Fractal Figure Explained with Construction Steps

A “fractaled” regular tetrahedron may be built of drinking straws or by using paper cutout patterns per Hopley (1994) and others.  These directions involve paper cutout patterns.

(1)  The initial regular tetrahedron was made from a paper cutout pattern with each edge four inches long. (2)  Four more smaller tetrahedra were made in the same way, with each edge being two inches.  These second-stage tetrahedra were attached to the faces of the initial tetrahedron, with the vertices of the smaller attached to the midpoints of the edges of the larger. 

A solid emerges, with eight vertices (which eventually become the vertices of the cube), and with 24 faces, all of which are equilateral triangles.  This figure has been given the name “stellar octangula”, and has been explored extensively.

(3)  Twenty-four more regular tetrahedra are now made, each with each edge one inch.  The vertices of these are attached to the midpoints of the edges of the 24 triangular faces.  The shape of the cube as a limiting figure becomes obvious at this third stage.

First generation is the large gold regular tetrahedron in the middle.  On its right side is a maroon second-generation regular tetrahedron, and two third-generation regular tetrahedra are on the right.