Relativity on Rotated Graph Paper (a graphical motivation)

(based mostly on https://www.physicsforums.com/threads/teaching-sr-without-simultaneity.1011051/post-6588952 and https://physics.stackexchange.com/a/689291/148184 )

Within the penultimate draft, I had a non-algebraic motivating argument (which additionally motivates time dilation and size contraction)
that needed to be omitted as a result of the article was already too lengthy.

This argument now seems in Introducing relativity on rotated graph paper (Ch 7), my contribution to a not too long ago revealed e book: Educating Einsteinian Physics in Faculties
Kersting and Blair, Routledge 2021, https://doi.org/10.4324/9781003161721
(has supplementary materials on the backside: https://www.routledge.com/Educating-Einsteinian-Physics-in-Faculties-An-Important-Information-for-Academics/Kersting-Blair/p/e book/9781003161721 )

The storyline… ranging from Einstein’s rules

The storyline goes like this:

• The Pace of Gentle Precept and Bob’s Velocity provides the form of transferring observer Bob’s light-clock diamond (with edges parallel to the sunshine cone, to the rotated graph paper).
  • building:
    Draw Bob’s worldline.
    Bob’s diamonds can have a diagonal alongside Bob’s worldline and edges parallel to the rotated grid.
    The scale of the diamonds corresponds to the spacing of the mirror worldlines, equidistant from and parallel to Bob’s worldline.
    However what determines the scale of the clock diamonds? (What occasion F on Bob’s worldline marks the signal-reflections?)
    

• The Relativity Precept determines the dimension (the scaling) of Bob’s light-clock diamond.
  (That is what I name the “Calibration Downside”.)
• SIGNAL-EXCHANGE EXPERIMENT: Two inertial observers meet at an occasion O.
  2 seconds after they meet, they ship a sign to the opposite.
  We count on they’ve the identical outcomes, in accord with the relativity precept.
  (I might have chosen “1 sec”… however the diagram is extra cluttered.)

An instance: the  ##v=(3/5)c## case

• Take ##v=(3/5)c## for simplicity.
  The makes an attempt:

  [* possible analysis: Since the round-trips are equal in both of the previous cases, we might
  expect the geometric mean ##sqrt{(3.2)(5)}=4## to be the expected result of their signal-exchange experiment.]

Do this for ##v=(4/5)c##.

Play with these utilizing
https://www.geogebra.org/m/HYD7hB9v#materials/UBXdQaz4 (make certain BOB’s diamonds are proven)
https://www.geogebra.org/m/kvfsq664 (up to date)… (make certain BOB’s diamonds are proven)
[You can manually adjust Bob’s velocity and “lengths [in the lab frame]” of the sunshine clocks.]

To acquire the textbook method relating the time-dilation elements and relative velocity, we’d to proceed alongside the strains of my AJP article.

However the level is that we will present (with a geometric building and with out algebra) how the Relativity Precept and the Pace of Gentle Precept indicate the incompatibilities with Absolute Time and Absolute Area, and recommend the necessity for the time-dilation and length-contraction results (as side-effects), on the way in which to establishing the equality of light-clock diamond areas (which is actually the invariance of the sq. interval).

Additional studying:

Relativity on Rotated Graph Paper 
Be taught About Relativity on Rotated Graph Paper
Be taught About Spacetime Diagrams of Gentle Clocks