Why does the travelling twin come back younger?

The travelling twin returns younger because the twins' journeys are not symmetrical. The stay-at-home twin spends the whole time in one inertial frame. The traveller uses two, one going out and one coming back, and switching between them is what tips the balance.

The thought experiment goes back to Einstein's 1905 paper, where he deduced that a clock carried from A to B would lag behind a clock waiting at B. In 1911 Paul Langevin told it as a story: a traveller at 99.995 percent of light speed ages two years and returns to find 200 years have passed on Earth.

Why "paradox"? Each twin sees the other as moving, so a naive application of time dilation suggests each should find the other aged less. Max von Laue was apparently the first to call it a paradox, in 1911.

Take numbers. A ship flies to a star 4 light years away at 0.8 of light speed. On Earth the round trip takes 10 years, and the ship's clocks run at 0.6 of Earth's rate, so the crew ages 6. The crew agrees: in their frame the distance shrinks to 2.4 light years, each leg takes 3 years, 6 in all. Twins born at launch meet again aged 6 and 10.

But on each leg the traveller may equally say that Earth's clocks run slow, at 0.6 of hers: 1.8 years per leg, 3.6 in all. Where are the other 6.4? Relativity has no absolute present. When she switches frames at the turnaround, what counts as "now" back on Earth jumps forward. Her brother never switches, so nothing like that happens on his side. Her acceleration, measurable with an accelerometer, shows who switched, but even with no acceleration, in a relay where two travellers pass and sync clocks, the gap remains. What counts is the path through spacetime: a straight, inertial one collects the most time.

And biological ageing is no different from clock time-keeping: the traveller really is younger.