Creature Physics — From Breathing to Walking

How animals move in Universe · The path from sin() to life

1. Current State

Every creature in Universe animates, but almost none of them move. The distinction matters: animation is a shader uniform feeding sin(uTime); movement is physics — velocity, heading, terrain contact, state transitions. Here is what each animal actually does right now.

Breathing

Universal baseline. sin(uTime * 0.8) * 0.015 scales the body mesh or SDF primitive. Every creature breathes. Nothing else is guaranteed.

all creatures 1 sin() call ~0 cost

Path Following

Lissajous curves for fish and swallows. Elliptical orbits for horses. The path is baked — the creature follows a mathematical curve, never reacting to anything.

fish, swallows, horses sin/cos pairs no awareness

Cave Fish (Most Sophisticated)

Stateful AI with separation avoidance. Fish detect neighbors, maintain minimum distance, freeze when player approaches, resume swimming after a timer. The only creature with a real state machine.

state machine separation player awareness

Horses

3-beat canter kinematics with lofted anatomy. The gait is hardcoded into the horse builder — legs move in a canter pattern, body bobs. But they follow a fixed elliptical path, never touch the ground.

canter gait elliptical path no terrain

Everything Else

Static meshes or simple sin() animation. Sheep sway. Deer stand still. Songbirds perch then fly on a timer. Rabbits hop on a timer. No physics, no awareness, no terrain contact.

static or sin() timer-based no interaction

2. The Physics Library

`shared/creature_physics.mjs`

The physics library provides the substrate for creature movement. Six systems, composable per-creature.

System	What it does	Key function	Cost
Ground following	Sample `terrainH(x,z)` each frame, set creature `y` to terrain height. Smooth with lerp to avoid popping.	`followTerrain(pos, terrainH)`	1 FBM sample/frame
Velocity locomotion	Integrate velocity: `pos += vel * dt`. Heading from `atan2(vel.z, vel.x)`. Speed clamped per species.	`integrate(pos, vel, dt)`	3 adds + 1 atan2
State machine	Finite states: idle, graze, walk, flee. Transitions on timers, proximity triggers, random chance. Each state sets target velocity.	`updateState(creature, dt)`	branch + timer
Boids flocking	Separation, alignment, cohesion with spatial hashing for O(n) neighbor lookup. Hash grid cell size = separation radius.	`flock(creatures, grid)`	O(n) with hash
Catmull-Rom splines	Smooth path through control points. Creatures follow spline with velocity proportional to arc length. Looping or one-shot.	`followSpline(t, points)`	4 lerps per eval
2-bone IK	Analytic IK solver for legs. Given hip position and foot target, compute knee angle via law of cosines. One `acos` call.	`solveIK2(hip, foot, L1, L2)`	1 acos + trig

Physics Pipeline

state machine→ target velocity→ integrate pos→ ground follow→ IK feet→ gait animation→ render

3. The Gait Library

`shared/creature_gait.mjs`

Gaits are sine waves at different phases per leg. That is the entire insight. A walk is four sines 90° apart. A trot is two sines 180° apart. A gallop is four sines with asymmetric timing.

Gait	Beats	Phase pattern	Foot timing	Body motion
Walk	4-beat	LH, LF, RH, RF — 90° apart	`sin(phase + i * π/2)`	Gentle lateral sway
Trot	2-beat diagonal	LH+RF, RH+LF — 180° apart	`sin(phase + i * π)`	Vertical bob, suspension
Canter	3-beat	RH, LH+RF, LF — asymmetric	`sin(phase + [0, 0.33, 0.33, 0.67] * 2π)`	Rolling pitch, lead change
Gallop	4-beat rotary	RH, LH, RF, LF — rapid sequence	`sin(phase + [0, 0.15, 0.5, 0.65] * 2π)`	Deep pitch, full suspension
Hop	2-beat symmetric	Both rear, both front — in phase	`sin(phase), sin(phase + π)`	Vertical launch, arc

Foot Height Curve

During swing phase, foot height follows a sinusoidal lift: y = liftHeight * sin(π * swingT). During stance phase, foot stays on ground. The swingT fraction (0→1 during swing) drives the entire animation.

Body Synchronization

Bob: body center oscillates vertically at stride frequency — sin(stridePhase * 2π) * bobAmplitude
Pitch: body tilts forward during acceleration, back during decel — proportional to d(speed)/dt
Breathing: rate increases with speed — breathRate = baseRate + speed * 0.3

4. Lithos Decomposition

Every creature physics operation decomposes into Lithos primitives. No new math is needed — the existing glyph table covers everything.

