A Miscellany of Polyhedra with Icosidodecahedral Symmetry

20 at 5 and 60 at 5 and 20 at 6 and 60 at 7242 faces mostly quadsConvex hull bstarballAll quads 240 facesAugmented DisdyakistriacontaConvex hull with 12 reg icosagonsConvex hulldecagons enneagons bowtie trapezoids stellated several timesDual of Convex hullashgdfasfdual of polyhedron with 12 pentadecagonsDual of Unnamed DualFaceted Dodeca duAL OF FIRST STELLATION OF ICOSAFaceted DodecaFaceted ICOSIDodecaHEDRAL POLYHEDRONStellated Convex hull 2Stellated Convex hullstellated Dual of Convex hullashgdfasfStellated Poly 4Stellated Poly 3.gifZonohedrified Poly.gifUnnamed Dual nc.gifZonohedrified Icosa.gif

All of these polyhedra were made using Stella 4d: Polyhedron Navigator. If you’d like to try this program yourself, simply visit http://www.software3d.com/Stella.php, where a free trial download is available.

Three Views of a Rotating Cluster of 33 Icosidodecahedra

33-icosidodeca

To make these three rotating cluster-polyhedra, I started with one icosidodecahedron in the center, then augmented each of its 32 faces with overlapping, additional icosidodecahedra, for a total of 33 icosidodecahedra per cluster. In the first image, only two colors are used: one for the triangular faces, and another for the pentagons. The second version, however, has the colors assigned by face-type, which is determined by each face’s placement in the overall cluster.

33-icosidodeca-ft

For the third version, I simply put Stella 4d (the program I use to make these images) into “rainbow color mode.” If you’d like to give Stella 4d a try, you can do so for free at this website.

33-icosidodeca-rc

 

A Gallery of Two Dozen Polyhedra with Icosidodecahedral Symmetry, with a Few of Them Chiral

Any of these rotating polyhedra may be made larger with a click. I created them using Stella 4d, a program you may try (as a free trial download) at http://www.software3d.com/Stella.php.

A Tetrahedral Exploration of the Icosahedron

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.

Icosa showing tet symm

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.

Icosa showing tet symm stellation 1

The second such stellation produces this polyhedron — a pyritohedral dodecahedron — by further-extending the red faces, and obscuring the blue triangles in the process.

Icosa showing tet symm stellation 2 pyritohedral dodecahedron

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.

Icosa showing tet symm stellation 3

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.

Icosa showing tet symm stellation 4 an octahedron with 2 face types

Next, I looked at the fifth tetrahedral stellation of the icosahedron, and was surprised at what I found.

Icosa showing tet symm stellation 5

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:

Icosa tet sym stellation 5's 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.

Icosa tet sym stellation 5's Convex hull ist stellation

Four Convex Polyhedra with Icosidodecahedral Symmetry

The smaller images above may be enlarged with a click. All these polyhedra were made using Stella 4d, available here.

Eight Chiral Polyhedra with Icosidodecahedral Symmetry

To see a larger version of any rotating model, simply click on it.

Each of these polyhedral images was created using a program called Stella 4d, which is available here.

Sixteen Convex Polyhedra Featuring Icosidodecahedral Symmetry

To enlarge any single image, simply click on it.

Of the five polyhedra above, all appear to feature decagons. Upon close inspection, though, one of them actually features icosagons — with half their sides very short. Can you spot this polyhedron?

The next set of three polyhedra all feature pentadecagons.

That’s eight so far. Not enough!

Here are eight more, to round out the set of all sixteen, each of which I made using Stella 4d: Polyhedron Navigator. This program may be tried for free at http://www.software3d.com/Stella.php.