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.

Two Different Forty-Part Polyhedral Compounds

Cubes 20 A

The polyhedron above is a compound of twenty cubes and twenty octahedra, colored by symmetry-based face-type. If the same compound is viewed in “rainbow color mode,” it looks like this:

Cubes 20 Octa 20 A

With this particular compound, though, there are two versions — without taking coloring into consideration at all. The other version simply has the twenty cubes and twenty octahedron in a different, but still symmetrical, arrangement:

Cubes 20 Octa 20 B

The compound above uses this second arrangement, colored by face type, and the next image is the same (second) compound, but in “rainbow color mode.”

Cubes 20 B

These rotating polyhedral images were made with Stella 4d, software you can try for yourself, right here.

Two Polyhedral Meta-Compounds

Compound of 3 Cubes and dual cube and otahedron compound

The polyhedral compound above is actually a compound of two compounds: the compound of three cubes (red, yellow, and blue), as well as the cube/octahedron base/dual compound (green and purple). The dual of this five-part compound is shown below, still with the cube/octahedron compound in green and purple (it is its own dual), and with the three parts of the compound of three octahedra in red, yellow, and blue.

Compound of 3 octahedra and dual cube and otahedron compound

I created these using the “add/blend from memory” function of Stella 4d: Polyhedron Navigator, one of this program’s capabilities which I have only recently begun to explore. You may try this software for yourself, for free, right here.

The Greatly Augmented Icosidodecahedron, and Its Dual

Augmented Icosidodeca

If a central polyhedron’s pentagonal and triangular faces are augmented by great dodecahedra and great icosahedra, I refer to it as a “greatly augmented” polyhedron. Here, this has been done with an icosidodecahedron. The same figure appears below, but in “rainbow color” mode.

Augmented Icosidodeca colored rainbow

In the next image, “color by face type,” based on symmetry, was used.

Augmented Icosidodeca colored by face type

The next image shows the dual of this polyhedral cluster, with face color chosen on the basis of number of sides.

Augmented Icosidodeca colored by whether sides have 5 or 16 sides

Here is another version of the dual, this one in “rainbow color” mode.

Augmented Icosidodeca colored rainbow DUAL

Finally, this image of the dual is colored based on face type.

Augmented Icosidodeca colored by face type DUAL

These six images were made with Stella 4d, which may be found here.

Two Polyhedral Compounds: the Dodecahedron / Truncated Octahedron, and Its Dual, the Icosahedron / Tetrakis Cube

Compound of Trunc Octa and Dodeca

That’s the compound of the dodecahedron and the truncated octahedron above. Shown next is its dual, the compound of the icosahedron and the tetrakis cube. Both compounds were made using Stella 4d: Polyhedron Navigator, which you may try here.

Compound of tetrakis cube and icosa

Firstflight, Lastflight (an illustrated short story)

Making show never did prize me when soberfied, but that undescribed me that day, and, for that mistake, payment was failsafely, fullwise, and painly made. Tranqued with Euphenol, selfbought at the official dispensing-machine on Convenience Corner, right after worktime, methought melooked mighty brave strolling wrongway homewise on my hands, feet toepointed at the otherseyes, down a steel walkway crowded with those farmore sensehaving, so neither of my LifeLine© MagnetShoes touched metal, but as soon as Unitility’s gravsynths fritzed surprising, I felt notimpressive, floating rapidly away from the crowd.

Weightlessness bit me mid-handspring, and sent me flying, but I was too headspun to realize predicamental situation until my homehalf of our Megalopolis already could be seen in entireness, several deathfalls away. Soon my home fragment of ShatteredEarth looked only littlemuch bigger than neighbor skyrocks, and the cities otherfolk in othercities had built on them, as our own beforefolk had built Megalopolis. Wheezing suddenly brought meself realizing: the thinning air would set me freesome if Unitility’s repairs took much longer. Time passed, and bleakness grew as hope thinned. I watched Megalopolis, overcrowded skyrock of ours, rotate in the . . . or was it me rotating? I couldn’t figure out how to tell, and this added “headache” to growing problemlist.

