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

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.


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


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.


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.


A Cluster-Polyhedron Formed By 15 Truncated Octahedra, Plus Variations

15 Trunc Octa

To form the cluster-polyhedron above, I started with one truncated octahedron in the center, and then augmented each of its fourteen faces with another truncated octahedron. Since the truncated octahedron is a space-filling polyhedron, this cluster-polyhedron has no gaps, nor overlaps. The same cluster-polyhedron is below, but colored differently:  each set of parallel faces gets a color of its own.

15 Trunc Octa color by face unless parallel

This is the cluster-polyhedron’s sixth stellation, using the same coloring-scheme as in the last image:

15 Trunc Octa color by face unless parallel 6th stellation

Here’s the sixth stellation again, but with the coloring scheme that Stella 4d:  Polyhedron Navigator (the program I use to make these images) calls “color by face type.” If you’d like to try Stella for yourself, you can do so here.

15 Trunc Octa color by face type 6th stellation

Also colored by face-type, here are the 12th, 19th, and 86th stellations.

15 Trunc Octa color by face type 12th stellation

15 Trunc Octa color by face type 19th stellation

15 Trunc Octa color by face type 86th stellation

Leaving stellations now, and returning to the original cluster-polyhedron, here is its dual.

15 Trunc Octa dual

This image reveals little about this dual, however, for much of its structure is internal. So that this internal structure may be seen, here is the same polyhedron, but with only its edges visible.

15 Trunc Octa dual wirre-frame

Finally, here is an edge-rendering of the original cluster-polyhedron, but with vertices shown as well — just not the faces.

15 Trunc Octa wirre-frame

Zonohedron Based On the Edges and Vertices of a Great Rhombcuboctahedron


Zonohedron Based On the Edges and Vertices of a Great Rhombcuboctahedron

This polyhedral monster has 578 faces of 26 types. In the image above, hexagons of any type are red, rhombi of any type (including squares) are yellow, and the blue faces are octagons. If each face-type is given a different color, though, this zonohedron looks like this:

Zonohedrified Trunc Cubocta

Another coloring-scheme — the best one, in my opinion — is like the first one here, except that regular hexagons are given their own color (purple), and squares are given their own as well (black):

Zonohedrified Trunc Cubocta 

All three images were created with Stella 4d, software available at http://www.software3d.com/Stella.php.

Octahedral Lattice of Truncated Octahedra, Meeting At Their Square Faces


Octahedral Lattice of Truncated Octahedra, Meeting At Their Square Faces

The second image here resulted from stellating the first one many times. It can be enlarged with a click.

sTELL Augmented Trunc Octa

The software used to create these rotating images, Stella 4d, may be tried for free at http://www.software3d.com/Stella.php.

A Fifty-Faced, Zonohedrified Form of the Truncated Octahedron


A Zonohedrified Form of the Truncated Octahedron

This zonohedron has fifty faces:

  • 6 regular octagons
  • 8 regular hexagons
  • 24 squares
  • 12 equilateral octagons, the only irregular polygons needed as faces of this polyhedron

(Image created with Stella 4d — software you can try yourself at http://www.software3d.com/Stella.php.)

A Close-Packing of Space, Using Three Different Polyhedra


A Close-Packing of Space, Using Three Different Polyhedra

This is like a tessellation, but in three dimensions, rather than two. The pattern can be repeated to fill all of space, using cubes (yellow), truncated octahedra (blue), and great rhombcuboctahedra, also known as truncated cuboctahedra (red).

Software credit: see www.software3d.com/stella.php to try or buy Stella 4d, the software I used to create this image.

A Cubic Arrangement of Truncated Octahedra


A Cubic Arrangement of Truncated Octahedra

This cubic arrangement of eight truncated octahedra has a hole in the center, and indentations in the center of each face of the cube. What would fit in these gaps? More truncated octahedra of the same size, that’s what. This wouldn’t be true for most polyhedra, but the truncated octahedron is unusual in that it can fill space without leaving gaps — much like hexagons can tile a plane, but in three dimensions.

Stella 4d was used to create this image, and you may try it for free at http://www.software3d.com/stella.php.

Cluster of Fifteen Decorated and Tightly-packed Truncated Octahedra


Cluster of Fifteen Decorated and Tightly-packed Truncated Octahedra

This cluster was made of the same polyhedron from the previous post, repeated in a space-filling pattern, similar to a tessellation, but in three dimensions. The truncated octahedron has the property, unusual among polyhedra, that it can fill space without leaving any gaps. One of the fifteen truncated octahedra is i the center of the cluster, while another is attached to each of the central polyhedron’s fourteen faces.

Software used to create this includes three separate programs: Geometer’s Sketchpad, MS-Paint, and Stella 4d. This third program may tried for free, and/or purchased, at http://www.software3d.com/stella.php.

Truncated Octahedron Carousel


Truncated Octahedron Carousel

The image on each face of this truncated octahedron is the one found in a previous post here, named Ten Circles, and was created with the use of two programs, Geometer’s Sketchpad and MS-Paint. As you’ll notice if you view other posts made today, though, the color scheme has been altered for this polyhedron.

Placing this image on each face of this polyhedron, as well as creating this rotating .gif file, required use of a third program, Stella 4d. This program may tried and/or purchased at http://www.software3d.com/stella.php. Unlike in the previous post, the images were “told” to stay upright while the polyhedron its rotates, creating a rotational effect in the yellow hexagonal faces, but a different effect in the red square faces. As far as I can tell, this is due to their different orientation in space, relative to the axis of rotation.