0.03 AU (~1650°C), only thermionic or thermophotovoltaic systems survive.

Pure Kepler: 1.9 d at 0.03 AU · 4.1 d at 0.05 AU · 365.25 d at 1 AU.

Note: the sim animates the light-pressure-modified period (see the Light-Pressure section): T = T_Kepler/√(1 − σ_statite/σ_ref) ≈ 1.087 × Kepler at σ_ref = 10 g/m² — 2.06 d at 0.03 AU, 140.3 d at 0.5 AU.

At 0.03 AU with 10 km captors:
Circumference = 2π × 4.49×10⁹ m = 28.2 million km
N_max = 28.2×10⁶ / 10 = 2.82 million captors per ring

The Fill slider controls what fraction of this gapless maximum is populated.

A full Dyson sphere would have f = 1 (100%). This simulation models a swarm—discrete collectors in orbit, not a solid shell.

Even at maximum density, a ring of 10 km captors covers only a thin band of the sphere. Multiple inclined rings increase coverage but remain far below 1%.

Example: 12 rings × 1.4M captors × 100 km² × 1500 kW/m² × 0.20
= 5×10²⁰ W = 500 EW (Kardashev ~1.47)

Type I (K=1.0): 10¹⁶ W — harness all energy reaching a planet
Type II (K=2.0): 10²⁶ W — harness entire stellar output
Earth today: ~18 TW → K ≈ 0.73

The Sun's total luminosity is 3.83×10²⁶ W. A complete Dyson sphere at 100% capture would reach K = 2.0.

Displayed in TWh (terawatt-hours).
1 TWh = 3.6×10¹⁵ J

Earth uses ~28,000 TWh of electricity per year. The "% of Earth Annual" shows how quickly the swarm would match this.

Sun-to-orbit ratio: Rendered to scale. At 0.03 AU, the Sun subtends ~18° (vs 0.5° from Earth).

Captors: True scale, 1:1. A 10 km collector at 0.5 AU is genuinely sub-pixel; its exact pixel coverage is preserved as alpha rather than enlarging it.

Orbital inclinations: Set by the chosen scheme (Greedy / Seashell, see the construction sections above). Inclination span and per-ring values are shown live in the Data tab.

ΔR stagger: Rings are at slightly different radii spanning ΔR·r centred on the mean radius. At ΔR = 0 the swarm is a single shell; at ΔR = 10% it is a fat toroidal shell. Visually subtle in the sim, but breaks line-of-sight occlusion mathematically.

• Sandberg, A. (2023). Grand Futures, §8.7 "Dyson Spheres" — toroidal-shell construction, Maxwell–Tisserand stability, light-pressure orbits, Fig 8.4–8.5.
• Maxwell, J.C. (1859). On the Stability of the Motion of Saturn's Rings. Adams Prize Essay.
• Pokrovsky, G.I. (1973). Spiral ring-spacing scheme cited in Sandberg 2023.
• Dyson, F. (1960). "Search for Artificial Stellar Sources of Infrared Radiation." Science 131(3414).
• Kardashev, N. (1964). "Transmission of Information by Extraterrestrial Civilizations." Soviet Astronomy 8(2).
• Sagan, C. (1973). The Cosmic Connection. K-scale interpolation.