Augmented Convex hull

My bubble of air biggifying as it bled away spacewise I could see, for this bubble was debrislittered, spaced evenish out here, but more crowded nearer our skyrock — paper, waterblobs, cloudpuffs, disoriented pigeons, half-eaten McFood left behind — but no other people out in the thinning bubble, just my sad self, now far away from safeness anykind. Apparently Euphenol overleaped me earlier, and, thus obliviated, I ended up the only fool I could see with sufficient maltimed stupidity to fall off the world. Most got away in shuttles, justvisible and receding to vanishment, or simply stayed inside the cubicles in our towers to safely outwait events, while a few on the surface I could only barelysee, with muchwise squinting. Each flailed arms, but all safely stuck to home skyrock by safety magnets at their feet. Unitility blackouts never lasted overlong, especially for gravservice; they’d likely live, not being me that day, for the air stayed thicker nearer to our skyrock, leaking only slowly from its insulasafed towers.

Luck having clearwise left me, my flightpath then entered the last raincloud in the thinning Megalopolis atmosphere, and instantly drenched I startled, from ends of my floating hair to toestips, moisture entering even my feetboots with their nowuseless safetymagnets that seemednow mocking me. It was frigid inside the cloud, and I watched in sockhorror as the water on my hand start freezing, ice spreading over my skin and standardissue clothiform, at about a centimeters per heartbeat, from multiple locations. Even though I wasn’t speaking, mouthchatter quicklike so bad made me fear teethcracks from the constantlike repeated impactstrikings. Remembering last dental torture-session determined me not to endure that level of pain while in my likely lastseconds, so clenched my teeth together determined, hardwise, to stopchatter, and this worked. At least one thing was on path, myway, now.

Not long, though. Leaving the cloud on the otherside, pressuregrowing inside me forced mouth fullopen, and more air than I knew I had speedily left, from both lungbottoms, up allway. I could hear bubbling from somewhere inside, wondered what and exactly where it was, then deciding not to know indeed was the better. Dental painmemories were now distanced as new pain eclipsed it hereandnow, from the vacuum conditions approaching. Newpain competed with growing dizzisorientation from samecause. Closed my eyes, notwishing seeing.

Suddenrealizing closing eyes seriously mistaken, started vomitmuch, then realized eyes were frozenshut while finishing uneating afterlunch’s, then lunch’s, then wakefast’s McFood. Panicked clawing at eyes quickly got one open partwise. It was enough permitting seeing, but liking what seen not happened. Big, bulging silver eye was growing towards me full of quickness, already filling most all of what I could see. It had iris lavished every color everseen and some hadn’t before, and the pupil, while small, clearly pointed in no other direction than me. Could an eye that large see all people? Surely, but at that moment its focus was clearly on me, leaving others forgotten for what moments remained of my consciousness. I shook my head, and the illusion was shattered; I thensaw what it had morphed from, which was the thing we still called the Moon, full bright now. Why did we still call it the Moon, I wondered, since the known pieces of ShatteredEarth now orbited it, not otherway aroundwise?

Coldifying broke as sudden heat grew backside my head, diverting focus from the Moon and all the skyrocks closeful enough to seeing. Turning around freefalling not close to easy, but I did it — was by swimming-motions against what air remained, making seen the heatsource: the Sunstar, unaffected by lastcentury’s Earthshatter. Sunstar then grewsize, as fading consciousness sent hallucinavision back myway, until the morphing orb grew even larger than my previous Moon-based vision of One Big Eye staring at onlyme. As I thought these weirdthoughts, the sun started changing its appearance, growing eyes everyway around.

Snub Dodeca

Feelinglike unexpectedly challenged, but voiceless with breathloss, I could but headshout at what I saw, but did so loudlymuch, enough to deafen a t’path, had one unlucked near: Die! Die! Die! I can stare anyone down! For a moment, triumph filled me as my headshouting seemlyworked — the sun greyed out, and then vanished altogether. I saw nothingness! The Sunstar itself was defeated!

