◄ UD0   VI · the hourglass   VII · the standing wave
the purple papers · VI ∞ VII · the fusion

The hourglass wave

The two sheets were always one picture. Here it is fused: a single hourglass, with the standing wave swept right through it. The wave pivots at the neck — the node, dead still — and swings widest at the two bulbs, the bellies, the antinodes. One hourglass is half a wavelength. Stack more, and you climb the harmonics — the compiler's multi-level waists.

the wave (living curve) node · the neck (never moves) antinode · the belly (max swing)

why it pivots at the neck

The hourglass outline is exactly the wave's envelope: half-width = A·|cos(Nπf)|. That's zero at the necks and maximal at the bellies — so the standing wave inside has its nodes at the necks (no motion) and its antinodes at the bellies (full swing). The shape and the wave are the same object.

stack them = climb the modes

One hourglass is a half-wavelength (1 neck, 2 bellies). Two is a full wavelength (2 necks, 3 bellies) — the standing-wave sheet. Drag the slider and you're walking up the harmonics, which is the same move as adding compiler waists: MLIR, the multi-level IR, hourglasses all the way down.