The Pythagorean theorem alone cannot directly derive the Lorentz transformation formula. The Pythagorean theorem relates to the geometry of right triangles and is not specifically applicable to the Lorentz transformation, which is a fundamental concept in special relativity.
The Lorentz transformation is derived from the principles of special relativity, which involve the constancy of the speed of light and the relativistic effects of time dilation and length contraction. The transformation equations express how space and time coordinates in one inertial reference frame are related to those in another moving reference frame.
The concept of a moving light clock or a tossed ball tracing out a sine wave is not directly related to the derivation of the Lorentz transformation. These examples might be used to illustrate certain aspects of relativistic effects, but the formal derivation of the Lorentz transformation involves mathematical equations, not geometric interpretations of specific objects in motion.
The Lorentz transformation equations involve terms such as time dilation, length contraction, and the relativistic addition of velocities. These equations were derived by Albert Einstein and Hermann Minkowski based on the principles of special relativity, and they provide a mathematical framework for understanding the behavior of space and time in relativistic scenarios.
In summary, while the Pythagorean theorem is a useful geometric concept, it is not directly applicable to deriving the Lorentz transformation or understanding the relativistic effects described by special relativity. The Lorentz transformation is based on the principles of special relativity, which involve the constancy of the speed of light and relativistic effects such as time dilation and length contraction.