No, idiot, you just selfblinded, staring at the Sunstar!  This thought, my most rational of allday, made me attempt screamreacting, but the mere wisps of air remaining were not enough to allow sonics from my effort. All I selfgained was an increase in the bodywide painstabbings, to levels I never knew everhappened.

In the darkness, another eye appeared, like the ones I had seen on the Sunstar, but based on nothing but rememberings now, since I could see nothingcept. There were changing swirlsparks everywhere within it, timed precisely with the growing pounding from within my skull and chest. Panic didn’t happen, but only because of the dizzycalm which sometimes happens from lackoxy. I got lost wondering what a headpop might feel like. Would I just puff away, like a candleblown, or would I painfully feel the bursting of each nerve and blood vessel? Detachment was now extreme, muchso that I carednot which. The eye got nearmuch, so that I should have been able to reachtouch the pupil, but my arms weren’t listening to brainorders to move. I fell then, tumbling, into eyecenter, a pupil far wider than my own height.

My contactmoment with its cornea’s thickslime covering provoked a spasm of the entire eye, scaring me to new levels. Meter-thick eyelids rushed toward me from twindirections opposite. My last heartbeat was deafening, in literality — I heard nothing more. Time remained for only a silent finalthought. What a way to lose a staring contest: death by Sunblink —

# # #

[The images above were made with Stella 4d, available here. Geometer’s Sketchpad and MS-Paint were also used, as well as a background image, which I altered, from this website.]

Four Non-Convex Polyhedra with Icosidodecahedral Symmetry

Unnamed Dhgual

pretty

Dual of Augmented Convex hull

unholy messl

All of these were made with Stella 4d, a program you can find at http://www.software3d.com/Stella.php.

An Unusual Presentation of the Icosahedron/Dodecahedron Base/Dual Compound

Leonardo Icosahedron

In this model, the usual presentation of the icosahedron/dodecahedron dual compound has been altered somewhat. The “arms” of star pentagons have been removed from the dodecahedron’s faces, and the icosahedron is rendered “Leonardo-style,” with smaller triangles removed from each of the faces of the icosahedron, with both these alterations made to enable you to see the model’s interior structure. Also, the dodecahedron is slightly larger than usual, so that its edges no longer intersect those of the icosahedron.

This model was made using Stella 4d, software you can obtain for yourself, with a free trial download available, at http://www.software3d.com/Stella.php.

Three Variations of Kepler’s Stella Octangula

The Stella Octangula was the name Johannes Kepler gave, centuries ago, to the compound of two tetrahedra. Here are three variations on it, all created using Stella 4d, a program you may try at http://www.software3d.com/Stella.php.

stella octangular variantstella octangular variant anotherstella octangular variant third

A Cluster of Nine Octahedra, and Related Polyhedra

If one starts with a central octahedron, then augments each of its eight triangular faces with identical octahedra, this is the result.

9 Octahedra

It is then possible to augment each visible triangle of this cluster with yet more octahedra, which produces this result, in which some octahedra overlap each other.

Meta-9 Octahedra

After making this, I wanted to see its convex hull:  the smallest, tightest-fitting convex polyhedron which can contain a given non-convex polyhedron. (I use Stella 4d: Polyhedron Navigator to perform these manipulations of polyhedra, and this program makes this a fast and easy process. If you’d like to try this software, even as a free trial download, the website to visit is http://www.software3d.com/Stella.php.) Here’s what this convex hull, which bears a resemblance to the rhombcuboctahedron, looks like.

Convex hull of meta-9-octahedron

Looking for previously-unseen, and interesting, polyhedra, I then starting stellating this convex hull. I did find something interesting — to me, anyway — after only two stellations.

2nd stellation of Convex hull of meta-9-octahedron

That concluded my latest polyhedral investigation, but I certainly don’t intend it to be my last.