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

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

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.

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.

An Expansion of the Rhombic Enneacontahedron with 422 Faces, Together with Its 360-Faced Dual

422 faces expansion of the REC

The polyhedron above had 422 faces and 360 vertices. In dual polyhedra, these numbers are reversed, so the next polyhedra (the dual of the first one) has 360 faces and 422 vertices. Both were created using Stella 4d, available here.

422 faces expansion of the REC the dual with 360 faces

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

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

 

The Snub Dodecahedron and Related Polyhedra, Including Compounds

Snub Dodeca

The dual of the snub dodecahedron (above) is called the pentagonal hexacontahedron (below, left). The compound of the two is shown below, at right. (Any of the smaller images here may be enlarged with a click.)

Like all chiral polyhedra, both these polyhedra can form compounds with their own mirror-images, as seen below.

Finally, all four polyhedra — two snub dodecahedra, and two pentagonal hexacontahedra — can be combined into a single compound.

Compound of enantiomorphic pair and base-dual compound snub dodeca

This polyhedral manipulation and .gif-making was performed using Stella 4d, a program you can find here.

A Torus and Its Dual, Part II

After I published the last post, which I did not originally intend to have two parts, this comment was left by one of my blog’s followers. My answer is also shown.

torus talk

A torus can be viewed as a flexible rectangle rolled into a donut shape, and I had used 24 small rectangles by 24 small rectangles as the settings for Stella 4 for the torus, and its dual, in the last post — which, due to the nature of that program, are actually rendered as toroidal polyhedra. To investigate my new question, I increased 24×24 to 90×90, and these three images show the results. The first shows a 90×90 torus, the second shows its dual, and the third shows the compound of the two.

Torus90.gif

 

Torus90dual

Torus90dualcompound

When I compare these images to those in the previous post, it is clear that these figures are approaching a limit as n, in the expression “nxn rectangle,” increases. What’s more, I recognize the dual now, of the true torus, at the limit, as n approaches infinity — it’s a cone. It’s not a finite-volume cone, but the infinite-volume cone one obtains by rotating a line around an axis which intersects that line. This figure, not a finite-volume cone, is the cone used to define the conic sections: the circle, ellipse, parabola, and hyperbola.

What’s more, I smell calculus afoot here. I do not yet know enough calculus.

“Learn a lot more about calculus” is definitely on my agenda for the coming Summer, for several reasons, not the least of which is that I plainly need it to make more headway in my understanding of geometry. 

[Note: Stella 4d, the program used to make these images, may be found at http://www.software3d.com/Stella.php.]