# Simple Inequalities in the Unit Cube

Simple Inequalities in the Unit Cube

Permute the triple , , in all six possible ways and form the six corresponding inequalities , , etc. Plot these relations in 3D with to split the unit cube into six congruent right-angled tetrahedra (called "orthoschemes").

x

y

z

x≤y≤z

x≤z≤y

0≤x,y,z≤1

In 2D there are two right-angled triangles, corresponding to and ; in 4D there are 24 simplices corresponding to , etc.

x≤y

y≤x

w≤x≤y≤z