Wave 55: Polyhedra Explorer, Fish School 3D & SPH Dam Break

Wave 55 at a glance

🔷 Polyhedra Explorer

Polyhedra — three-dimensional solids bounded by flat polygonal faces — are among the most studied objects in mathematics. The ancient Greeks classified the five Platonic solids: convex polyhedra whose faces are all congruent regular polygons and whose vertices are all equivalent. Archimedes extended this to 13 Archimedean solids: convex polyhedra with two or more types of regular polygon faces but still vertex-transitive.

Euler's polyhedron formula

Leonhard Euler (1752) discovered that for any convex polyhedron — indeed for any simply-connected polyhedral surface homeomorphic to a sphere — the counts of vertices V, edges E, and faces F satisfy:

The formula generalises: for a surface of genus g (a torus has g=1,) the Euler characteristic is 2−2g. A torus satisfies V−E+F=0. This connection between combinatorial properties (vertex/edge/face counts) and topological properties (genus) is a cornerstone of algebraic topology.

The five Platonic solids and their symmetry groups

The Platonic solids correspond to the finite rotation groups of 3D space. Their symmetry groups are the same as those of the rotational symmetries of these solids:

• Tetrahedron — T symmetry (chiral tetrahedral), 12 rotations.
• Cube and Octahedron — O symmetry (octahedral), 24 rotations. These two are duals of each other.
• Dodecahedron and Icosahedron — I symmetry (icosahedral), 60 rotations. Also duals.

There are only five because the requirement for a regular convex solid limits the possible combinations of regular polygon faces meeting at a vertex: the sum of face angles at each vertex must be strictly less than 360°. Only triangles, squares, and pentagons produce valid solutions before this angular deficit runs out.

Perspective projection on Canvas 2D

The simulator uses a pure Canvas 2D perspective projection: each 3D vertex (x, y, z) is first rotated by two quaternion-derived matrices (yaw and pitch, updated each drag tick), then projected to screen with sx = cx + f·X/(Z+d), sy = cy + f·Y/(Z+d) where f is the focal length and d is the camera offset. Faces are painter-sorted by their centroid Z coordinate to give correct depth ordering.

🐟 Fish School 3D — Boids with Predator

In 1986 Craig Reynolds published Boids — a deceptively simple model that produces strikingly lifelike flocking behaviour from just three local rules. No global coordinator directs the flock; each agent reacts only to its neighbours within a perception radius. The emergent result looks indistinguishable from a real starling murmuration or anchovy school.

The three Boids rules

The net steering force at each tick is a weighted sum:

Depth layering for pseudo-3D

Canvas 2D runs a 2D Boids simulation, but three depth layers simulate the three-dimensional structure of a real fish school. Surface fish (layer 0) are drawn larger and in lighter cyan; mid-water fish (layer 1) are medium-sized in ocean blue; deep fish (layer 2) are smaller and dark navy. The layers use slightly different Boids radii so they maintain distinct sub-schools while still influencing each other through the cohesion and alignment radii.

The predator shark

The shark is a single agent with a dedicated hunting behaviour:

• Each tick the shark steers toward the nearest fish within huntRadius.
• Fish within panicRadius (110 px) enter panic mode: their separation weight spikes and they add a strong flee vector directly away from the shark.
• Panicking fish turn yellow — in real anchovy schools, pale coloration under stress is a documented phenomenon possibly related to reduced iridophore activation.
• If the shark catches a fish (distance < 8 px), the fish respawns at a random edge, mimicking the statistical near-constancy of school size despite predation.

Cluster counting

The stats panel reports the number of distinct clusters each frame. Cluster membership is determined by a simple union-find: two fish are in the same cluster if their mutual distance is below the cohesion radius. During normal flocking the count is typically 1–3; during a shark attack it often spikes to 6–10 as the school fragments.

💧 SPH Dam Break — Smoothed Particle Hydrodynamics

Smoothed Particle Hydrodynamics (SPH) was invented simultaneously and independently by Lucy (1977) and Gingold & Monaghan (1977) to simulate astrophysical phenomena such as stellar collisions. Unlike grid-based (Eulerian) methods, SPH is fully Lagrangian: the fluid is discretised into particles that carry mass, velocity, and thermodynamic variables. No mesh is required; the particles themselves define the computational domain.

The SPH approximation

Any field quantity A at position r can be approximated as a weighted sum over neighbouring particle contributions:

Pressure and viscosity forces

First, density is computed for every particle from the Poly6 sum. Then pressure is derived from a simple equation of state:

The pressure force on particle i from all neighbours j is:

Stability: substeps and damping

The simulation runs 3 substeps per animation frame with a fixed dt = 0.004 s. Boundary elastic reflection uses a damping factor of 0.4 so kinetic energy is removed on wall contact — modelling the inelastic nature of real concrete/fluid boundaries. For N ≤ 600 the O(N²) neighbour search runs in a single JS loop at 60 fps on modern hardware; grid-acceleration (spatial hashing) would extend the cap to ~5000 particles.