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