# Physical Computing

Physical computing is how the physical world is made to run inside a
computer. A waving flag, a strand of hair, smoke curling off a flame, a
bridge under load — each is turned into numbers a machine can step forward
in time.


_Figure 001 — Cloth: a sheet discretized into point masses linked by springs, stepped
forward each frame.
_

_Figure 002 — Fluid: a buoyant smoke plume, advected through an incompressible flow.
_

The central step is **discretization**: a continuous object
becomes a finite set of points, and the laws of physics — forces,
velocities, constraints — become arithmetic a machine repeats millions of
times a second.


Most soft things start as a mass-spring system. Pin point masses together
with springs that pull when stretched and push when squeezed, add gravity
and a little damping, and a flat grid of points begins to drape, swing,
and ripple like real cloth.


_Figure 003 — Finite elements: a truss bridge under a rolling load, members shaded by
tension and compression.
_

Time advances by **integration**. Knowing each point’s
position and the forces on it, you take a small step forward — explicit
Euler is simplest; Verlet and implicit schemes trade cost for stability so
the simulation doesn’t blow up.


Fluids are the hard case. The Navier–Stokes equations govern every flow,
from smoke to breaking waves, and solving them on a grid or with
particles is what turns code into water, fire, and moving air.


_Figure 004 — Hair: strands of linked segments, swaying and settling under gravity.
_

Stiffer matter calls for the **finite element method**: break
a beam or a wing into small cells, solve how each one deforms under load,
and stitch the answers into the stress and strain of the whole structure.


_Figure 005 — Aerodynamics: a spinning cylinder — circulation packs the flow over the
top, lifting it by the Magnus effect.
_

Aerodynamics is physical computing aimed at air. Stream a flow past a
wing, resolve the pressure above and below, and the same equations that
animate smoke predict lift, drag, and whether a shape will fly.


Mathematics, physics, and code meet under a compute budget: you choose
what to approximate — and a good approximation, stepped fast enough, can
be indistinguishable from the real thing.


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_Coming soon — notes for this subject are in progress._