The dual to this cluster-polyhedron appears below. Both virtual models were created using Stella 4d: Polyhedron Navigator, software available here.
A Rhombic Enneacontahedron, Augmented with Sixty Rhombic Dodecahedra, Along with Its Dual
Reply
Tag Archives: polyhedron
A Kite-Faced Polyhedron Based on the Cube, Octahedron, and Rhombic Dodecahedron
Above is the entire figure, showing all three set of kites. The yellow set below, though, lie along the edges of a rhombic dodecahedron.
The next set, the blue kites, lie along the edges of an octahedron.
Finally, the red set of kites lies along the edges of a cube — the dual to the octahedron delineated by the blue kites.
These images were made using Stella 4d, which is available here.
“Fractured” Octahedra
These are variations of the octahedron. I made them all with Stella 4d, which is available here. “Fractured” isn’t an official term, as “truncated or stellated” are, but I can’t come up with a better description, at least not yet. Other suggestions are welcome.
A Variant of the Octahedron Which Features Regular Dodecagons and Quadrilaterals of Three Types
(See here for more information on Stella 4d, the software used to create this image.)
Six Hexagons and Four Triangles, As Faces of a Small Polyhedron
One could call this a half-truncated cube. A fully truncated cube has eight triangular faces, created by truncation, and this has half as many.
(See here for more information on Stella 4d, the software used to create this image.)
A Survey of Polyhedra with Pyritohedral Symmetry
The simplest way for many to understand pyritohedral symmetry is simply to realize that it is the symmetry of the seams in a volleyball. The first time I encountered this unusual symmetry-type was in the golden icosahedron I blogged about here, a figure which much resembles this pyritohedral icosahedron, except the dozen isosceles triangles in this one have a leg-to-base ratio which is not the golden ratio.
Earlier today, I went on a search for polyhedra with pyritohedral symmetry. I found several, but the worthwhile findings from the search are far from exhausted. Here are some others I found, exploring and manipulating polyhedra using Stella 4d, which you can try at this website.
In the version of the pyritohedral icosahedron above, the twelve green triangles have become heptagons which use very short sides to approximate triangles. The one below is of a similar figure, but one in which truncations has happened, so I call it a truncated pyritohedral icosahedron.
There also exist many pyritohedral polyhedra based, more or less, on the cube. These are a few I have found:
Now, is this next one a pyritohedral cube, or a pyritohedral dodecahedron? A case could be made for either, so it inhabits a “gray zone” between varying categories.
Here is a pyritohedral icosidodecahedron:
This one could probably be described in multiple ways, also, but it looks, to me, like a rhombic dodecahedron with its six four-valent vertices being double-truncated in a pyritohedral manner, with pairs of isosceles trapezoids appearing where the truncations took place.
One thing that this one, and the last, have in common is that the largest faces are heptagons. It appears to be a pyritohedral dodecahedron which has been only partially truncated.
This survey could not have been performed without a program called Stella 4d, which I rely on heavily for polyhedral investigations. It may be purchased, or tried for free, at http://www.software3d.com/Stella.php.
If You Have Enough Platonic Dodecahedra Around, and Glue Them Together Just Right, You Can Make a Rhombic Triacontahedron.
Aren’t you glad to know that? As soon as I found out icosahedra can form a rhombic dodecahedron (see last post), I knew this would be true as well. Why? Zome explains why, actually. It’s at http://www.zometool.com. Anything buildable with yellow Zome can be built out of icosahedra. Dodecahedra con form anything buildable with red Zome. Finally, if you can make it with blue Zome, it can be built out of rhombic triacontahedra. It follows that rhombicosidodecahedra can build anything Zome-constructible — but one look at a Zomeball makes that easy to believe, since Zomeballs are modified rhombicosidodecahedra.
Anyway, here’s the rhombic triacontahedron, made of dodecahedra:
[Image created with Stella 4d; see http://www.software3d.com/Stella.php for more info re: this program.]
A Rhombic Dodecahedral Lattice, Made of Icosahedra
I used Stella 4d: Polyhedron Navigator to make this. You can find this program at http://www.software3d.com/Stella.php.
A Compound of Three Square Dipyramids
The reason I am not calling this a compound of three octahedra is that the faces of the dipyramids aren’t quite equilateral. They are, however, isosceles.
This was created with Stella 4d, which you can buy, or try for free, right here.
A Partially-Invisible Rhombicosidodecahedron, and One of Its Stellations
The polyhedron above originally had thirty yellow square faces, but I rendered them invisible so that the interior structure of this polyhedron could be seen.
When stellating such a partially-invisible figure, the new faces “inherited” from the “parent polyhedron” are either visible or invisible, depending on which type of face they are derived from. This makes for a very unusual look for some stellations, such as this, the rhombicosidodecahedron’s 50th:
I created these images using a program called Stella 4d: Polyhedron Navigator. You may try it for yourself at http://www.software3d.com/Stella.php.
RobertLovesPi.net 