considering two Galilean reference frames, S and S', in standard configuration, as shown in Fig. 2-1. Fig. 3-1a. Draw the and axes of frame S. The axis is horizontal and the (actually ) axis is vertical, which is the opposite of the usual convention in kinematics. The axis is scaled by a factor of so that both axes have common units of length. In the diagram shown, the gridlines are spaced one unit distance apart. The 45° diagonal lines represent the worldlines of two photons passing through the origin at time The slope of these worldlines is 1 because the photons advance one unit in space per unit of time. Two events, and have been plotted on this graph so that their coordinates may be compared in the S and S' frames. Fig. 3-1b. Draw the and axes of frame S'. The axis represents the worldline of the origin of the S' coordinate system as measured in frame S. In this figure, Both the and axes are tilted from the unprimed axes by an angle where The primed and unprimed axes share a common origin because frames S and S' had been set up in standard configuration, so that when Fig. 3-1c. Units in the primed axes have a different scale from units in the unprimed axes. From the Lorentz transformations, we observe that coordinates of in the primed coordinate system transform to in the unprimed coordinate system. Likewise, coordinates of in the primed coordinate system transform to in the unprimed system. Draw gridlines parallel with the axis through points as measured in the unprimed frame, where is an integer. Likewise, draw gridlines parallel with the axis through as measured in the unprimed frame. Using the Pythagorean theorem, we observe that the spacing between units equals times the spacing between units, as measured in frame S. This ratio is always greater than 1, and ultimately it approaches infinity as Fig. 3-1d. Since the speed of light is an invariant, the worldlines of two photons passing through the origin at time still plot as 45° diagonal lines. The primed coordinates of and are related to the unprimed coordinates through the Lorentz transformations and could be approximately measured from the graph (assuming that it has been plotted accurately enough), but the real merit of a Minkowski diagram is its granting us a geometric view of the scenario. For example, in this figure,