Symmetrohedra are polyhedra with some form of polyhedral symmetry, and many (not necessarily all) regular faces. The first two symmetrohedra here each include four regular enneagons as faces.
The next two symmetrohedra each include four regular dodecagons as faces.
All four of these were made using Stella 4d, which you can try out for free at http://www.software3d.com/Stella.php.
I created these with Stella 4d, which you may try for free at this website. To make a given polyhedral stellation appear larger, simply click on it.
Each of these has a tetrahedron hidden from view in the center.
These were made using Stella 4d, which you may try for yourself here.
Caltrops, when resting on a horizontal surface, have a sharp, narrow point sticking straight up. Stepping on such objects is painful. Most polyhedra do not have such a shape; the most well-known example of an exception to this is the tetrahedron. This fact is well-known to many players of role-playing games, who often use the term “d4” for tetrahedral dice, and who usually try to avoid stepping on them. Here are some other polyhedra which resemble caltrops. All were made using Stella 4d, software available at this website. The first two images may be made larger by simply clicking on them.
The third example, made with the same program, varies this idea somewhat: in physical form, resting on a floor, this caltrop-polyhedron would have three, not just one, potentially foot-damaging “spikes” sticking straight up.
This polyhedron has sixteen faces: four equilateral triangles, and a dozen kites. It was created using Stella 4d, which may be found at http://www.software3d.com/Stella.php.
Mathematicians have discovered more than one set of rules for polyhedral stellation. The software I use for rapidly manipulating polyhedra (Stella 4d, available here, including as a free trial download) lets the user choose between different sets of stellation criteria, but I generally favor what are called the “fully supported” stellation rules.
For this exercise, I still used the fully supported stellation rules, but set Stella to view these polyhedra as having only tetrahedral symmetry, rather than icosidodecahedral (or “icosahedral”) symmetry. For the icosahedron, this tetrahedral symmetry can be seen in this coloring-pattern.
The next image shows what the icosahedron looks like after a single stellation, when performed through the “lens” of tetrahedral symmetry. This stellation extends the red triangles as kites, and hides the yellow triangles from view in the process.
The second such stellation produces this polyhedron — a pyritohedral dodecahedron — by further-extending the red faces, and obscuring the blue triangles in the process.
The third tetrahedral stellation of the icosahedron produces another pyritohedral figure, which further demonstrates that pyritohedral symmetry is related to both icosidodecahedral and tetrahedral symmetry.
The fourth such stellation produces a Platonic octahedron, but one where the coloring-scheme makes it plain that Stella is still viewing this figure as having tetrahedral symmetry. Given that the octahedron itself has cuboctahedral (or “octahedral”) symmetry, this is an increase in the number of polyhedral symmetry-types which have appeared, so far, in this brief survey.
Next, I looked at the fifth tetrahedral stellation of the icosahedron, and was surprised at what I found.
While I was curious about what would happen if I continued stellating this polyhedron, I also wanted to see this fifth stellation’s convex hull, since I could already tell it would have only hexagons and triangles as faces. Here is that convex hull:
For the last step in this survey, I performed one more tetrahedral stellation, this time on the convex hull I had just produced.
To enlarge any individual image, simply click on it.
These polyhedral images were created using Stella 4d: Polyhedron Navigator, a program you can find at http://www.software3d.com/Stella.php, with a free trial download available.
The polyhedron above is a tetrahedrally-symmetric polyhedron featuring regular dodecagons and triangles, as well as two types of trapezoidal faces.
To make this second polyhedron from the first one, I first augmented each dodecagonal face with an antiprism, took the convex hull of the result, and then used the “try to make faces regular” function of the polyhedron-manipulation software I use, Stella 4d, which can be tried for free right here. The result is a polyhedron which maintains tetrahedral symmetry, and has, as faces, regular dodecagons and hexagons, as well as trapezoids and rectangles.
In addition to the four regular enneagons, the polyhedron above also has rhombi and isosceles triangles as faces. The next one, however, adds equilateral triangles, instead, to the four regular enneagons, along with trapezoids and rectangles.
Only the last of these three truly deserves to be called a symmetrohedron, in my opinion, for both its hexagons and enneagons are regular. Only the “bowtie trapezoid” pairs are irregular.
All three of these polyhedra were created using software called Stella 4d: Polyhedron Navigator, which I use frequently for the blog-posts here. You can try it for free at this website.
I’ve been shown, by the program’s creator, a function of Stella 4d which was previously unknown to me, and I’ve been having fun playing around with it. It works like this: you start with a polyhedron with, say, icosidodecahedral symmetry, set the program to view it as a figure with only tetrahedral symmetry (that’s the part which is new to me), and then stellate the polyhedron repeatedly. (Note: you can try a free trial download of this program here.) Several recent posts here have featured polyhedra created using this method. For this one, I started with the snub dodecahedron, one of two Archimedean solids which is chiral.
Using typical stellation (as opposed to this new variety), stellating the snub dodecahedron once turns all of the yellow triangles in the figure above into kites, covering each of the red triangles in the process. With “tetrahedral stellation,” though, this can be done in stages, producing a greater variety of snub-dodecahedron variants which feature kites. As it turns out, the kites appear twelve at a time, in four sets of three, with positions corresponding to the vertices (or the faces) of a tetrahedron. Here’s the first one, featuring one dozen kites.
Having done this once (and also changing the colors, just for fun), I did it again, resulting in a snub-dodecahedron-variant featuring two dozen kites. At this level, the positions of the kite-triads correspond to those of the vertices of a cube.
You probably know what’s coming next: adding another dozen kites, for a total of 36, in twelve sets of three kites each. At this point, it is the remaining, non-stellated four-triangle panels, not the kite triads, which have positions corresponding to those of the vertices of a cube (or the faces of an octahedron, if you prefer).
Incoming next: another dozen kites, for a total of 48 kites, or 16 kite-triads. The four remaining non-stellated panels of four triangles each are now arranged tetrahedrally, just as the kite-triads were, when the first dozen kites were added.
With one more iteration of this process, no triangles remain, for all have been replaced by kites — sixty (five dozen) in all. This is also the first “normal” stellation of the snub dodecahedron, as mentioned near the beginning of this post.
From beginning to end, these polyhedra never lost their chirality, nor had it reversed.
RobertLovesPi.net 