Octahedra and Truncated Cubes Can Fill Space Without Leaving Any Gaps

Augmented Trunc Cube

I created this using Stella 4d, which you can try for free right here. It’s much like a tessellation, but in three dimensions instead of two.

A Faceted Version of the Truncated Cube

This is the truncated cube, which is one of the Archimedean solids.

Convex hull.gif

To make a faceted version of this solid, one must connect at least some of the vertices in different ways. Doing that creates new faces.

Faceted Trunc Cube 8 hexagons blue and yellow triangles.gif

This faceted version of the truncated cube includes eight blue equilateral triangles, eight larger, yellow equilateral triangles, and eight irregular, red hexagons. It’s easy to spot the yellow and blue triangles, but seeing the red hexagons is harder. In the final picture here, I have hidden all faces except for three of the hexagons, so that their positions can be more easily seen.

Faceted Trunc Cube 8 hexagons blue and yellow triangles some parts hidden.gif

I made all three of these images using Robert Webb’s program called Stella 4d: Polyhedron Navigator. It is available for purchase, or as a free trial download, at http://www.software3d.com/Stella.php.

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

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.

Trunc Cubocta honeycomb core

Next, I augment the hexagonal faces with truncated tetrahedra.

Trunc Cubocta honeycomb core plus 1.gif

The next polyhedra to be added are truncated cubes.

Trunc Cubocta honeycomb core plus 2

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

Trunc Cubocta honeycomb core plus 3

Now more truncated tetrahedra are added.

Trunc Cubocta honeycomb core plus 4

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

Trunc Cubocta honeycomb core plus 5

Next come more truncated cubes.

Trunc Cubocta honeycomb core plus 6

More great rhombcuboctahedra come next.

Trunc Cubocta honeycomb core plus 7

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.

Three Stellations of the Truncated Cube

12TH STELLATION OF Trunc Cube.gif

The polyhedron above is the 12th stellation of the truncated cube. The one below is the 14th.

14th-stellation-of-trunc-cube

The next one shown is the 18th and final stellation. If stellated again, the result is an ordinary truncated cube.

18th-and-final-stellation-of-trunc-cube

These virtual models were made using Stella 4d, software you may try for yourself at http://www.software3d.com/Stella.php.

 

Two Polyhedral Compounds: the Icosidodecahedron with the Truncated Cube, and the Rhombic Triacontahedron with the Triakis Octahedron

Compound of Icosidodeca and Trunc Cube

These two compounds, above and below, are duals. Also, in each of them, one polyhedron with icosidodecahedral symmetry is combined with a second polyhedron with cuboctahedral symmetry to form a compound with pyritohedral symmetry: the symmetry of a standard volleyball.

Compound of RTC and Triakis octahedron also pyritohedral

A program called Stella 4d was used to make these compounds, and create these images. It may be purchased, or tried for free, at this website.

A Space-Filling Arrangement of Polyhedra Using Truncated Cubes, Rhombcuboctahedra, Cubes, and Octagonal Prisms

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms

This image above has only one polyhedron-type hidden from view, in the center:  a red truncated cube. Next, more of this pattern I just found will be added.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 2

The next step will be to add another layer of blue octagonal prisms.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 3And now, more yellow cubes.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 4This was an accidental discovery I made, just messing around with Stella 4d, a program you may try for yourself at http://www.software3d.com/Stella.php. The next cells added will be red truncated cubes.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 5

Next up, I’ll add a set of pink rhombcuboctahedra.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 6The next set of polyhedra added: some yellow cubes, and blue octagonal prisms.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 7Now I’ll add more of the red truncated cubes.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 8At this point, more yellow cubes are needed.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 9The next polyhedra added will be pink rhombcuboctahedra.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 10

And now, more of the blue octagonal prisms.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 11

As long as this pattern is followed, this may be continued without limit, filling space, without leaving any gaps.