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# Symmetric transformations

What follows are Lorentz and Ignatowski transformations and their duals with symmetric and vector forms for reference.

For the (3+1) Lorentz transformation there are length spaceaxes x, y, and, z; temporal axis t (time line), velocity v, and maximum velocity c, with β = v/c and γ = 1/√(1 − β²):

The symmetric form is

The symmetric Lorentz transformation for vectors is

For the (1+3) dual Lorentz transformation there are duration space axes x, y, and, z; stantial axis r (stanceline), lenticity u, and maximum lenticity 1/c, with ζ = cu and λ = 1/√(1 − ζ²):

The dual symmetric form is

The dual symmetric Lorentz transformation for vectors is

For the (3+1) Ignatowski transformation there are length space axes x, y, and, z; temporal axis t (time line), velocity v, and maximum velocity V, with β = v/V:

The symmetric form is

For the (1+3) dual Ignatowski transformation there are duration space axes x, y, and, z; stantial axis r (stance line), lenticity u, and maximum lenticity U, with ζ = u/U:

The dual symmetric form is

The Galilean transformation has V → ∞:

The dual Galilean transformation has U → ∞: