Theorem of Dual Tetrahedron

The dual of a convex geometric solid is obtained by taking the center of gravity for all faces as vertices. The faces of the dual are associated with the original vertices: for each vertex consider the center of adjacent faces and use them to generate a face of the dual. Obviously, the convex hull of these vertices will be used (not the star polygons).

Theorem of Dual Tetrahedron

The dual of a regular tetrahedron has the side length equal to 1/3 of the side length of the original tetrahedron.


Of course, we prefer a visual solution.

First, we construct the dual (red) of a tetrahedron (green).

Secondly, we truncate the green tetrahedron (cutting edges at 1/3 from each vertex). To convert this into a hull body, we have to generate its convex hull and to compact the faces of the resulting object.

And last, we promote the dual to the default Build module and add copies of it to all missing corners of the previously constructed object, thus obtaining a shape similar to the original.