Boosts preserve the interval
A Lorentz transformation is the rule that converts one inertial observer's coordinates (ct,x,y,z) into another's (ct',x',y',z'). The demand that fixes it is simple: it must leave the interval s^2 unchanged, just as an ordinary rotation leaves length unchanged. For a boost of speed v along the x-axis, the transformation is:
The standard boost, with \beta=v/c and the Lorentz factor \gamma=1/\sqrt{1-\beta^2}. Notice time and space mix — x appears in t' and t appears in x'.
You can check by direct substitution that c^2t'^2-x'^2 = c^2t^2-x^2: the \gamma^2 and cross terms cancel exactly because \gamma^2(1-\beta^2)=1. So a boost really is the spacetime analogue of a rotation. But it is a strange rotation — it does not preserve c^2t^2+x^2 (that would be a circle) but c^2t^2-x^2 (a hyperbola). That is the clue to its true nature.
Rapidity: the hyperbolic angle
An ordinary rotation by angle \theta uses \cos\theta and \sin\theta, and angles simply add. A boost uses hyperbolic functions and an angle called the rapidity \varphi, defined by \tanh\varphi=\beta. Then \gamma=\cosh\varphi and \gamma\beta=\sinh\varphi, and the boost takes exactly the form of a rotation:
A boost is a hyperbolic rotation through rapidity \varphi. Compare an ordinary rotation, which has \cos/\sin and a plus sign where this has \cosh/\sinh and a minus sign.
Relativistic velocity addition falls straight out of rapidity addition and the identity \tanh(a+b)=\frac{\tanh a+\tanh b}{1+\tanh a\tanh b}. Two half-light-speed boosts give $0.8c$, not c.
Reading the effects off the diagram
On a spacetime diagram, the boosted axes tilt: the t'-axis (the moving observer's worldline) rotates toward the light line, and the x'-axis (their `now`) rotates up to meet it. The two axes close like a pair of scissors, always straddling the light cone, which stays fixed because light has \beta=1 for everyone. Slide the boost slider below and watch all three famous effects appear at once.
Relativity of simultaneity is the tilt of the x'-axis: events on it (all `now` for the moving observer) are not horizontal, so they happen at different times for the stationary observer. Time dilation is read where a worldline crosses the invariant hyperbola c^2t^2-x^2=\text{const}: the moving clock's tick reaches the hyperbola `later` in stationary time. Length contraction is the same hyperbola read on the space side. Crucially, all three are reciprocal — each observer sees the other's clocks slow and rulers short — which is only paradox-free because they disagree about simultaneity.
The Lorentz group
The full set of transformations that preserve the interval — boosts in any direction and ordinary spatial rotations — forms the Lorentz group, written O(1,3). Composing two of its members always gives another member; that closure is what `group` means. The subtle part: two boosts in different directions compose into a boost plus a rotation. That leftover rotation is the Thomas precession, and it is not a mathematical curiosity — it produces a real, measurable shift in the spin precession of electrons in atoms.