Tag Archives: convex hull

The Compound of the Truncated Isocahedron and the Pentakis Dodecahedron, with Related Polyhedra

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The yellow-and-red polyhedron in the compound below is the truncated icosahedron, one of the Archimedean solids. The blue figure is its dual, the pentakis dodecahedron, which is one of the Catalan solids.

Pentakis dodecahedron and truncated icosahedron

The next image shows the convex hull of this base/dual compound. Its faces are kites and rhombi.

Convex hull of trunctaed icosahedron slash pentakis dodecahedron compound

Shown next is the dual of this convex hull, which features regular hexagons, regular pentagons, and isosceles triangles.

dual of Convex hull of trunctaed icosahedron slash pentakis dodecahedron compound

Next, here is the compound of the last two polyhedra shown.

dual and base compound of Convex hull of trunctaed icosahedron slash pentakis dodecahedron compound

Continuing this process, here is the convex hull of the compound shown immediately above.

Convex hull

This latest convex hull has an interesting dual, which is shown below. It blends characteristics of several Archimedean solids, including the rhombicosidodecahedron, the truncated icosahedron, and the great rhombicosidodecahedron.

Dual of Convex hull

This process could be continued indefinitely — making a compound of the last two polyhedra shown, then forming its convex hull, then creating that convex hull’s dual, and so on.

All these polyhedra were made using Stella 4d: Polyhedron Navigator, which you can purchase (or try for free) at http://www.software3d.com/Stella.php. 

Eight Kite-Rhombus Solids, Plus Five All-Kite Polyhedra — the Convex Hulls of the Thirteen Archimedean-Catalan Compounds

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In a kite-rhombus solid, or KRS, all faces are either kites or rhombi, and there are at least some of both of these quadrilateral-types as faces. I have found eight such polyhedra, all of which are formed by creating the convex hull of different Archimedean-Catalan base-dual compounds. Not all Archimedean-Catalan compounds produce kite-rhombus solids, but one of the eight that does is derived from the truncated dodecahedron, as explained below.

Trunc Dodeca

The next step is to create the compound of this solid and its dual, the triakis icosahedron. In the image below, this dual is the blue polyhedron.

Trunc Dodeca dual the triakis icosahedron

The convex hull of this compound, below, I’m simply calling “the KRS derived from the truncated dodecahedron,” until and unless someone invents a better name for it.

KR solid based on the truncated dodecahedron

The next KRS shown is derived, in the same manner, from the truncated tetrahedron.

KR solid based on the truncated tetrahedron

Here is the KRS derived from the truncated cube.

KR solid based on the truncated cube

The truncated icosahedron is the “seed” from which the next KRS shown is derived. This KRS is a “stretched” form of a zonohedron called the rhombic enneacontahedron.

KR solid based on the truncated icosahedron

Another of these kite-rhombus solids, shown below, is based on the truncated octahedron.

KR solid based on the truncated octahedron

The next KRS shown is based on the rhombcuboctahedron.

KR solid derived from the rhombcuboctahedron

Two of the Archimedeans are chiral, and they both produce chiral kite-rhombus solids. This one is derived from the snub cube.

KR solid based on the snube cube

Finally, to complete this set of eight, here is the KRS based on the snub dodecahedron.

KR solid based on the snub dodecahedron

You may be wondering what happens when this same process is applied to the other five Archimedean solids. The answer is that all-kite polyhedra are produced; they have no rhombic faces. Two are “stretched” forms of Catalan solids, and are derived from the cuboctahedron and the icosidodecahedron:

If this procedure is applied to the rhombicosidodecahedron, the result is an all-kite polyhedron with two different face-types, as seen below.

all kited based on the RID

The two remaining Archimedean solids are the great rhombcuboctahedron and the great rhombicosidodecahedron, each of which produces a polyhedron with three different types of kites as faces.

The polyhedron-manipulation and image-production for this post was performed using Stella 4d: Polyhedron Navigator, which may be purchased or tried for free at http://www.software3d.com/Stella.php.

A Polyhedral Journey, Beginning With an Expansion of the Rhombic Triacontahedron

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The blue figure below is the rhombic triacontahedron. It has thirty identical faces, and is one of the Catalan solids, also known as Archimedean duals. This particular Catalan solid’s dual is the icosidodecahedron.

Rhombic Triaconta

I use a program called Stella 4d (available here) to transform polyhedra, and the next step here was to augment each face of this polyhedron with a prism, keeping all edge lengths the same.

