Consider the following snapshot from a Geometry-Expressions (GX) (http://www.geometryexpressions.com) file:
The blue coordinate-frame lines represent our inertial frame, at rest with respect to us. The red coordinate-frame lines represent another inertial frame, one moving at speed v with respect to us. The red x-axis has slope v (actually v / c^2, but c = 1 here) and the red y-axis has slope 1/v, reflecting the space-time symmetry of the Special Theory of Relativity. B is an event in space-time, that is, a single, fixed point in space and time whose coordinates depend on the frame of reference in which the coordinates are measured. The cyan lines, parallel to the blue axes, measure the coordinates in our frame. The magenta lines, parallel to the red axes, measure the coordinates in the moving frame. Geometry Expressions calculates the Lorentz transformation for us. Minkowski geometry for free. Is that cool, or what? Here's the GX source file: http://home.comcast.net/~brianbec/Minkowski1.gx . (Speed of light = 1, and every place you see v * x, mentally substitute v * x / c^2.)