Some Ten-Part Polyhedral Compounds

Posted on 8 June 2018 by RobertLovesPi

While examining different facetings of the dodecahedron, I stumbled across one which is also a compound of ten elongated octahedra.

Here’s what this compound looks like with the edges and vertices hidden:

Next, I’ll put the edges and vertices back, but hide nine of the ten components of the compound. This makes it easier to see the single elongated octahedron which is still shown.

Here’s what this elongated octahedron looks like with all those vertices and edges hidden from view.

I made all these polyhedral transformations using Stella 4d, a program you can try for yourself at this website. Stella includes a “measurement mode,” and, using that, I was able to determine that the short edge to long edge ratio in these elongated octahedra is 1:sqrt(2).

The next thing I wanted to try was to make the octahedra regular. Stella has a function for that, too, and here’s the result: a compound of ten regular octahedra.

My last step in this polyhedral exploration was to form the dual of this solid. Since the octahedron’s dual is the cube, this dual is a compound of ten cubes.

Augmenting the Dodecahedron with Great Dodecahedra

Posted on 7 June 2018 by RobertLovesPi

These two polyhedra are the dodecahedron (left), and the great dodecahedron (right).

Since the faces of both of these polyhedra are regular pentagons, it is possible to augment each of the dodecahedron’s twelve faces with a great dodecahedron. Here is the result.

I used Stella 4d to make these images. You may try this program for yourself at http://www.software3d.com/Stella.php.

Using Rhombic Triacontahedra to Build an Icosidodecahedron

Posted on 4 June 2018 by RobertLovesPi

These two polyhedra are the icosidodecahedron (left), and its dual, the rhombic triacontahedron (right).

One nice thing about these two polyhedra is that one of them, the rhombic triacontahedron, can be used repeatedly, as a building-block, to build the other one, the icosidodecahedron. To get this started, I first constructed one edge of the icosidodecahedron, simply by lining up four rhombic triacontahedra.

Three of these lines of rhombic triacontahedra make one of the icosidodecahedron’s triangular faces.

Next, a pentagon is attached to this triangle.

Next, the pentagonal ring is surrounded by triangles.

More triangles and pentagons bring this process to the half-way point. If we were building a pentagonal rotunda (one of the Johnson solids), this would be the finished product.

Adding the other half completes the icosidodecahedron.

All of these images were created using Stella 4d: Polyhedron Navigator. You may try this program yourself, for free, at http://www.software3d.com/Stella.php. The last thing I did with Stella, for this post, was to put the finished model into rainbow color mode.

Augmenting the Icosahedron with Great Icosahedra

Posted on 3 June 2018 by RobertLovesPi

These two polyhedra are the icosahedron (left), and the great icosahedron (right).

Since the faces of both of these polyhedra are equilateral triangles, it is possible to augment each of the icosahedron’s twenty faces with a great icosahedron. Here is the result.

I used Stella 4d to make these images. You may try this program for yourself at http://www.software3d.com/Stella.php.

Three Archimedean Solids Which Fill Space Together: The Great Rhombcuboctahedron, the Truncated Tetrahedron, and the Truncated Cube

Posted on 2 June 2018 by RobertLovesPi

To start building this space-filling honeycomb of three Archimedean solids, I begin with a great rhombcuboctahedron. This polyhedron is also called the great rhombicuboctahedron, as well as the truncated cuboctahedron.

Next, I augment the hexagonal faces with truncated tetrahedra.

The next polyhedra to be added are truncated cubes.

Now it’s time for another layer of great rhombcuboctahedra.

Now more truncated tetrahedra are added.

Now it’s time for a few more great rhombcuboctahedra.

Next come more truncated cubes.

More great rhombcuboctahedra come next.

More augmentations using these three Archimedean solids can be continued, in this manner, indefinitely. The images above were created with Stella 4d: Polyhedron Navigator, a program you may try for yourself at http://www.software3d.com/Stella.php.