Rhombic Triaconta augmented

After that, I created the convex hull of this prism-augmented rhombic triacontahedron, which is the smallest convex figure which can enclose a given polyhedron.

Convex hull

Another ability of Stella is the “try to make faces regular” function. Throwing this function at this four-color polyhedron above produced the altered version below, in which edge lengths are brought as close together as possible. It isn’t possible to do this perfectly, though, and that is most easily seen in the yellow faces. While close to being squares, they are actually trapedoids.

ch after ttmfr

For the next transformation, I looked at the dual of this polyhedron. If I had to name it, I would call it the trikaipentakis icosidodecahedron. It has two face types: sixty of the larger kites, and sixty of the smaller ones, also.

ch after ttmfr dual

Next, I used prisms, again, to augment each face. The height used for these prisms is the length of the edges where orange kites meet purple kites.

aug ch after ttmfr dual

Lastly, I made the convex hull of the polyhedron above. This convex hull appears below.

Convex hull again

 

Thirty-Four Rotating, Convex, and Non-Chiral Polyhedra with Icosidodecahedral Symmetry

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Most in the field call this type of symmetry “icosahedral,” but I prefer the term George Hart uses — along with “cuboctahedral” in place of “octahedral.”

Each polyhedral image here was created with Stella 4d: Polyhedron Navigator. At this linked page, you can try a trial version of that program for free.

12 icosagons and stuff

Dual of Convex huull

272 faces in 12 reg dec & 120 scalene triangles and 60 isos triangles and 60 isos traps and 20 reg hexagons

Convex hnbghjull

Convex hull 20 8 and 12

By the way, when I described these polyhedra, in this post’s title, as non-chiral, I was not referring to the coloring-schemes used here, many of which obviously are chiral, but only the shapes themselves.

Convex hull LP

convex hull of synthetic comet nucleus

Dual of Convex hull

Convexxx hull

Convhgfehgx hull

That 10 GB space-upgrade, which most bloggers don’t ever need, is really coming in handy right now. In other words, some of these .gif files are huge!

Convhgfex hull

Dual of Chgonvex hulgfl

Dual of Chgonvex hull

Dual of Cjhfjonvex hull

Dualhhc of Convex hull

Why, yes, I am including some words after every fifth polyhedron. That will help, later, when I count them for the title of this post.

Dual of Conjhvex hull

enneagons and kites

Enneagons, Pentagon, isostraps, and rectangles

exansiaon Convex hull

Expanded GRID

I’m not sure why that last one is spinning the opposite direction from the others. Perhaps this polyhedron is trying to start a trend. On the other hand, it could just simply upside-down.

Faceted Stellated Dual of Convex hull

metaexpanded RID

ID variation

intermediate form

jdshfgs

That’s twenty-five so far.

metaRID

pentadecagons two types hexagons and trapezoids

RID variant

RTC and RID blend

Stellated Poly

Clearly, I should have checked the number of files in that file folder before deciding to simply post them all together, based on what they have in common. That’s thirty so far.

sixty pentagons and twenty hexagons

twenty regular dodecagons

Unnabsghfmed

what is this thing

A Polyhedral Journey, Beginning with the Snub Cube / Pentagonal Isositetrahedron Base/Dual Compound

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The snub cube and its dual make an attractive compound. Since the snub cube is chiral, its chirality is preserved in this compound.

Penta Icositetra & snub cube compound

If you examine the convex hull of this compound, you will find it to be chiral as well.

Convex hull of snub cube& dual compound

Here is the mirror image of that convex hull:

Convex hull mirror image

These two convex hulls, of course, have twin, chiral, duals:

dual of Convex hull of snub cube& dual compound

Dual of Convex hull mirror image

The two chiral convex hulls above (the red, blue, and yellow ones), made an interesting compound, as well.

Compound of enantiomorphic pair not dual

This is also true of their chiral duals:

Compound of enantiomorphic pair

I next stellated this last figure numerous times (I stopped counting at ~200), to obtain this polyhedron:

Stellated Compound of enantiomorphic pair dual

After seeing this, I wanted to know what its dual would look like — and it was a nice polyhedron on which to end this particular polyhedral journey.

dual of Stellated Compound of enantiomorphic pair dual

I  make these transformations of polyhedra, and create these virtual models, using a program called Stella 4d. It may be purchased, or tried for free, at http://www.software3d.com/Stella.php.

