A Collection of Unusual Polyhedra

In the post directly before this one, the third image was an icosahedral cluster of icosahedra. Curious about what its convex hull would look like, I made it, and thereby saw the first polyhedron I have encountered which has 68 triangular faces.

68 triangles Convex hull

Still curious, I next examined this polyhedron’s dual. The result was an unusual 36-faced polyhedron, with a dozen irregular heptagons, and two different sets of a dozen irregular pentagons.

dual of 68 triangles Convex hull -- this dual has 36 faces including 12 heptagons and 12 each of two types of pentagon

Stella 4d, which is available at http://www.software3d.com/Stella.php, has a “try to make faces regular” function, and I tried to use it on this 36-faced polyhedron. When making the faces regular is not possible, as was the case this time, it sometimes produce surprising results — and this turned out to be one of these times.

dual of the 68-triangle polyhedron after 'try to make faces regular' used

The next thing I did was to examine the dual of this latest polyhedron. The result, a cluster of tetrahedra and triangles, was completely unexpected.

dual of the dual of the 68-triangle polyhedron after 'try to make faces regular' used

The next alteration I performed was to create the convex hull of this cluster of triangles and tetrahedra.

Convex hull of that triangular mess

Having seen that, I wanted to see its dual, so I made it. It turned out to have a dozen faces which are kites, plus another dozen which are irregaular pentagons.

dual of the Convex hull of that triangular mess 12 kites and 12 irregular pentagons

Next, I tried the “try to make faces regular” function again — and, once more, was surprised by the result.

dozen kites and dozen pentagons after 'try to make faces regular' used

Out of curiosity, I then created this latest polyhedron’s convex hull. It turned out to have four faces which are equilateral triangles, a dozen other faces which are isosceles triangles, and a dozen faces which are irregular pentagons.

Convex hull Z

Next, I created the dual of this polyhedron, and it turns out to have faces which, while not identical, can be described the same way:  four equilateral triangles, a dozen other isosceles triangles, and a dozen irregular pentagons — again. To find such similarity between a polyhedron and its dual is quite uncommon.

dual of Convex hull Z

I next attempted the “try to make faces regular” function, once more. Stella 4d, this time, was able to make the pentagons regular, and the triangles which were already regular stayed that way, as well. However, to accomplish this, the twelve other isosceles triangles not only changed shape a bit, but also shifted their orientation inward, making the overall result a non-convex polyhedron.

TTMFR

Having a non-convex polyhedron on my hands, the next step was obvious:  create its convex hull. One more, I saw a polyhedron with faces which were four equilateral triangles, a dozen other isosceles triangles, and a dozen regular pentagons.

Convex hull

I then created the dual of this polyhedron, and, again, found myself looking at a polyhedron with, as faces, a dozen irregaular pentagons, a dozen identical isosceles triangles, and four regular triangles. However, the arrangement of these faces was noticeably different than before.

latest Convex hull

Given this difference in face-arrangement, I decided, once more, to use the “try to make faces regular” function of Stella 4d. The results were, as before, unexpected.

TTMFRA

Next, I created this latest polyhedron’s dual.

TTMFRA dual

At no point in this particular “polyhedral journey,” as I call them, had I used stellation — so I decided to make that my next step. After stellating this last polyhedron 109 times, I found this:

109 stellationsTTMFRA dual

I then created the dual of this polyhedron. The result, unexpectedly, had a cuboctahedral appearance.

Faceted Dual

A single stellation of this latest polyhedron radically altered its appearance.

stellation Faceted Dual

My next step was to create the dual of this polyhedron.

dual Faceted Stellated Poly

This seemed like a good place to stop, and so I did.

About RobertLovesPi

I go by RobertLovesPi on-line, and am interested in many things. The majority of these things 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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