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

About RobertLovesPi

I go by RobertLovesPi on-line, and am interested in many things, a large portion of which are geometrical. Welcome to my little slice of the Internet. The viewpoints and opinions expressed on this website are my own. They should not be confused with the views of my employer, nor any other organization, nor institution, of any kind.
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3 Responses to A Second Type of Double Icosahedron, and Related Polyhedra

  1. Hi there
    I have just nominated you for a Liebster Award for your blogging.
    To take a look at how to accept and nominate others please see my blog post https://photosociology.wordpress.com/2018/06/24/liebster-award/

    I hope you accept and I look forward to reading your answers.
    Best wishes
    Ricard PhotoSociology.

    Liked by 2 people

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