“Stella” is on Sale!

Stella 4d: Polyhedron Navigator is selling at 25% off to help cure the boredom of people in isolation because of COVID-19. This software is used to make almost all of the polyhedral images on this blog. Trying the program to see if you like it is free. The website to visit to try Stella is http://www.software3d.com/Stella.php.

Five Variants of the Compound of Two Tetrahedra

This is the compound of two tetrahedra, also known as Johannes Kepler’s Stella Octangula.

I found the five variations of this polyhedral compound shown below, located deep within the stellation-series of the great rhombicuboctahedron.

These .gif images were all made using Stella 4d, a program you can try for free at http://www.software3d.com/Stella.php.

Compound of the Great and Small Stellated Dodecahedra

Compound of Great Stellated Dodeca and Small Stellated Dodeca color scheme 2

In this compound, as shown above, the small stellated dodecahedron is yellow, while the red polyhedron is the great stellated dodecahedron. Below, the same compound is colored differently; each face has its own color, unless faces are in parallel planes, in which case they have the same color.

Compound of Great Stellated Dodeca and Small Stellated Dodeca color scheme 1

Making a physical model of this compound would have taken most of the day, if I did it using such things as posterboard or card stock, compass, ruler, tape, scissors, and pencils. For the first several years I built models of polyhedra, starting about nineteen years ago, that was how I built such models. The virtual polyhedra shown above, by contrast, took about ten minutes to make, using Stella 4d: Polyhedron Navigator, which you can try for free, or purchase, here.

There’s also a middle path: using Stella to print out nets on cardstock, cutting them out, and then taping or gluing these Stella-generated nets together to make physical models. I haven’t spent much time on this road myself, but I have several friends who have, including the creator of Stella. You can see some of his incredible models here, and some amazing photographs of other Stella users’ paper models, as well as some in other media, at this website.

An Experiment Involving Augmentation of Octahedra with More Octahedra, Etc.

I’m going to start this experiment with a single octahedron, with faces in two colors, placed so that two faces which share an edge are always of different colors.


Next, I will augment the red faces — and only the red faces — with identical octahedra.


The regions with four blue, adjacent faces look as though they might hold icosahedra — but I checked, and they don’t quite fit. I will therefore continue the same process — augmenting only the red faces with more octahedra of the original type.


I’ve now decided that I definitely like this game, so I’ll keep playing it.


Immediately above, at the fourth of these images, some of the octahedra have started to overlap slightly, but I’m choosing to not be bothered by that — I’m continuing the now-established pattern, just in order to see where it takes me.


The regions of overlap are now far more obvious, but I’m continuing, anyway. Why? Because this is fun, that’s why! Right now, Stella 4d, the program I use to do these polyhedral manipulations, is chugging away on the next one. (This program is avilable at http://www.software3.com/Stella.php.) Ah, it’s ready — here it is!


Rather than repeat this process again, I now have another question: what would the convex hull of this figure look like? (A convex hull of a non-convex polyhedron is the smallest convex polyhedron which can contain a given non-convex polyhedron.) With Stella 4d, that’s easily answered.

Convex hull

I must admit this: that was nothing like what I expected — but such unexpected discoveries are a large part of what makes these polyhedral investigations with Stella 4d so much fun. And now, to close this particular polyhedral journey, I will have Stella 4d produce, for me, the dual of the convex hull shown above. (In case you aren’t familiar with duality regarding polyhedra, it describes the relationship between the octahedron, with which this post began, and the familar cube. Basically, with duals, faces and verticies are “flipped” over edges, although that is an extremely informal and imprecise way to describe the at the process.)

dual of Convex hull

And with that, my friends, I bid you good night!

Three Different Stellations of the Rhombic Triacontahedron

Rhombic Triaconta stellation

These polyhedra are selected from the the (long) stellation-series of the rhombic triacontahedron. Stellating polyhedra, and manipulating them in other ways, is easy with Stella 4d, which you may try, as a free trial download, here:   www.software3d.com/Stella.php.

stellated RTC one of many

stellated RTC another of many

42-Faced Polyhedron Featuring Heptagons


42-Faced Polyhedron Featuring Heptagons

The blue faces are irregular heptagons, and are twenty-four in number. There are twelve of the green rhombi, and six of the red squares. This was made using Stella 4d, which you may try or buy at http://www.software3d.com/Stella.php.

A Wire-Frame Zonohedron Based On the Faces, Edges, and Vertices of an Icosahedron


A Wire-Frame Zonohedron Based On the Faces, Edges, and Vertices of an Icosahedron

This is the shape of the largest zonohedron one can make with red, yellow and blue Zome (see http://www.zometool.com for more on that product for 3-d real-world polyhedron modeling). This image was made using Stella 4d, which you can find at http://www.software3d.com/stella.php.

24 Kites, Flying in a Whirlwind


24 Kites, Flying in a Whirlwind

To create this using Stella 4d (see http://www.software3d.com/stella.php for free trial demo), I started with a snub cube, added it to its own mirror image, stellated it several times, and then rendered the square faces invisible.

A Pulsating Zonohedron


A Zonohedron

This polyhedron is an example of a zonohedron — a polyhedron with faces which are all zonogons, each of which is congruent and parallel to another zonogon on the opposite side of the polyhedron.

Of course, this just raises the question, “What’s a zonogon?” It’s a special type of polygon, one with an even number of sides, and with opposite sides congruent and parallel.

The pulsating effect is just for fun — a feature which is optional in Stella 4d, the program I used to make this. I started with an icosahedron/dodecahedron compound, stellated it many times, and then zonohedrified the result. This software makes all these operations easy, and may be tried for free at http://www.software3d.com/stella.php.

Dodecahedron with Mandalas


Dodecahedron with Mandalas

This rotating image took three different programs to create. First, I made the mandala (seen in the previous post) using Geometer’s Sketchpad. Next, I used MS-Paint to complete the colorization of it. Finally, I used Stella 4d (see http://www.software3d.com/stella.php if you’d like to try or buy it) to put this mandala on each face of a dodecahedron, and then create the rotating .gif you see here.

I find both Stella 4d and Geometer’s Sketchpad to be indispensable tools for mathematical investigations and the creation of geometrical art, and highly recommend both programs.