Shown above: the compound of the icosahedron and the rhombic dodecahedron. Below is its dual, the compound of the dodecahedron and the cuboctahedron.
Both these compounds were created using the “add/blend polyhedron from memory” function in Stella 4d: Polyhedron Navigator. To check out this program for yourself, just follow this link.
This is the rhombic dodecahedron, the dual of the Archimedean cuboctahedron.
While the rhombic dodecahedron has 12 faces, there are many other polyhedra made entirely out of rhombi, and most of them have more than twelve faces. An example is the rhombic enneacontahedron, which has two face-types: sixty wide rhombi, and thirty narrow ones. It is one of several possible zonohedrified dodecahedra.
As the next figure shows, the wide rhombi of the rhombic enneacontahedron have exactly the same shape as the rhombic dodecahedron’s faces, so the two polyhedra can be stuck together (augmented) at those faces. These wide rhombi have diagonals with lengths in a ratio of one to the square root of two.
The next picture shows what happens if you take one central rhombic enneacontahedron, and augment all sixty of its wide faces with rhombic dodecahedra.
Since this polyhedral cluster in non-convex, it can be changed by creating its convex hull, which can the thought of as pulling a rubber sheet tightly around the entire polyhedron. Here’s the convex hull of the augmented polyhedron above.
The program I use to make these rotating images, Stella 4d (which you can try here), has a function called “try to make faces regular.” If applied to the convex hull above, this function leaves the triangles and pentagon regular, and makes the octagons regular as well. However, the rhombi become kites. The rectangles merely change, getting slightly longer, while rotating 90º, but they do remain rectangles.
After creating this last polyhedron, I started stellating it. After stellating it eight times, I obtained this polyhedron:
Once more, I applied the “try to make faces regular” function.
This polyhedron has five-valent vertices where the shorter edges of the kites meet. These are also the vertices of pentagonal pyramids which use kite-diagonals as base edges. By using faceting (the inverse operation of stellation), I next removed these pyramids, exposing their regular pentagonal faces.
In this polyhedron, all faces are regular, except for the red triangles, which are isosceles. (There are also triangles — the pink ones — which are regular.) Each of these isosceles triangles has ~63.2º base angles, and a ~53.6º vertex angle, with legs just under 11% longer than the base. This is a judgement call, for “near-miss” to the Johnson solids has not been precisely defined, but I see an ~11% edge-length difference as too great for this to be classified as a “near-miss,” even though I would love to claim discovery of another near-miss to the Johnson solids (if it even turns out I am the first one to find this polyhedron, which may not be the case). It is close to being a near-miss, though, so it belongs in the even-less-precisely defined group of polyhedra which are called, quite informally, “near near-misses.”
For the sake of comparison, here is a similar polyhedron (included in Stella 4d‘s enormous, built-in library of polyhedra) which is recognized as a near-miss to the Johnson solids. (I do not know the name of the person who discovered it, or I would include it here — I only know it wasn’t me.) It’s called the “half-truncated truncated icosahedron,” and its longest ledges are just over 7% longer than its shorter edges, with the non-regularity of faces also limited to isosceles triangles. However, this irregularity appears in all of the triangles in the polyhedron below — and in the “near near-miss above,” the irregularity only appeared in some of the triangular faces.
It is well-known that the cuboctahedron and the rhombic dodecahedron are dual polyhedra. However, until I stumbled upon this, I was unaware that rhombic dodecahedra could actually be arranged into a cluster with the overall shape of a cuboctahedron.
[Software credit: see http://www.software3d.com/Stella.php for more information about Stella 4d, the program I use to make these rotating images. A free trial download is available at that website.]
One of the thirteen rhombic dodecahedra in this cluster cannot be seen, for it is hidden in the middle. The other twelve are each attached to a face of the central rhombic dodecahedron.
If one then creates the convex hull of this cluster — the smallest convex polyhedron which can contain it — this is the result:
This polyhedron has fifty faces: the six square faces of a cube, the eight triangular faces of an octahedron, the twelve rhombic faces of a rhombic dodecahedron, and twenty-four rectangles to fill the gaps between the other faces.
This fifty-faced polyhedron also has an interesting dual, with 48 faces, all of which are kites. Half of these 48 kites are of one type, and arranged into eight panels of three kites each, while the other half are arranged into six panels of four kites each:
Returning to the fifty-faced polyhedron, two images above, here is what happens if one tries to make each face as regular as possible:
In this polyhedron, the six squares are still squares, the eight triangles are still regular, and the twelve rhombi are still rhombi, although these rhombi are wider than before. The 24 rectangles, however, have now been transformed into isosceles trapezoids.
[Software credit: see http://www.software3d.com/Stella.php for more information about Stella 4d, the program I use to make these rotating images. A free trial download is available at that website.]
The faces of this polyhedron are decorated with the same type of curvy tessellation seen in the last post here, and it was created using Geometer’s Sketchpad and MS-Paint. Projecting these images onto the faces of this rhombic dodecahedron, in different colors, and then creating this rotating images of it, required a third program, Stella 4d: Polyhedron Navigator. This latter program, an indispensable tool for polyhedral investiagations, may be tried for free, as a trial version, at http://www.software3d.com/Stella.php.
The original polyhedral cluster I built using Stella 4d (available here) is above. Below is its 29th stellation.
And the 30th stellation, as well:
This is the original polyhedral cluster’s dual:
The next image is a variant of the original polyhedral cluster, rendered with only its edges, but not faces or vertices, visible. I wish I could remember exactly how I made this variant, but I simply cannot recall the exact methods I used.
This is the dual of the polyhedron shown immediately above, rendered in the same manner:
This is a compound of the two dual polyhedra right before this sentence.
RobertLovesPi.net 