A compound will be vertex-transitive if all the vertexes of its components correspond to the vertexes of a regular polyhedron and will be face-transitive if the plane-faces of its components coincide with the plane-faces of a single regular polyhedron.
Stella Octangula is one of the five regular compounds that exist, because, as we will see:
- The vertexes of its reciprocal tetrahedral correspond to the vertexes of a cube;
- Its faces belong to the plane-faces of a regular octahedron
- The edges of both tetrahedra coincide with the diagonals of the faces of a cube.
On the other hand, Stella Octangula can also be considered as the only stellation of the octahedron.
To understand the stellation of a polyhedron, one must imagine its faces being prolonged symmetrically until they intersect each other, defining new edges and the faces of a new polyhedron. Performing this operation on the regular octahedron, we will obtain two tetrahedra that, intersecting each other, will generate Stella Octangula.
To draw Stella Octangula, we can start with the cube and, drawing its twelve facial diagonals, obtain two regular tetrahedra (this is also the procedure to draw a regular octahedron inside a cube, as the intersection of the diagonals of each face define the six vertex of a regular octahedron).
We may also draw Stella Octangula inside a regular dodecahedron, drawing its edges from eight of its twenty vertexes.
There´s a discussion about Stella Octangula that might also be of interest: four of its vertexes (those that define the edges drawn in black, parallel to the horizontal coordinate plane), correspond to the midpoints of the edges of the large tetrahedra. By this reason, some authors considerer Stella Octangula not as a polyhedron, in the real sense of the word, but a compound polyhedra, because every polyhedron vertex must be defined by the intersection of three edges.
For reasons more or less similar to these, some authors do not consider "Figura de Skilling" as a polyhedron, because some of its edges are defined by the intersection of four faces.
Mathematician Luca Pacioli studied the two tetrahedra set, naming it “Octaedrum Elevatum”, because it can also be obtained by juxtaposing regular tetrahedra to the faces of a regular octahedron. Stella Octangula was drawn by Wentzel Jamnitzer in 1568, in the extraordinary “Perspectiva Corporum Regularium” and later rediscovered by Johannes Kepler in 1619, in “Harmonices Mundi”.
For a better understanding of the previous remarks, in the following orthogonal axonometry, the (visible) edges that Stella Octangula shares with a regular octahedron were drawn in black, one of the tetrahedron in green and the other one in red.
These drawings correspond to the representation of Stella Octangula in Axonometric Orthogonal Projection, beginning with the previous construction of a cube (drawn in dashed line, although not corresponding to any edges invisibility).
In this drawing, we have only Stella Octangula and the axonometric axes:
All construction lines corresponding to each pair of coordinated rebated axes were drawn in different colors: ochre for those corresponding to the frontal coordinate plan and green for the lateral coordinate plan.
The angle between the axonometrical projection of axis x and y measures 120º, while the angle defined by the later with the axonometric projection of z axis varies between 92º and 158º.
Advisable reading:
"Regular Polytopes" - H.S.M. Coxeter
“Tudo o que há num cubo...” - Eduardo Veloso (Educação & Matemática nº 26, 1993)
Stella Octangula in WolframMathworld
Software Stella
Virtual Polyhedra - George Hart
