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.
Zonohedra are a subset of polyhedra with all faces in pairs of parallel and congruent zonogons. Zonogons are polygons with sides which occur only as parallel and congruent pairs of line segments. As a consequence of this, the faces of zonohedra must have even numbers of sides.
Considering all the restrictions on zonohedra, it may be surprising that there is so much variety among them. Every polyhedron shown in this post is a zonohedron. The colors are chosen so that all four-sided zonogons have one color, all six-sided zonogons have a second color, and so on.
I made all of these using Stella 4d: Polyhedron Navigator. This program may be tried for free at this website.
The polyhedron above is the truncated icosahedron, widely known as the pattern for most soccer balls. In the image below, the faces and edges have been hidden, leaving only the vertices.
To make a faceted version of this polyhedron, these vertices must be connected in novel ways, creating new edges and faces. There are many faceted versions of this polyhedron, of which seven are shown below.
I used Stella 4d to make these polyhedral images, and you’re invited to try the program for yourself at http://www.software3d.com/Stella.php.
Symmetrohedra are polyhedra with some form of polyhedral symmetry, all faces convex, and many (but not all) faces regular. Here are four I have found using Stella 4d, a polyhedron-manipulation program you can try for yourself at http://www.software3d.com/Stella.php.
The second of these symmetrohedra is also a zonohedron, and is colored the way I usually color zonohedra, coloring faces simply by number of sides per face. That is why some of the red octagons in that solid are regular, while others are elongated. The other three symmetrohedra are colored by face type, with the modification that the fourth one’s scalene triangles are all given the same color.
These symmetrohedra were all generated using Stella 4d, a program you may try for yourself at http://www.software3d.com/Stella.php.
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.
These two polyhedra are the dodecahedron (left), and the great dodecahedron (right).
Since the faces of both of these polyhedra are regular pentagons, it is possible to augment each of the dodecahedron’s twelve faces with a great dodecahedron. Here is the result.
I used Stella 4d to make these images. You may try this program for yourself at http://www.software3d.com/Stella.php.
These two polyhedra are the icosidodecahedron (left), and its dual, the rhombic triacontahedron (right).
One nice thing about these two polyhedra is that one of them, the rhombic triacontahedron, can be used repeatedly, as a building-block, to build the other one, the icosidodecahedron. To get this started, I first constructed one edge of the icosidodecahedron, simply by lining up four rhombic triacontahedra.
Three of these lines of rhombic triacontahedra make one of the icosidodecahedron’s triangular faces.
Next, a pentagon is attached to this triangle.
Next, the pentagonal ring is surrounded by triangles.
More triangles and pentagons bring this process to the half-way point. If we were building a pentagonal rotunda (one of the Johnson solids), this would be the finished product.
Adding the other half completes the icosidodecahedron.
All of these images were created using Stella 4d: Polyhedron Navigator. You may try this program yourself, for free, at http://www.software3d.com/Stella.php. The last thing I did with Stella, for this post, was to put the finished model into rainbow color mode.
These two polyhedra are the icosahedron (left), and the great icosahedron (right).
Since the faces of both of these polyhedra are equilateral triangles, it is possible to augment each of the icosahedron’s twenty faces with a great icosahedron. Here is the result.
I used Stella 4d to make these images. You may try this program for yourself at http://www.software3d.com/Stella.php.
To start building this space-filling honeycomb of three Archimedean solids, I begin with a great rhombcuboctahedron. This polyhedron is also called the great rhombicuboctahedron, as well as the truncated cuboctahedron.
Next, I augment the hexagonal faces with truncated tetrahedra.
The next polyhedra to be added are truncated cubes.
Now it’s time for another layer of great rhombcuboctahedra.
Now more truncated tetrahedra are added.
Now it’s time for a few more great rhombcuboctahedra.
Next come more truncated cubes.
More great rhombcuboctahedra come next.
More augmentations using these three Archimedean solids can be continued, in this manner, indefinitely. The images above were created with Stella 4d: Polyhedron Navigator, a program you may try for yourself at http://www.software3d.com/Stella.php.
I made this using Stella 4d: Polyhedron Navigator. You can try this program for free at http://www.software3d.com/Stella.php.
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