Functional Equation of Rectangular Graphs

The equation $$ \max\{|x_1|, |x_2|, \dots, |x_d|\} = 1 $$ gives us a hypercube of side length $2$ centered at the origin. This can be simplified to an expression with absolute values only, but a messy one: we have $\max\{A,B\} = \frac{A+B +|A-B|}{2}$, and we can chain this together to write a max of $d$ values using absolute values. When $d=2$, we have a nicer expression with absolute values: $$|x_1 + x_2| + |x_1 - x_2| = 2.$$ This gives us a square of side length $2$ centered at $(0,0)$

By replacing each $x_i$ with some linear function of $x_i$, we can move the hypercube (or square) to another center, and scale the sides independently, so we get an arbitrary cuboid (or rectangle).