Relativity (3 of n): The Interval

My explanation of Special Relativity, trying to keep it almost completely geometric, continued from part 2 wherein I describe

• what the Moving People spacetime grid looks like when mapped out by the Stationary People,
  how all we have left to do is figure out their unit spacing, and
  
  
• how there can exist Intermediate People who see the Moving and the Stationary People moving in opposite directions at the same speed.
  
  

We now take a moment to introduce, out of left field, a new definition that will turn out to be very useful:

Give a pair of events, separated by a distance $\mathrm{\Delta z}$ and elapsed time $\mathrm{\Delta t}$, the interval between them is defined as the quantity ${\mathrm{\Delta z}}^2-{\mathrm{\Delta t}}^2$

which looks like a squared distance except for an annoying minus sign,
which seems to depend on whose coordinates we're using (Stationary People vs. Moving People vs. somebody else), but we will now show that:

For any given pair of distinct events, the interval between them is an invariant

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