Some Enantiomorphic-Pair Compounds

In the last post here, three different color-versions of the same cluster-polyhedron were shown. Since this cluster-polyhedron is chiral, it is possible to make a compound of it, and its own enantiomer (or “mirror-image,” if you prefer). This first image shows that, with the face-color chosen by the number of sides of each face.

c5c augmented with snub cubes Compound of enantiomorphic pair

Shown next is the dual of this figure, also colored by the number of sides of each face.

c5c augmented with snub cubes Compound of enantiomorphic pair dual colored by number of sides of each face

Next, another image of the first compound shown here, but with the colors chosen by face-type (referring to each face’s position in the overall polyhedron).

c5c augmented with snub cubes Compound of enantiomorphic pair colored by face type

Finally, here is the dual, again, also with colors chosen by face-type.

c5c augmented with snub cubes Compound of enantiomorphic pair dual colored by face type

All four of these images were generated with Stella 4d, a computer program available at http://www.software3d.com/Stella.php.

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

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.

exp sn dodeca 2

Exp Sn Dodaca

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.

exp sn dodeca 2 compound of enantiomophic pair

exp sn dodaca Compound of enantiomorphic pair exp snub dodeca

All four of these images were created using Stella 4d: Polyhedron Navigator, software available (including a free trial download) at this website.

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.

Compounds of Enantiamorphic Archimedean Solid Duals

An enantiomorphic-pair compound requires a chiral polyhedron, for it is a compound of a polyhedron and its mirror image. Among the Archimedeans, only the snub cube and snub dodecahedron are chiral. For this reason, only threir duals are chiral, among the Archimedean duals, also known as the Catalan solids.

Compound of enantiomorphic pair snub cube duals

That’s a compound of two mirror-image snub cube duals (pentagonal icositetrahedra) above; the similar compound for the snub dodecahedron duals (pentagonal hexacontahedra) is below.

Compound of enantiomorphic pair

Both these compounds were made with Stella 4d, which is available at http://www.software3d.com/Stella.php.

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 Pentagonal Icositetrahedron, Decorated with Rippled Tessellations, Along with Its Compound with Its Own Mirror-Image

Image

A Pentagonal Icositetrahedron, Decorated with Rippled Tessellations

The decorations on each face were created using the design, made using Geometer’s Sketchpad and MS_Paint, from this post: https://robertlovespi.wordpress.com/2014/05/28/rippling-tessellation-using-squares-regular-octagons-and-octaconcave-equilateral-hexadecagons/. I then used Stella 4d, available at http://www.software3d.com/Stella.php, to project this flat image onto each face of this chiral polyhedron, the dual of the snub cube, and make this rotating image.

Next, I used Stella to add this figure to its own mirror-image, to make a compound — something that is always possible with chiral polyhedra. Here is the result.

Compound of enantiomorphic pair

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.