A Zonish Icosahedron, and Some of Its “Relatives”

To begin this, I used Stella 4d (available here) to create a zonish polyhedron from the icosahedron, by adding zones along the x-, y-, and z-axes. The result has less symmetry than the original, but it is symmetry of a type I find particularly interesting.

zonohedrified icosahedron xyz

After making that figure, I began stellating it, and found a number of interesting polyhedra in this polyhedron’s stellation-series. This is the second such stellation:

zonohedrified icosahedron xyz 2nd stellation

This is the 18th stellation:

zonohedrified icosahedron xyz 18th stellation

The next one, the 20th stellation, is simply a distorted version of the Platonic dodecahedron.

zonohedrified icosahedron xyz 20th stellation

This one is the 22nd stellation:

zonohedrified icosahedron xyz 22nd stellation

This is the 30th stellation:

zonohedrified icosahedron xyz 30th stellation

The next really interesting stellation I found was the 69th:

zonohedrified icosahedron xyz 69th stellation

At this point, I returned to the original polyhedron at the top of this post, and examined its dual. It has 24 faces, all of which are quadrilaterals.

zonohedrified icosahedron xyz dual

This is the third stellation of this dual — and another distorted Platonic dodecahedron.

zonohedrified icosahedron xyz dual 3rd stellation

This is the dual’s 7th stellation:

zonohedrified icosahedron xyz dual 7th stellation

And this one is the dual’s 18th stellation:

zonohedrified icosahedron xyz dual 18th stellation

At this point, I took the convex hull of this 18th stellation of the original polyhedron’s dual, and here’s what appeared:

Convex hull of 18th stellation of dual of zonish icosahedron xyz

Here is this convex hull’s dual:

dual of Convex hull of 18th stellation of dual of zonish icosahedron xyz

Stella 4d, the program I use to make these (available here), has a built-in “try to make faces regular” function. When possible, it works quite well, but making the faces of a polyhedron regular, or even close to regular, is not always possible. I tried it on the polyhedron immediately above, and obtained this interesting result:

spring model of Dual of convex hull of stellation of zonish xyz icosahedron

While interesting, this also struck me as a dead end, so I returned to the red-and-yellow convex hull which is the third image above, from right here, and started stellating it. At the 19th stellation of this convex hull, I found this:

19th stellation of Convex hull of 18th stellation of dual of zonish icosahedron xyz

I also found an interesting polyhedron as the 19th stellation of the dual which is three images above:

19th stellation of dual of Convex hull of 18th stellation of dual of zonish icosahedron xyz

About RobertLovesPi

I go by RobertLovesPi on-line, and am interested in many things. 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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