Indexing the archive…
Your Universe of Digital Possibilities
A mouse’s heart races near 600 beats a minute; a blue whale’s can fall to two. Metabolism scales as M3/4, heart rate as M−1/4, and the two cancel into roughly the same lifetime of heartbeats for most mammals. Turn the dial from below a shrew to a whale: the engine grows a space-filling vascular tree down to the same 4 µm capillary every time, and the exponent you fit is counted out of that tree — then change its branching, and the exponent changes with it.
Basal metabolic rate grows slower than mass and faster than surface: a whale is a hundred million times a mouse’s mass and burns a million times its power. B0 is Kleiber’s ≈ 70 kcal a day per kg3/4.
A warm body loses heat through its skin, so it should burn in proportion to its surface. The exponent Kleiber displaced — and that later re-analyses keep bringing back.
Branch n ways at every level; radii shrink by β, lengths by γ, down to a capillary that is the same in every mammal. Area-preserving (a = ½) and space-filling (b = ⅓) give 1/(2a+b) = 3/4: the number of capillaries — the power — grows as blood volume to the 3/4.
Cardiac output follows metabolism (M3/4); the stroke volume is a fixed share of blood volume (M). Their ratio is the heart rate: a mouse near 600 beats a minute, a whale near ten.
Vessel radius against the viscous skin depth of one heartbeat. Above one, blood travels as a reflected-free pulse wave and area-preserving branches match impedance; below one it is viscous Poiseuille flow, and radii should shrink as n−1/3 instead.
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 third size moves the square–cube race inside the body: what grows with mass is not a surface or a cross-section but a supply network, and its geometry — area kept at every junction, space filled at every level, one capillary size for all — is what fixes the quarter powers. Lifespan here is a consequence, read off the plot: slower hearts, longer lives, about the same count. The same networks, rewired as roads and wires, are The City’s; the branching of trunks instead of arteries is The Tree’s; and where The Hub grows a network by preferential attachment, this one is measured against the body it must fill.