Max Born (1920) and systematically
Paul Gruner (1921) introduced symmetric Minkowski diagrams in German and French papers, where the ct'-axis is perpendicular to the x-axis, as well as the ct-axis perpendicular to the x'-axis (for sources and historical details, see
Loedel diagram). In 1948 and in subsequent papers, Loedel independently rediscovered such diagrams. They were again rediscovered in 1955 by Henri Amar, who subsequently wrote in 1957 in
American Journal of Physics: "I regret my unfamiliarity with South American literature and wish to acknowledge the priority of Professor Loedel's work", along with a note by Loedel Palumbo citing his publications on the geometrical representation of Lorentz transformations. Those diagrams are therefore called "Loedel diagrams", and have been cited by some
textbook authors on the subject. Suppose there are two collinear velocities
v and
w. How does one find the frame of reference in which the velocities become equal speeds in opposite directions? One solution uses modern algebra to find it: Suppose \tanh \ a \ =\ v/c and \tanh \ b \ = \ w/c, so that
a and
b are
rapidities corresponding to velocities
v and
w. Let
m = (
a +
b)/2, the midpoint rapidity. The transformation : z \mapsto z e^{-mj} of the
split-complex number plane represents the required transformation since e^{aj} \mapsto e^{(a-b)j/2} and e^{bj} \mapsto e^{(b-a)j/2}. As the exponents are
additive inverses of each other, the images represent equal speeds in opposite directions. ==Publications==