A Chiral Polyhedron with 120 Pentagonal Faces, Together with Its Dual

120 pentagons half of each type

In this chiral polyhedron, sixty faces are the small, purple pentagons, while the other sixty are the larger, orange pentagons. The next image shows its dual.

120 pentagons half of each type the dual

Both images were created with Stella 4d, a program you can buy, or try for free, at this website: http://www.software3d.com/Stella.php.

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

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.

Flying Kites into the Snub Dodecahedron, a Dozen at a Time, Using Tetrahedral Stellation

I’ve been shown, by the program’s creator, a function of Stella 4d which was previously unknown to me, and I’ve been having fun playing around with it. It works like this: you start with a polyhedron with, say, icosidodecahedral symmetry, set the program to view it as a figure with only tetrahedral symmetry (that’s the part which is new to me), and then stellate the polyhedron repeatedly. (Note: you can try a free trial download of this program here.) Several recent posts here have featured polyhedra created using this method. For this one, I started with the snub dodecahedron, one of two Archimedean solids which is chiral.

Snub Dodeca

Using typical stellation (as opposed to this new variety), stellating the snub dodecahedron once turns all of the yellow triangles in the figure above into kites, covering each of the red triangles in the process. With “tetrahedral stellation,” though, this can be done in stages, producing a greater variety of snub-dodecahedron variants which feature kites. As it turns out, the kites appear twelve at a time, in four sets of three, with positions corresponding to the vertices (or the faces) of a tetrahedron. Here’s the first one, featuring one dozen kites.

Snub Dodeca variant with kites

Having done this once (and also changing the colors, just for fun), I did it again, resulting in a snub-dodecahedron-variant featuring two dozen kites. At this level, the positions of the kite-triads correspond to those of the vertices of a cube.

Snub Dodeca variant with kites 1

You probably know what’s coming next: adding another dozen kites, for a total of 36, in twelve sets of three kites each. At this point, it is the remaining, non-stellated four-triangle panels, not the kite triads, which have positions corresponding to those of the vertices of a cube (or the faces of an octahedron, if you prefer).

Snub Dodeca variant with kites 2

Incoming next: another dozen kites, for a total of 48 kites, or 16 kite-triads. The four remaining non-stellated panels of four triangles each are now arranged tetrahedrally, just as the kite-triads were, when the first dozen kites were added.

Snub Dodeca variant with kites 3

With one more iteration of this process, no triangles remain, for all have been replaced by kites — sixty (five dozen) in all. This is also the first “normal” stellation of the snub dodecahedron, as mentioned near the beginning of this post.

Snub Dodeca variant with kites 4

From beginning to end, these polyhedra never lost their chirality, nor had it reversed.

A Chiral Tessellation, Using Regular Dodecagons, Regular Hexagons, Squares, and Rhombi (from 2012)

tess chiral 2012I have several “lost works” that I’m slowly finding and posting, from old jumpdrives, computers, little-known blogs, etc., and this is one of them. I made it in 2012, but few have seen it before now.

On Consistent and Inconsistent Combining of Chiralities, Using Polyhedral Augmentation

For any given chiral polyhedron, a way already exists to combine it with its own mirror-image — by creating a compound. However, using augmentation, rather than compounding, opens up new possibilities.

The most well-known chiral polyhedron is the snub cube. This reflection of it will be referred to here using the letter “A.”

Snub Cube ATo avoid unnecessary confusion, the same direction of rotation is used throughout this post. Apart from that, though, the image below, “Snub Cube B,” is the reflection of the first snub cube shown.

Snub Cube B

There are many ways to modify polyhedra, and one of them is augmentation. One way to augment a snub cube is to attach additional snub cubes to each square face of a central snub cube, creating a cluster of seven snub cubes. In the next image, all seven are of the “A” variety.

Snub Cube seven of them  AA

If one examines the reflection of this cluster of seven “A” snub cubes, all seven, in the reflection, are of the “B” variety, as shown here:

Snub Cube seven of them BB

Even though one is the reflection of the other, both clusters of seven snub cubes above have something in common: consistent chirality. As the next image shows, inconsistent chirality is also possible.

Snub Cube A augmented with B

The cluster shown immediately above has a central snub cube of the “A” variety, but is augmented with six “B”-variety snub cubes. It therefore exhibits inconsistent chirality, as does its reflection, a “B” snub cube augmented with “A” snub cubes:

Snub Cube B augmented with A

With simple seven-part snub-cube  clusters formed by augmentation of a central snub cube’s square faces by six snub cubes of identical chirality to each other, this exhausts the four possibilities. However, multiplying the possibilities would be easy, by adding more components, using other polyhedra, mixing chiralities within the set of polyhedra added during an augmentation, and/or mixing consistent and inconsistent chirality, at different stages of the growth of a polyhedral cluster formed via repeated augmentation.

All the images in this post were created using Stella 4d, which you can try for yourself at this website.

A Chiral Solution to the Zome Cryptocube Puzzle

This is my second solution to the Zome Cryptocube puzzle. In this puzzle, you start with a black cube, build a white, symmetrical, aethetically-pleasing geometrical structure which incorporates it, and then, finally, remove the cube. In addition, I added a rule of my own, this time around: I wanted a solution which is chiral; that is, it exists in left- and right-handed forms.

image (34)

It took a long time, but I finally found such a chiral solution, one with tetrahedral symmetry. Above, it appears with the original black cube; below, you can see what it looks like without the black cube’s edges.

image (35)

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.