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It kind of bothers me that the derivation for the Lorentz transformation relies on two dimensions of space. (Here I am referring to the standard derivation where one person is using a vertical light clock in a trolley traveling horizontally at speed v, and an observer outside is watching). My problem is that this derivation relies on the Pythagorean theorem which is an equation only meaningful in (at least) 2 spatial dimensions. So, how can the Lorentz transformation (and the resulting formula for time dilation) be used in one-dimensional derivations (e.g., the 1D relativistic Doppler effect)?