This compound has three parts: two tetrahedra, plus one smaller cube. I made it using Stella 4d: Polyhedron Navigator, which you can try for free at http://www.software3d.com/Stella.php.
In the picture above, each component of this compound has its own color. In the one below, each set of parallel faces is given a color of its own.
These images were made using Stella 4d, software you may try for yourself at this website.
I came across this little beauty while exploring stellations of the triakis octahedron, which is the dual of the truncated cube. Its three components are each eight-faced trapezohedra, and it showed up as the sixth in that stellation-series.
Stella 4d: Polyhedron Navigator was used to make this rotating image. You may try it for free right here.
The 18th stellation of the rhombicosidodecahedron, shown above, is also an interesting compound. The yellow component of this compound is the rhombic triacontahedron, and the blue-and-red component is a “stretched” form of the truncated icosahedron.
This was made using Stella 4d, which you can try for free right here.
The yellow-and-red polyhedron in the compound below is the truncated icosahedron, one of the Archimedean solids. The blue figure is its dual, the pentakis dodecahedron, which is one of the Catalan solids.
The next image shows the convex hull of this base/dual compound. Its faces are kites and rhombi.
Shown next is the dual of this convex hull, which features regular hexagons, regular pentagons, and isosceles triangles.
Next, here is the compound of the last two polyhedra shown.
Continuing this process, here is the convex hull of the compound shown immediately above.
This latest convex hull has an interesting dual, which is shown below. It blends characteristics of several Archimedean solids, including the rhombicosidodecahedron, the truncated icosahedron, and the great rhombicosidodecahedron.
This process could be continued indefinitely — making a compound of the last two polyhedra shown, then forming its convex hull, then creating that convex hull’s dual, and so on.
All these polyhedra were made using Stella 4d: Polyhedron Navigator, which you can purchase (or try for free) at http://www.software3d.com/Stella.php.
I made this rotating virtual model using Stella 4d: Polyhedron Navigator, which you can try for yourself at http://www.software3d.com/Stella.php. This solid is different from most two-part polyhedral compounds because an unusually high fraction of one polyhedron, the yellow octahedron, is hidden inside the compound’s other component.
While examining different facetings of the dodecahedron, I stumbled across one which is also a compound of ten elongated octahedra.
Here’s what this compound looks like with the edges and vertices hidden:
Next, I’ll put the edges and vertices back, but hide nine of the ten components of the compound. This makes it easier to see the single elongated octahedron which is still shown.
Here’s what this elongated octahedron looks like with all those vertices and edges hidden from view.
I made all these polyhedral transformations using Stella 4d, a program you can try for yourself at this website. Stella includes a “measurement mode,” and, using that, I was able to determine that the short edge to long edge ratio in these elongated octahedra is 1:sqrt(2).
The next thing I wanted to try was to make the octahedra regular. Stella has a function for that, too, and here’s the result: a compound of ten regular octahedra.
My last step in this polyhedral exploration was to form the dual of this solid. Since the octahedron’s dual is the cube, this dual is a compound of ten cubes.
I made this using Stella 4d: Polyhedron Navigator. You can try this program for free at http://www.software3d.com/Stella.php.
This can also be called the compound of two truncated tetrahedra.
This image was created using Stella 4d, which you can try at this website.