A Black-on-Black Polyhedron: The Final Stellation of the Icosahedron

Posted on 28 July 2017 by RobertLovesPi

Icosa

I made this using Stella 4d: Polyhedron Navigator, which you may try for free right here.

Selected Stellations of the Truncated Dodecahedron

Posted on 17 July 2017 by RobertLovesPi

This is the truncated dodecahedron. It is one of the Archimedean solids.

Trunc Dodeca

This polyhedron has a long stellation-series, from which I selected several on aesthetic grounds. The figure immediately below is the truncated dodecahedron’s 16th stellation.

16th stellation of Trunc Dodeca

Here is the 21st stellation.

21st stellation of Trunc Dodeca

It’s easy to stellate polyhedra rapidly, and make many other changes to them, with Stella 4d: Polyhedron Navigator. You can try it for free at http://www.software3d.com/Stella.php.

25th stellation of Trunc Dodeca.gif

The stellation shown immediately above is the 25th, and the one shown immediately below is the 27th.

27th stellation of Trunc Dodeca

Here is the next stellation: the 28th. Unlike the ones shown above, it is chiral.

28th stellation of Trunc Dodeca.gif

This is the truncated dodecahedron’s 31st stellation.

31st stellation of Trunc Dodeca.gif

This one is the 38th stellation.

38th stellation of Trunc Dodeca.gif

This one is the 44th.

44th stellation of Trunc Dodeca.gif

The last one shown here is called the truncated dodecahedron’s final stellation because, if it is stellated once more, it returns to the original truncated dodecahedron.

Final stellation of Trunc Dodeca

A Compound of Two Stellated Polyhedra

Posted on 18 February 2017 by RobertLovesPi

compound-of-great-and-small-stellated-dodecahedron I made this using Stella 4d, software you can try for free at http://www.software3d.com/Stella.php.

Some Tetrahedral Stellations of the Truncated Cube

Posted on 19 January 2017 by RobertLovesPi

a-tetrahedral-stellation-of-the-trunc-cube

another-tetrahedral-stellation-of-the-trunc-cube-2

another-tetrahedral-stellation-of-the-trunc-cube-5

another-tetrahedral-stellation-of-the-trunc-cube-3

another-tetrahedral-stellation-of-the-trunc-cube-6

another-tetrahedral-stellation-of-the-trunc-cube-4

another-tetrahedral-stellation-of-the-trunc-cube

I created these with Stella 4d, which you may try for free at this website. To make a given polyhedral stellation appear larger, simply click on it.

Three Stellations of the Truncated Cube

Posted on 31 December 2016 by RobertLovesPi

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

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

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

A Decorated Pentagonal Hexacontahedron, with Three of Its Stellations

Posted on 2 October 2016 by RobertLovesPi

penta-hexeconta

This is a pentagonal hexacontahedron, the dual of the snub dodecahedron. It’s decorated with mandalas of the the type I blogged here, two posts ago. The mandalas do interesting things when this polyhedron is stellated, as you can see below.

penta-hexeconta-stel-1

That was the first stellation, and here is the second:

penta-hexeconta-stel-2

The sixth stellation was the last one I found interesting enough to post.

penta-hexeconta-stel-6

All four polyhedral images above were created using Stella 4d: Polyhedron Navigator, software you may buy, or try for free, at this website.

Building a “Polyhedral Porcupine”

Posted on 24 September 2016 by RobertLovesPi

This is the icosahedron, followed by its first stellation.

The first stellation of the icosahedron can be stellated again, and again, and so on. The “final stellation” of the icosahedron is the one right before the stellation-series “wraps around,” back to where it started:

This final stellation of the icosahedron would serve pretty well as a “polyhedral porcupine,” but I was seeking something even better, so I turned my attention to polyhedral compounds. This is the compound of five icosahedra:

The program I use to manipulate these solids is called Stella 4d: Polyhedron Navigator (free trial download available here). My next move, using Stella, was to create the final stellation of this five-icosahedron compound . . . and, when I saw it, I knew I had found my “polyhedral porcupine.”

Some Stellations of the Truncated Dodecahedron

Posted on 31 July 2016 by RobertLovesPi

The stellation-series of the truncated dodecahedron contains some interesting polyhedra. Selections from this series appear below.

24th Stellated Trunc Dodeca chiral

The polyhedron above is the 24th stellation of the truncated dodecahedron, while the one below is the 25th stellation.

25th stellation of Trunc Dodeca

27th Stellated Trunc Dodeca chiral

The polyhedron immediately above is the truncated dodecahedron’s 27th stellation. The one shown below is the 29th stellation.

29th Stellated Trunc Dodeca chiral

36th Stellated Trunc Dodeca chiral

The last two polyhedra in this post are the truncated dodecahedron’s 36th stellation (above), and its 70th stellation (below).

70th Stellated Trunc Dodeca

These images were created using Stella 4d, software available here.

The 11th, 13th, and 15th Stellations of the Icosahedron

Posted on 31 July 2016 by RobertLovesPi

First, this is the 11th stellation.

Stellated Icosa the 11th

Next, the 13th:

13th Stellated Icosa

And, finally, the 15th stellation of the icosahedron:

15th Stellated Icosa

I used Stella 4d, which you can find here, to make these.

A Hybrid of the Tetrahedron and the Great Dodecahedron

Posted on 18 January 2016 by RobertLovesPi

I made this by stellating a dodecahedron repeatedly, but doing so with Stella 4d, the polyhedral-manipulation software I use (available here), set to use tetrahedral symmetry, rather than the higher-order icosahedral symmetry (which I often call “icosidodecahedral” symmetry) inherent to Platonic dodecahedra.

Hybrid of great Dodeca and the tet