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What are some alternative ways of describing a Tetrahedron rather than using 4 points?

This question made me wonder besides using the four points following methods could also be used to describe a Tetrahedron :

1.Intersection of 4 cubes ( or some other solids having at least 4 flat surfaces in total between them)

2.Set of points that satisfy some inequities that is bounded between 4 planes.

But still the 4 four point description of Tetrahedron seems by far the simplest.

Is there a list known alternative methods used for prescribing a 2D or 3D object? (varying the coordinate systems does not count)

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People in CAD (and other fields) spend their lives figuring out how to represent 2D and 3D objects in computer systems. For 3D objects, the discipline is often called "solid modeling". The wikipedia article at includes a list of representation schemes. Your original 4-point scheme is a form of "boundary representation". Your intersection idea is a form of "CSG" representation.

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You can use its tetrahedral symmetry, which is further invariant under coordinate changes.

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