Math CentralQuandaries & Queries


Question from Breon, a student:

what is the formula for a triangular prism?

A triangular prism can be given by a set of inequalities
such as (for a right angled isosceles triangle:)

x >= 0
y >= 0
x+y <= 1
z >= 0
z <= 1

This is the usual way to express the region.

While it is rather artificial, each of these can be expressed as an equality using the absolute value function or something similar:

x = |x|
y = |y|,
1-x-y = |1-x-y|

These can be turned into a single inequality. If any of these is NOT true, the difference of its sides will be nonzero, otherwise (eg) x-|x| will be negative. If we add all these up, the sum will be zero only if all the equalities above hold. The equation

x-|x| + y-|y| + 1-x-y-|1-x-y| + z-|z| + 1-z-|1-z| = 0

is true exactly for (x,y,z) in the prism above. It can be simplified to

2-|x|-|y|-|1-x-y|-|z|-|1-z| = 0

which also works. This can be done - in principle - for any polyhedron.

Similar tricks can be used to obtain an equation that exactly characterizes the boundary of a polyhedron but the expressions will be messier! It is NOT possible to do this using only polynomials; an absolute value function or something similar has to be used.

Good Hunting!

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