A Compound of an Octahedron and a Pyritohedral Dodecahedron

Posted on 28 May 2018 by RobertLovesPi

compound of a pyritohedral dodecahedron and an octahedron

I made this using Stella 4d: Polyhedron Navigator. You can try this program for free at http://www.software3d.com/Stella.php.

An Offspring of a Dodecahedron and a Tetrahedron

Posted on 26 May 2018 by RobertLovesPi

Dodeca tetrahedrally stellated mutliple times

Stellated Dodeca.gif

Stellated Dodeca rb

To make this polyhedron, I first changed the symmetry-type of a dodecahedron from icosahedral to tetrahedral, then stellated it twice. This was done using Stella 4d, a program you may try for free at http://www.software3d.com/Stella.php.

An Expansion of the Rhombicosidodecahedron

Posted on 6 May 2018 by RobertLovesPi

An expansion of the RID with 122 faces 30 rhombi 60 almostsquares 12 pentagons and twenty triangles

This expanded version of the rhombicosidodecahedron has, as faces, 30 rhombi, 60 almost-square trapezoids, twelve regular pentagons, and twenty equilateral triangles, for a total of 122 faces. I made it using Stella 4d, software you may try for free at http://www.software3d.com/Stella.php.

I call this an “expansion of the rhombicosidodecahedron” because it is similar in appearance to that Archimedean solid. However, it is formed by augmenting the thirty faces of a rhombic triacontahedron with prisms, making the convex hull of the result, and then using Stella‘s “try to make faces regular” function on that convex hull.

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

Posted on 29 April 2018 by RobertLovesPi

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.

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.

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.

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

Here is the KRS derived from 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.

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

The next KRS shown is based on the rhombicuboctahedron.

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

Finally, to complete this set of eight, here is the KRS 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:

all kites based on the cuboctahedron also a stretching of the strombic icositetrahedron which is the dual of the rhombcuboctahedron

all kites based on the icosidodechadedron and a stretching of the strombic hexecontahedron which is the dual of the RID

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

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

all kites based on the great rhombcuboctahedron

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.

Ten Different Facetings of the Rhombicosidodecahedron

Posted on 28 April 2018 by RobertLovesPi

This is the rhombicosidodecahedron. It is considered by many people, including me, to be the most attractive Archimedean solid.

To create a faceted polyhedron, the first step is to get rid of all the faces and edges, leaving only the vertices, as shown below.

In the case of this polyhedron, there are sixty vertices. To create a faceted version of this polyhedron, these vertices are connected by edges in ways which are different than in the original polyhedron. The new positioning of edges defines new faces, often in the interior of the original polyhedron. Here is one such faceting, with the red hexagonal faces in the interior of the now-removed original polyhedron.

The rhombicosidodecahedron can be faceted in many different ways. I don’t know how many possible facetings this polyhedron has, but it is a finite number much larger than the ten shown in this post. Here’s another one.

In faceted polyhedra, many faces intersect other faces, as is the case with the red and yellow faces above. The next faceting demonstrates that faceted polyhedra are sometimes incredibly complex.

Faceted polyhedra can even contain holes that go all the way through the solid, as seen in the next image.

Sometimes, a faceting of a non-chiral polyhedron can be chiral, as seen below. Chiral polyhedra are those which exist in “left-handed” and “right-handed” reflections of each other.

Any chiral polyhedron may be fused with its mirror image to form a compound, and that’s exactly what was done to produce the next image. In addition to being a polyhedral compound, it is also, itself, another faceted version of the rhombicosidodecahedron.

All these polyhedral manipulations and gif-creations were performed using a program called Stella 4d: Polyhedron Navigator. If you’d like to try Stella for yourself, please visit http://www.software3d.com/Stella.php, where a free trial download is available.

The rest of the rhombicosidodecahedron-facetings needed to round out this set of ten are shown below, without further comment.