Indexing the archive…
Your Universe of Digital Possibilities
Purcell, 1977: at the size of a bacterium, water is thicker than honey feels to you, and inertia is gone — a swimmer that only opens and closes goes nowhere, exactly nowhere, however fast it closes. So bacteria corkscrew. Turn the dial from E. coli to a blue whale and watch the rule hold until a Reynolds number of one, then break: the scallop starts to coast. In the corner, stirred syrup un-stirs when you turn the handle back.
One dimensionless number decides everything: the ratio of inertia to viscosity. Low Re, the fluid moves in smooth sheets; past a critical value (~2000 in a pipe) it shatters into eddies on every scale. The readout above is this number, in the instrument’s own units.
Navier–Stokes with inertia struck out. No time appears: the flow is fixed by the boundary motion at this instant, linear in it, and reversed exactly when the boundary retraces its path.
Under Stokes flow the body’s velocity is linear in the shape’s rate of change, so displacement depends only on the path through shape space, never on speed. A one-hinge stroke goes out and back along the same path — and the displacement cancels exactly.
Each piece of a slender body feels local drag, twice as stiff sideways as lengthways. Demand zero total force and torque and the body’s velocity follows from its shape change alone — the engine behind the scallop, Purcell’s swimmer and the bacterial helix here.
The Lemuel grammar: a quantity as a power of body size. On log–log axes every allometric law is a straight line whose slope is the exponent b — fitted, on every instrument of the edition, from the sizes you visit.
The fourth size turns The Fall’s crossover into a place to live. Below Re = 1 there is no coasting and no momentum: force is velocity, the flow is reversible, and only a stroke that is not its own mirror image gets anywhere — a corkscrew, a travelling wave, Purcell’s two hinges. Above it, the scallop’s jet works and wakes linger. The same reversibility G. I. Taylor demonstrated with syrup is the microscopic law The Arrow builds its asymmetry on; the size at which drops stop being round is The Drop’s.