A Second Type of Double Icosahedron, and Related Polyhedra

After seeing my post about what I called the “double icosahedron,” which is two complete icosahedra joined at one common triangular face, my friend Tom Ruen brought my attention to a similar figure he likes. This second type of double icosahedron is made of two icosahedra which meet at an internal pentagon, rather than a triangular face. Tom jokingly referred to this figure as “a double patty pentagonal antiprism in a pentagonal pyramid bun.”

Augmented Gyroelongated Penta Pyramid one color

It wasn’t hard to make this figure using Stella 4d, the program I use for polyhedral manipulation and image-creation (you can try it for free here), but I didn’t make it out of icosahedra. It was easier to make this figure from gyroelongated pentagonal pyramids, or “J11s” for short. This polyhedron is one of the 92 Johnson solids.

J11.gif

To make the polyhedron Tom had brought to my attention, I simply augmented one J11 with another J11, joining them at their pentagonal faces. Curious about what the dual of this solid would look like, I generated it with Stella.

Augmented Gyroelongated Penta Pyramid dual.gif

The dual of the double J11 appears to be a modification of a dodecahedron, which is no surprise, for the dodecahedron is the dual of the icosahedron.

I next explored the stellation-series of the double J11, and found several attractive polyhedra there. This one is the double J11’s 4th stellation.

Augmented Gyroelongated Penta Pyramid 4th stellation.gif

The next polyhedron shown is the double J11’s 16th stellation.

Augmented Gyroelongated Penta Pyramid 16th stellation.gif

Here is the 30th stellation:

Augmented Gyroelongated Penta Pyramid 30th stellation.gif

I also liked the 43rd:

Augmented Gyroelongated Penta Pyramid 43rd stellation.gif

The next one shown is the double J11’s 55th stellation.

Augmented Gyroelongated Penta Pyramid 55th stellation.gif

Finally, the 56th stellation is shown below. These stellations, as well as the double J11 itself, and its dual, all have five-fold dihedral symmetry.

Augmented Gyroelongated Penta Pyramid 56th stellation

Having “mined” the double J11’s stellation-series for interesting polyhedra, I next turned to zonohedrification of this solid. The next image shows the zonohedron based on the double J11’s faces. It has many rhombic faces in two “hemispheres,” separated by a belt of octagonal zonogons. This zonohedron, as well as the others which follow, all have ten-fold dihedral symmetry.

Zonohedrified Augmented Gyroelongated Penta Pyramid f.gif

Zonohedrification based on vertices produced this result:

Zonohedrified Augmented Gyroelongated Penta Pyramid v.gif

The next zonohedron shown was formed based on the edges of the double J11.

Zonohedrified Augmented Gyroelongated Penta Pyramid e.gif

Next, I tried zonohedrification based on vertices and edges, both.

Zonohedrified Augmented Gyroelongated Penta Pyramid v e.gif

Next, vertices and faces:

Zonohedrified Augmented Gyroelongated Penta Pyramid v f.gif

The next zonohedrification-combination I tried was to add zones based on the double J11’s edges and faces.

Zonohedrified Augmented Gyroelongated Penta Pyramid e f.gif

Finally, I ended this exploration of the double J11’s “family” by adding zones to build a zonohedron based on all three of these polyhedron characteristics: vertices, edges, and faces.

Zonohedrified Augmented Gyroelongated Penta Pyramid v e f.gif

An Offspring of a Dodecahedron and a Tetrahedron

Dodeca tetrahedrally stellated mutliple times

Stellated Dodeca.gif

Stellated Dodeca rb

To make this polyhedron, I first changed the symmetry-type of a dodecahedron from icosahedral to tetrahedral, then stellated it twice. This was done using Stella 4d, a program you may try for free at http://www.software3d.com/Stella.php.

The 43rd Stellation of the Snub Dodecahedron, and Related Polyhedra, Part One

If you stellate the snub dodecahedron 43 times, this is the result. The yellow faces are kites, not rhombi.

Stellated Snub Dodeca refl

Like the snub dodecahedron itself, this polyhedron is chiral. Here is the mirror-image of the polyhedron shown above.

Stellated Snub Dodeca 43rd mirror image

Any chiral polyhedron may be combined with its own mirror-image to create a compound.

Compound of enantiomorphic pair x

This is the dual of the snub dodecahedron’s 43rd stellation.

Stellated Snub Dodeca refl chiral dual

This dual is also chiral. Here is its reflection.

43rd stellation snub dodeca dual reflection

Finally, here is the compound of both duals.

Compound of enantiomorphic pair duals

I used Stella 4d: Polyhedron Navigator to create these images. You may try this program for yourself at http://www.software3d.com/Stella.php.

Selected Stellations of the Truncated Dodecahedron

This is the truncated dodecahedron. It is one of the Archimedean solids.

Trunc Dodeca

This polyhedron has a long stellation-series, from which I selected several on aesthetic grounds. The figure immediately below is the truncated dodecahedron’s 16th stellation.

16th stellation of Trunc Dodeca

Here is the 21st stellation.

21st stellation of Trunc Dodeca

It’s easy to stellate polyhedra rapidly, and make many other changes to them, with Stella 4d: Polyhedron Navigator. You can try it for free at http://www.software3d.com/Stella.php.

25th stellation of Trunc Dodeca.gif

The stellation shown immediately above is the 25th, and the one shown immediately below is the 27th.

27th stellation of Trunc Dodeca

Here is the next stellation: the 28th. Unlike the ones shown above, it is chiral.

28th stellation of Trunc Dodeca.gif

This is the truncated dodecahedron’s 31st stellation.

31st stellation of Trunc Dodeca.gif

This one is the 38th stellation.

38th stellation of Trunc Dodeca.gif

This one is the 44th.

44th stellation of Trunc Dodeca.gif

The last one shown here is called the truncated dodecahedron’s final stellation because, if it is stellated once more, it returns to the original truncated dodecahedron.

Final stellation of Trunc Dodeca

Some Tetrahedral Stellations of the Truncated Cube

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.

Three Stellations of the Truncated Cube

12TH STELLATION OF Trunc Cube.gif

The polyhedron above is the 12th stellation of the truncated cube. The one below is the 14th.

14th-stellation-of-trunc-cube

The next one shown is the 18th and final stellation. If stellated again, the result is an ordinary truncated cube.

18th-and-final-stellation-of-trunc-cube

These virtual models were made using Stella 4d, software you may try for yourself at http://www.software3d.com/Stella.php.