Devlog #56 – Wave 36: Ant Colony, Band Structure & Lorenz Waterwheel

Release Stats

New Simulations

Ant Colony — Design Notes

ACO Algorithm

Ant Colony Optimization is a probabilistic metaheuristic inspired by the stigmergic behaviour of real ants. Each “ant” in the simulation builds a complete tour through all 16 cities by repeatedly choosing its next city using a roulette-wheel selection weighted by:

• p(i → j) = τijα · ηijβ / ∑ τikα · ηikβ

where τij is the pheromone level on edge (i, j), ηij = 1 / dist(i, j) is the heuristic (inverse distance), and α and β are user-controlled weights. After all ants complete their tours, pheromone evaporates:

• τij ← (1 − ρ) · τij (floored at 0.1 to prevent extinction)

then each ant deposits Q / tour_length on its edges, rewarding shorter paths. Within 50–200 iterations the colony reliably converges to a near-optimal tour.

Pheromone Rendering

Edges are drawn with width and opacity proportional to the normalised pheromone level t = τij / τmax: line width = 0.5 + t × 3.5 px and alpha = t × 0.7 in the emerald colour #10b981. The current best tour is redrawn each frame as a solid #a7f3d0 line at width 2, making the winning path immediately legible against the faint pheromone background.

Controls & Stats

• Ants (10–120): colony size per iteration.
• α (0.5–4): pheromone weight — higher values amplify exploitation of existing trails.
• β (0.5–5): heuristic weight — higher values favour greedy nearest-neighbour moves.
• ρ (0.01–0.3): evaporation rate — higher values forget old routes faster.
• Q (10–200): deposit strength — scales deposit per unit length.
• Speed (1–8×): ACO iterations per animation frame.

Stats panel shows best tour length (px), iteration count, current average tour length, and improvement percentage vs. the initial random tour.

Band Structure — Design Notes

E-k Diagram

The Brillouin zone runs from −π/a to +π/a on the k-axis. For semiconductor materials the simulation draws two valence bands (heavy hole and light hole, both parabolas curving downward from k=0) plus a split-off band shifted −0.3 eV below the valence band top. For indirect-gap materials (Si, Ge) the conduction band minimum is displaced to k ≈ ±0.45π/a; for direct-gap GaAs the minimum sits at k=0.

Fermi-Dirac & Doping

The Fermi level for an intrinsic semiconductor is pinned near mid-gap (E F = Eg/2). The doping slider shifts it linearly ±0.4 eV: “n-type” doping (+) raises E F toward the conduction band, increasing electron concentration; “p-type” doping (−) lowers it toward the valence band, increasing hole concentration. The Fermi-Dirac panel visualises f(E) = 1 / (1 + exp((E − E F) / k T)) as a curve over the same energy range: at 0 K it is a step function; at 1000 K it is broadly rounded.

Density of States

The density-of-states (DOS) panel to the right of the E-k diagram shows a simplified parabolic DOS per energy slice. For semiconductors the valence-band DOS is proportional to √(−E) below 0 eV and the conduction-band DOS is proportional to √(E − Eg) above Eg. Each row pixel is coloured by Fermi-Dirac occupancy: filled states appear blue, empty states appear dark red, and the Fermi-smeared transition region near E F blends between them.

Lorenz Waterwheel — Design Notes

Equations of Motion

Lorenz (1963) derived his famous three-ODE system while modelling Rayleigh–Bénard convection. Malkus & Howard (1972) showed that the same equations describe a leaky waterwheel: if bucket water a_n and b_n are the Fourier amplitudes of the azimuthal mass distribution, and x, y, z are appropriate linear combinations, the system reduces exactly to:

• dx/dt = σ(y − x)
• dy/dt = x(r − z) − y
• dz/dt = xy − bz

where x is proportional to wheel angular velocity, y & z encode the asymmetric water distribution. The integration uses fourth-order Runge-Kutta at dt = 0.005.

Attractor Regimes

The system has three qualitatively different regimes depending on the Rayleigh number r:

• r ≤ 1: single fixed point at the origin — wheel always comes to rest.
• 1 < r < 24.74: two stable non-trivial fixed points — wheel spins steadily in one direction.
• r > 24.74: chaotic attractor — rotation direction reverses unpredictably, tracing the double-lobed butterfly in the x–z phase plane.

The three preset buttons set r = 28 (chaotic), 14 (stable fixed points), and 24.7 (edge of chaos) respectively.

Waterwheel Visualisation

Twelve buckets are equally spaced around a circle of radius 140 px. Each bucket has a fill level (0–1) shown as a blue arc clipped to the bucket circle. Every integration step, water drips into the bucket nearest the top-centre of the wheel, while all buckets leak proportional to their current fill level (mimicking real bucket holes). The wheel rotation angle increments by x × dt × 0.15 each step, so the visual angle is a direct integral of the Lorenz x variable. A green/red arc around the rim indicates the current rotation direction (±x).

Ukrainian Translations

All three simulations are available in Ukrainian at /uk/ant-colony/, /uk/band-structure/, and /uk/lorenz-waterwheel/. Key translated strings include:

• Ant Colony: “Мурашина Колонія”, “Вага феромону”, “Вага відстані”, “Випаровування”, “Осадження”, “Новий граф”, “Скинути”.
• Band Structure: “Зонна структура”, “Валентна зона”, “Зона провідності”, “Заборонена зона”, “Рівень Фермі”, “Легування”, “Власний”.
• Lorenz Waterwheel: “Колесо Лоренца”, “Атрактор Лоренца”, “Число Релея”, “Пауза”/“Грати”, “Скинути”, “Хаотичний”.

What’s Next — Wave 37 Preview

The backlog continues to grow. Candidates for Wave 37 include:

• SVD Compression — Singular Value Decomposition applied to image compression; watch reconstruction quality improve as rank increases (linear algebra / signal processing).
• Z-Pinch — Self-compressing plasma column under a self-generated magnetic field; visualise the Rayleigh–Taylor instability (plasma physics).
• Potential Energy Surface — 2D contour map of a molecular potential energy landscape with reaction path optimisation (physical chemistry / computational chemistry).

All Wave 37 simulations will ship with EN + UK pages on launch day.