An Icosahedron, Augmented by Snub Dodecahedra, Plus Two Versions of a Related Polyhedral Cluster

Icosa augmented by snub dodecahedra

Because the snub dodecahedron is chiral, the polyhedral cluster, above, is also chiral, as only one enantiomer of the snub dodecahedron was used when augmenting the single icosahedron, which is hidden at the center of the cluster.

As is the case with all chiral polyhedra, this cluster can be used to make a compound of itself, and its own enantiomer (or “mirror-image”):

Compound of enantiomorphic pair of snub-dodeca-augemented icosahedra

The image above uses the same coloring-scheme as the first image shown in this post. If, however, the two enantiomorphic components are each given a different overall color, this second cluster looks quite different:

Compound of enantiomorphic pair of snub-dodeca-augemented icosahedra colored by chirality

All three of these virtual models were created using Stella 4d, software available at this website.

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

60 and 60 -- chiral

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.

60 and 60 chiral --Compound of enantiomorphic pair

What Are Chiral Polyhedra? An Explanation, with Examples

Convex hhgfull

Two polyhedra are shown in this post — one which is chiral, and a similar one which is not. The non-chiral polyhedron in this pair is above. Its mirror-image is not any different from itself, except if you consider the direction of rotation.

The similar polyhedron below, however, features an overall “twist,” causing it to qualify as a chiral polyhedron. In its mirror-image (not shown, unless you use a mirror to make it visible), the “twisting” goes in the opposite direction. The direction of rotation would be reversed as well, of course, in a reflected image.

Codjfhnvexsdjag hhgfull

Multiple terms exist for mirror-image pairs of chiral polyhedra, the most well-known of which are the snub cube ansd snub dodecahedron, two of the thirteen Archimedean Solids. Some prefer to call them “enantiomers,” but many others prefer the more familiar term “reflections,” which I often use. I’ve also seen such polyhedra referred to as “left-handed” and “right-handed” forms, but I avoid these anthropomorphic terms related to handedness, simply because, if there is an established rule which would let me know whether any given chiral polyhedron is left- or right-handed, I’m not familiar with it. (Also, polyhedra do not have hands.) I could not, therefore, tell you if the example shown above would be correctly described as left- or right-handed — either because no such rule exists, or there is such a rule, but it is unknown to me. If the latter, I would appreciate it if someone would provide the details in a comment.

Both images above were created with Stella 4d, software you can try, for free, right here.

Two Chiral Polyhedra

Image

Two Chiral Polyhedra

To make this, I started with the dual of the great rhombicosidodecahedron, a polyhedron known as the dysdyakis triacontahedron. I then augmented half of its faces with tall prisms (thereby creating the chirality in this polyhedron), and took the convex hull of the result. The sixty red triangles are the tops of the augmentation-prisms.

A stellation of the above polyhedron, and a color-change, produced this result, also chiral. It may be enlarged with a click.

Stellated Convex hull based on expanded RTC

These polyhedra were created using Stella 4d, a program which you may buy — or try for free, as a trial download — at http://www.software3d.com/Stella.php.

A Polyhedron Featuring Sixty Octagons and Sixty Triangles

Image

A Polyhedron Featuring Sixty Octagons and Sixty Triangles

If someone had asked me if it were possible to form a symmetric polyhedra out of irregular triangles and octagons, using exactly sixty of one type each, I would have guessed that it were not possible. Why does it work here? Part of the reason is that each triangle borders three octagons, and each octagon borders three triangles — a necessary, but not sufficient, condition. This is a partial truncation of an isomorph of the pentagonal hexacontahedron, the dual of the snub dodecahedron. As such, no surprise — it’s chiral.

This was made while stumbling about in the wilderness of the infinite number of possible polyhedra using Stella 4d: Polyhedron Navigator. You can get it here: http://www.software3d.com/Stella.php.

A Chiral Tessellation

A Chiral Tessellation

In this chiral tessellation, the blue triangles and green hexagons are regular. The yellow hexagons are “Golden Hexagons,” which are what you get if you reflect a regular pentagon over one of its own diagonals, then unify the two reflections. The pink and purple quadrilaterals are two types of rhombi, and the red hexagons are a third type of equilateral hexagon. All of the edges of all polygons here have the same length.

There are three different types of points of three-fold rotational symmetry repeated here. Two of these types are centered in the middle of blue triangles, while the third is centered in the middle of some of the green hexagons — specifically, the ones surrounded only by alternating red and yellow hexagons.

When I try to generate the mirror-image of this tessellation, it overloads Geometer’s Sketchpad, and crashes the program. However, inverting the colors of the same reflection, in MS-Paint, to make a color-variant, is easy:

weird tessellation

Polyhedron Featuring Eighty Regular Hexacontagons in the Pattern of the Triangles of a Snub Dodecahedron

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Polyhedron Featuring Eighty Regular Hexacontagons in the Pattern of the Triangles of a Snub Dodecahedron

To make this, I attached tall pyramids (by their vertices) to the centers of the triangular faces of a snub dodecahedron. These pyramids have bases which are regular polygons with sixty sides each. After that modification of a snub dodecahedron, I took the convex hull of the result.

Just like the snub dodecahedron upon which this is based, this polyhedron is chiral. For any chiral polyhedron, Stella 4d (the software I use to make most of the images on this blog) will allow you to quickly make a compound of the polyhedron and its mirror image. When I did that, I obtained this result.

Compound of enantiomorphic pair

Stella 4d may be tried and/or bought at www.software3d.com/Stella.php.