A Polyhedron Featuring 180 Kites as Faces, Plus Related Polyhedra

Posted on 3 October 2015 by RobertLovesPi

If one starts with the great rhombicosidodecahedron, then makes a compound of it, and its dual, and then forms the convex hull of that compound, this is the result:

This polyhedron has 180 faces, all of them kites. What’s more, there are equal numbers — sixty each — of the three different types of kites in this polyhedron.

It also has an interesting dual:

These virtual polyhedral models were created using Stella 4d: Polyhedron Navigator, which you can buy, or try for free, right here. Stella contains a “try to make faces regular” function, and here is what appears if that operation is applied to the dual shown above:

The dual of this figure is similar to the original polyhedron at the top of this post, featuring 180 kites, again: sixty each, of three different types:

Five Polar Polyhedra

Posted on 27 September 2015 by RobertLovesPi

Most polyhedra I post have cuboctahedral, tetrahedral, or icosidodecahedral symmetry, or some pyritohedral or chiral variation of one of these symmetry-types. These, however, are exceptions. I call them “polar polyhedra” because they each have an identifiable “North Pole” and “South Pole,” which are, in three of these five images, at the ends of their axes of rotation.

These rotating images were created using Stella 4d, software you may try for yourself, right here.

Sixty and Sixty: A Chiral Polyhedron, as well as the Compound of It, and Its Own Reflection

Posted on 19 September 2015 by RobertLovesPi

This polyhedron is chiral, meaning that (unlike many well-known polyhedra) it exists in “left-handed” and “right-handed” forms — reflections of each other. These “reflections” are also called enantiomers. I call this polyhedron “sixty and sixty” because there are sixty faces which are irregular, purple quadrilaterals, as well as sixty faces which are irregular, orange pentagons.

I stumbled upon this polyhedron while playing around with Stella 4d: Polyhedron Navigator, software you can try right here. For those who research polyhedra, I know of no better tool.

To see the other enantiomer, there is a simple way — just hold a mirror in front of your computer screen, with it showing the image above, and look in the mirror!

With any chiral polyhedron, it is possible to make a compound out of the two enantiomers. Here is what the compound looks like, for this “sixty and sixty” polyhedron cannot be seen this way, so here is an image of it, also created using Stella 4d.

One of Many Faceted Rhombicosidodecahedra

Posted on 16 September 2015 by RobertLovesPi

This was created by making the dual of the 32nd stellation of the strombic hexacontahedron, which is itself the dual of the rhombicosidodecahedron. This technique for finding facetings works because faceting is the reciprocal function of polyhedral stellation.

I did this using Stella 4d, which you can try for yourself, for free, at http://www.software3d.com/Stella.php.

A Rhombic Ring of Icosahedra, Leading to a Rhombic Dodecahedron Made of Icosahedra

Posted on 15 September 2015 by RobertLovesPi

As it turns out, eight icosahedra form this rhombic ring, by augmentation:

Measured from the centers of these icosahedra, the long and short diagonal of this rhombus are in a (√2):1 ratio. How do I know this? Because that’s the only rhombus which can made this polyhedron, a rhombic dodecahedron, dual to the cuboctahedron.

This rhombic dodecahedral cluster of icosahedra could be extended to fill space, since the rhombic dodecahedron itself has this property, an unusual property for polyhedra. Whether space-filling or not, the number of icosahedron per rhombic-dodecahedron edge could be increased to 5, 7, 9, or any greater odd number. Why would even numbers not work? This is a consequence of the fact that opposite faces of an icosahedron are inverted, relative to each other; a pair of icosahedra (or more than one pair, producing odd numbers > 1 when added to the vertex-icosahedron) must be attached to the one at a rhombic-dodecahedron-vertex to make these two inversions bring the triangular face back around to its original orientation, via an even number of half-rotations, without which this consruction of these icosahedral rhombi cannot happen.

Here’s another view of this rhombic dodecahedron, in “rainbow color” mode:

All images above were produced using Stella 4d, software which may be tried for free right here.

