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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.
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