- The **Bondi k-calculus** (an algebra-based method) develops the basic ideas of special-relativity in $tx$-plane (without using the transverse direction). 
This approach is presented in his book “Relativity and Common Sense” (1962, 1964). In addition, Bondi presented “E=mc2: Thinking Relativity Through”, a series of ten lectures on BBC TV running from Oct 5 to Dec 7, 1963.
(Read more at my contributed article to a blog at https://www.physicsforums.com/insights/relativity-using-bondi-k-calculus/ )

I have used it to develop the "longitudinal light clock" without appealing to the standard textbook "transverse light clock". In particular, I draw the light signals in the longitudinal light clock to form the "**light-clock diamonds**" (the "**causal diamond**" between consecutive tick events).  All light-clock diamonds for all inertial observers have equal area since the Lorentz boost transformation has determinant one.
This is developed using the Bondi k-calculus in my article:<BR> 

*Relativity on rotated graph paper*,<BR> AmJPhy 84, 344 (2016); https://doi.org/10.1119/1.4943251 .<BR> 
An early draft is at https://arxiv.org/abs/1111.7254 .

Rather than use "time-dilation" from the transverse clock,<BR>
I use the Principle of Relativity in the $tx$-plane.

The key figure (using $v=(3/5)c$) is<BR>

[![robphy-RRGP-diamonds][1]][1]

Once the light-clock diamond size is established (using the Bondi k-calculus in the above), the spacelike-diagonal of the light-clock diamond determines
the reflection events on the longitudinal light-clock. So, draw through those
events parallels to the timelike diagonal. This shows length-contraction
as viewed in the lab frame. (The construction can be shown to be symmetric between the observers.)


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 - I have a different approach (that doesn't use the Bondi $k$-calculus directly) which appears in my recent contributed chapter<P>
*Introducing relativity on rotated graph paper*<BR>
Ch 7 in 
**Teaching Einsteinian Physics in Schools**<BR>
Kersting and Blair, Routledge 2021, https://doi.org/10.4324/9781003161721 

I will describe it below.<BR>
You can play with the ideas in this visualization:
https://www.geogebra.org/m/HYD7hB9v#material/UBXdQaz4
<BR>

The key idea is that the diamond **size** is determined by the Principle of Relativity. (The diamond **shape** is determined by the Speed of Light Principle and the velocity of the observer.)
The two observers perform the same experiment
and should expect the same results:<BR> 
**2 seconds after they meet, send a light signal to the other.**

Assuming absolute time and absolute space fails to satisfy the principle, 
but the third configuration works.

Assuming absolute time (so the elapsed times are equal),<BR> 
Alice (red) receives Bob's signal at 3.2, whereas Bob (blue) receives it at 5.<BR>
Bob's ticks need to be scaled up from this size. 
This hints that there is time-dilation... but by how much?<BR>
[![robphy-RRGP-absoluteTime][2]][2]

Assuming absolute space (so the length of the light-clock is the same),<BR> 
Alice (red) receives Bob's signal at 5, whereas Bob (blue) receives it at 3.<BR>
Bob's ticks need to be scaled down from this size. 
This hints that there is length-contraction... but by how much?<BR>
[![robphy-RRGP-absoluteSpace][3]][3]

By playing around (taking a hint from the geometric mean?),<BR>
we get agreement with the Principle of Relativity by each receiving the other signal at 4 ticks.<BR>
<BR>
[![robphy-RRGP-relativity][4]][4]
<BR>
The ratio $(4\mbox{ ticks})/(2\mbox{ ticks})$ is the Doppler factor $k=2$ for $v=(3/5)c$,
where we have used the Principle of Relativity and the Speed of Light Principle...
<BR>
...and, as a consequence, we now know the factor for time-dilation and length-contraction.
<BR>
Try it for $v=(4/5)c$.
<BR>
https://www.geogebra.org/m/kvfsq664 (updated)... make sure BOB's diamonds are shown

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For more information, consult my article and chapter above.<BR>
See also
<BR> https://www.physicsforums.com/insights/spacetime-diagrams-light-clocks/
<BR> https://www.physicsforums.com/insights/relativity-rotated-graph-paper/



  [1]: https://i.sstatic.net/AOhaVm.png
  [2]: https://i.sstatic.net/WIQsdm.png
  [3]: https://i.sstatic.net/9tzrym.png
  [4]: https://i.sstatic.net/IVKczm.png