An irregular tetrahedron

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Question from RAUL:

I am looking an expression for an edge length as function of the other five edge lengths of irregular tetrahedron.

This irregular tetrahedron exhibits the following characteristics:

All its six edge or side lengths are different
There are known the lengths of four of its sides o edges
There is a linear relation between the other two side lengths, like L(i) = K*L(i+1), where:
L(i) = Length of side i
L(i+1) = Length of side i+1
K = constant
L(i) and L(i+1) are opposite sides
L(i) and L(i+1) are the unknowns I am looking for.

This seems to be a difficulty.

The sixth edge is independent of the other five, except in the extreme case(s) where the four points are coplanar. Think of the five sides forming two triangles, hinged along the 'side' opposite the missing side. This hinge is flexible and and rotation the triangles around this hinge changes the missing side length.

There can be no formula - regardless of the relationships among the other five.
(The 2-D analog is trying to find the third side of a triangle from the other two sides.)

Walter Whiteley
York University

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