Physics	Primitives	Ops
Ground following	`terrainH ⇒ fbm2` (∼)	~40 (FBM)
Heading	`atan2 → ∼ ≃`	~4
Velocity	`+ position * velocity dt`	3
Boids separation	`- pos_i pos_j → diff, √ diff → dist, · diff / dist`	~8 per pair
IK (law of cosines)	`≃ (a²+b²-c²)/(2ab)` — one cos¹ call	~6
Gait timing	`∼(phase × 2π)` — sine at different phases	1 per leg
State machine	Branch + compare — not math, just control flow	~2
Spatial hash	`floor(pos / cellSize)` — quantize to grid	2

The most expensive operation is ground following (FBM = 6 octaves of noise = ~40 ops). Everything else is trivially cheap. The total per-creature per-frame cost is dominated by terrain sampling.

5. Creature Catalog

Animal	Home	Moves	Follows terrain	Has gait	Flocks	State machine
Bull	Taurus	Breathes	No	No	No	No
Horses	Virgo	Elliptical path	No	Canter	No	No
Sheep	Virgo	Sway	No	No	No	No
Cave fish	Cenote	Swims	N/A	N/A	Separation	Freeze/swim
Rabbit	Virgo	Hops	No	Hop	No	Timer-based
Songbirds	Virgo	Perch/fly	No	No	No	Timer-based
Fish schools	Pisces	Lissajous	N/A	Tail wag	Pseudo	No
Deer	Virgo	Static	No	No	No	No
Swallows	Virgo	Lissajous flight	No	No	No	No

The catalog shows the gap clearly: only cave fish have real behavior. Everything else is on rails or static. The physics and gait libraries exist to close this gap systematically.

6. The Path to Life

Phase A — Now

Breathing + Path Following

Every creature breathes via sin(uTime). Some follow fixed curves (Lissajous, ellipse). No terrain contact, no awareness, no state. Cave fish are the exception — separation + freeze state.

Phase B — Libraries

Ground Following + Locomotion + State Machine

Wire creature_physics.mjs into each creature. Terrain sampling via terrainH. Velocity integration. State machines: idle → graze → walk → flee. Creatures touch the ground and respond to proximity.

Phase C — Integration

IK Foot Placement + Gait Animation

Wire creature_gait.mjs into each quadruped. 2-bone IK places feet on terrain. Gait phase drives leg animation. Body bob and pitch sync to stride. Horses walk, trot, canter. Sheep amble. Deer bound.

Phase D — Emergence

Flocking + Inter-Creature Awareness

Boids for sheep herds and bird flocks. Spatial hashing for efficient neighbor queries. Creatures see each other — horses avoid sheep, birds scatter when player approaches, deer flee as a group.

Integration Pipeline

Phase A→ sin(uTime)→ breathe

Phase B→ state machine→ velocity→ terrainH→ ground contact

Phase C→ gait phase→ IK solve→ foot placement→ body bob

Phase D→ spatial hash→ boids→ flock velocity→ emergent herding

7. Connection to ∼

All creature movement is sine. This is not a simplification — it is the literal mathematical content. Every behavior in the creature system reduces to sin() at different frequencies, phases, and amplitudes.

Breathing

sin(t * 0.8) * 0.015 — one sine, one creature, forever

Gait

sin(phase + leg_i * offset) — same sine, different phase per leg. Walk = 4 sines. Trot = 2. Gallop = 4 with asymmetry.

Flocking

Oscillation around flock center — each agent is a damped sine responding to neighbors. The flock IS a superposition of sines.

Ground following

terrainH = fbm(p) = Σ sin(freq_i * p + phase_i) / amp_i — terrain IS sine. Following terrain IS following sine.

Path curves

Lissajous = sin(a*t), sin(b*t). Ellipse = cos(t), sin(t). Every path is a pair of sines.

Body bob

sin(stridePhase * 2π) * amplitude — the body rides a sine wave generated by its own feet

Tail wag

sin(t * wagFreq) * wagAmp — a fish tail is a sine. A dog tail is a sine. Every tail is a sine.

The Lithos glyph ∼ is not just a rendering primitive. It is the atom of life. Breathing is sine. Walking is sine. Flocking is sine. The entire creature physics system is a composition of ∼ at different scales.