Simple Inequalities in the Unit Cube

separation

dimensions

2

3

Permute the triple

x

,

y

,

z

in all six possible ways and form the six corresponding inequalities

x≤y≤z

,

x≤z≤y

, etc. Plot these relations in 3D with

0≤x,y,z≤1

to split the unit cube into six congruent right-angled tetrahedra (called "orthoschemes").

In 2D there are two right-angled triangles, corresponding to

x≤y

and

y≤x

; in 4D there are 24 simplices corresponding to

w≤x≤y≤z

, etc.