Two Versions of a Slowly Rotating Rhombic Triacontahedron, Adorned with Spectral Patterns on Each Face

Posted on 13 September 2015 by RobertLovesPi

It took three programs to make this. First, outlines of the “double rainbow” patterns on each face were constructed using Geometer’s Sketchpad. A screenshot from that program was then pasted into MS-Paint, which was used to add color to the outline of the pattern on each face. Next, the colorized image was projected onto each face of a rhombic triacontahedron, using Stella 4d: Polyhedron Navigator — the program that put this all together, and what I used to generate the rotating .gif above. Stella is available at http://www.software3d.com/Stella.php, with a free trial download available.

Interestingly, while this polyhedron itself is not chiral, the coloring-pattern of it, shown above, is.

With only small modifications, Stella can produce a very different version:

Which one do you like better?

On Binary Planets, and Binary Polyhedra

Posted on 26 July 2015 by RobertLovesPi

This image of binary polyhedra of unequal size was, obviously, inspired by the double dwarf planet at the center of the Pluto / Charon system. The outer satellites also orbit Pluto and Charon’s common center of mass, or barycenter, which lies above Pluto’s surface. In the similar case of the Earth / Moon system, the barycenter stays within the interior of the larger body, the Earth.

I know of one other quasi-binary system in this solar system which involves a barycenter outside the larger body, but it isn’t one many would expect: it’s the Sun / Jupiter system. Both orbit their barycenter (or that of the whole solar system, more properly, but they are pretty much in the same place), Jupiter doing so at an average orbital radius of 5.2 AU — and the Sun doing so, staying opposite Jupiter, with an orbital radius which is slightly larger than the visible Sun itself. The Sun, therefore, orbits a point outside itself which is the gravitational center of the entire solar system.

Why don’t we notice this “wobble” in the Sun’s motion? Well, orbiting binary objects orbit their barycenters with equal orbital periods, as seen in the image above, where the orbital period of both the large, tightly-orbiting rhombicosidodecahedron, and the small, large-orbit icosahedron, is precisely eight seconds. In the case of the Sun / Jupiter system, the sun completes one complete Jupiter-induced wobble, in a tight ellipse, with their barycenter at one focus, but with an orbital period of one jovian year, which is just under twelve Earth years. If the Jovian-induced solar wobble were faster, it would be much more noticeable.

[Image credit: the picture of the orbiting polyhedra above was made with software called Stella 4d, available at this website.]

Thirty-Four Rotating, Convex, and Non-Chiral Polyhedra with Icosidodecahedral Symmetry

Posted on 22 July 2015 by RobertLovesPi

Most in the field call this type of symmetry “icosahedral,” but I prefer the term George Hart uses — along with “cuboctahedral” in place of “octahedral.”

Each polyhedral image here was created with Stella 4d: Polyhedron Navigator. At this linked page, you can try a trial version of that program for free.

By the way, when I described these polyhedra, in this post’s title, as non-chiral, I was not referring to the coloring-schemes used here, many of which obviously are chiral, but only the shapes themselves.

That 10 GB space-upgrade, which most bloggers don’t ever need, is really coming in handy right now. In other words, some of these .gif files are huge!

Why, yes, I am including some words after every fifth polyhedron. That will help, later, when I count them for the title of this post.

I’m not sure why that last one is spinning the opposite direction from the others. Perhaps this polyhedron is trying to start a trend. On the other hand, it could just simply upside-down.

That’s twenty-five so far.

Clearly, I should have checked the number of files in that file folder before deciding to simply post them all together, based on what they have in common. That’s thirty so far.

Two Different Versions of an Expanded Snub Dodecahedron, Both of Which Feature Regular Decagons

Posted on 18 July 2015 by RobertLovesPi

The snub dodecahedron, one of the Archimedean solids, can be expanded in multiple ways, two of which are shown below. In each of these expanded versions, regular decagons replace each of the twelve regular pentagons of a snub dodecahedron.

Like the snub dodecahedron itself, both of these polyhedra are chiral, and any chiral polyhedron can be used to create a compound of itself and its own mirror-image, Below, you’ll find these enantiomorphic-pair compounds, each made from one of the two polyhedra above, together with its own reflection.