A Collection of Unusual Polyhedra

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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 (the program I used to make all these images), 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 irregular 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 irregular 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.

A Polyhedral Investigation, Starting with an Augmentation of the Truncated Octahedron

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If one starts with a central truncated octahedron, leaves its six square faces untouched, and augments its eight hexagonal faces with trianglular cupolae, this is the result.

AUGMENTED TRUNCTAED OCTAHEDRON

Seeing this, I did a quick check of its dual, and found it quite interesting:

DUAL OF AUGMENTED TRUNCATED OCTAHEDRON

After seeing this dual, I next created its convex hull.

Convex hull x

After seeing this convex hull, I next creating its dual:  one of several 48-faced polyhedra I have found with two different sets of twenty-four kites as faces, one set in six panels of four kites each, and the other set consisting of eight sets of three kites each. I think of these recurring 48-kite-faced polyhedra as polyhedral expressions of a simple fact of arithmetic: (6)(4) = (8)(3) = 24.

48 KITES AGAiN

I use Stella 4d (available at http://www.software3d.com/Stella.php) to perform these polyhedral transformations. The last one I created in this particular “polyhedral journey” is shown below — but, unfortunately, I cannot recall exactly what I did, to which of the above polyhedra, to create it.

Convex hull OF AUGMENTED CUBOCTAHEDRON

A Cluster of Nine Octahedra, and Related Polyhedra

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If one starts with a central octahedron, then augments each of its eight triangular faces with identical octahedra, this is the result.

9 Octahedra

It is then possible to augment each visible triangle of this cluster with yet more octahedra, which produces this result, in which some octahedra overlap each other.

Meta-9 Octahedra

After making this, I wanted to see its convex hull:  the smallest, tightest-fitting convex polyhedron which can contain a given non-convex polyhedron. (I use Stella 4d: Polyhedron Navigator to perform these manipulations of polyhedra, and this program makes this a fast and easy process. If you’d like to try this software, even as a free trial download, the website to visit is http://www.software3d.com/Stella.php.) Here’s what this convex hull, which bears a resemblance to the rhombcuboctahedron, looks like.

Convex hull of meta-9-octahedron

Looking for previously-unseen, and interesting, polyhedra, I then starting stellating this convex hull. I did find something interesting — to me, anyway — after only two stellations.

2nd stellation of Convex hull of meta-9-octahedron

That concluded my latest polyhedral investigation, but I certainly don’t intend it to be my last.

An Alteration of the Icosahedron/Dodecahedron Compound

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Dual of Convex hull

The dual of the icosahedron is the dodecahedron, and a compound can be made of those two solids. If one then takes the convex hull of this solid, the result is a rhombic triacontahedron. One can then made a compound of the rhombic triacontahedron and its dual, the icosidodecahedron — and then take the convex hull of that compound. If one then makes another compound of that convex hull and its dual, and then makes a convex hull of that compound, the dual of this latest convex hull is the polyhedron you see above.

I did try to make the faces of this solid regular, but that attempt did not succeed.

All of these polyhedral manipulations were were performed with Stella 4d:  Polyhedron Navigator, available at http://www.software3d.com/Stella.php.

A Cluster of Thirteen Rhombic Dodecahedra, and Three Other Related Polyhedra

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13 Rhombic Dodeca

One of the thirteen rhombic dodecahedra in this cluster cannot be seen, for it is hidden in the middle. The other twelve are each attached to a face of the central rhombic dodecahedron.

If one then creates the convex hull of this cluster — the smallest convex polyhedron which can contain it — this is the result:

Convex hull before TTMFR

This polyhedron has fifty faces:  the six square faces of a cube, the eight triangular faces of an octahedron, the twelve rhombic faces of a rhombic dodecahedron, and twenty-four rectangles to fill the gaps between the other faces.

This fifty-faced polyhedron also has an interesting dual, with 48 faces, all of which are kites. Half of these 48 kites are of one type, and arranged into eight panels of three kites each, while the other half are arranged into six panels of four kites each:

48 kites

Returning to the fifty-faced polyhedron, two images above, here is what happens if one tries to make each face as regular as possible:

Unnamed

In this polyhedron, the six squares are still squares, the eight triangles are still regular, and the twelve rhombi are still rhombi, although these rhombi are wider than before. The 24 rectangles, however, have now been transformed into isosceles trapezoids.

[Software credit:  see http://www.software3d.com/Stella.php for more information about Stella 4d, the program I use to make these rotating images. A free trial download is available at that website.]