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About RobertLovesPi

I go by RobertLovesPi on-line, and am interested in many things, a large portion of which are geometrical. Welcome to my own little slice of the Internet. The viewpoints and opinions expressed on this website are my own. They should not be confused with those of my employer, nor any other organization, nor institution, of any kind.

A Box of Heptagons

It is hard to find polyhedra which contain regular heptagons, but that simply makes the search more challenging, and therefore more fun. This polyhedron features six regular heptagons, and was created with Stella 4d, software available here.

Box of Heptagons

A Large Collection of Polyhedra with Pyritohedral Symmetry, Some of Them Chiral

pyritohedral polyhedron featuring octagons

pyritohedral polyhedron featuring heptagons

14 hexagons of two types 12 heptagons 24 pentagons 50 faces in all and pyritohedral

hourglasses

hexagons and pyritohedral rectangles

Dual of Convex hull

idea pyrito

interesting pyritohedral convex hull from 8Apr15 blog post

kites and paired irregular pentagons pyritohedral

pyrito and interesting

Pyrito Compound of enantiomorphic pair

jsfasdkf

pyritic Convex hull

pyrito Dual of Convex hull

pyrito RCO

pyrito stellation of great Dodeca 2 actuaaly these are tetstells

pyrito stellation of great Dodeca

pyrito

pyritohedral and chiral 3

pyritohedral and chiral

idea stellated

pyritohedral Convex hull

pyritohedral double snub cube

pyritohedral dual

Pyritohedral Icosahedral with added octagons

pyritohedral triangles and heptagons again

pyritohedral triangles and heptagons

pyritol

pyritoonvex hull

pyritowrinkles by no of sides

Unmessnamed Dual

Faceted Dual pyritohedral and chiral

Final Stellation of Compound of Icosidodeca and Trunc Cube also pyritohedral

chiral Dual of Convex hull

Unnamed Dual

It’s quite an informal way to define it, but pyritohedral symmetry is the symmetry-type of a standard volleyball. These images of pyritohedral polyhedra were made using Stella 4d, software available at http://software3d.com/Stella.php.

The Small Ditrigonal Icosidodecahedron, Together with Its Fifth Stellation

Faceted Dodeca

I made the polyhedron above by performing a faceting of the dodecahedron, and only realized, after the fact, that I had stumbled upon one of the uniform polyhedra, a set of polyhedra I have not yet studied extensively. It is called the small ditrigonal icosidodecahedron, and its faces are twelve star pentagons and twenty equilateral triangles, with the triangles intersecting each other. Below is its fifth stellation, which appears to be a compound of a yellow dodecahedron and a red polyhedron which I do not (yet) recognize, although it does look quite familiar.

5th stellation of the Small Ditrigonal Icosidodeca

Both images were created using Stella 4d, software you can try right here.

Partially Truncated Platonic and Rhombic Dodecahedra

chiral polyhedron featuring a dozen hexagons and four triangles

Each of these dodecahedra were modified by truncations  at exactly four of their three-valent vertices. As a result, each has four equilateral triangles as faces. In the one above, the Platonic dodecahedron’s pentagonal faces are modified into a dozen irregular hexagons by these truncations, while, in the one below, the rhombic dodecahedron’s faces are modified into twelve irregular pentagons.

dozen pents 4 triangles

Both of these polyhedra were created using Stella 4d, software you can try for yourself at this website.

A Mandala Made of Hexagons, Enneagons, and Dodecagons

recreational math from 2011

I recently re-discovered this “lost work,” which I made using Geometer’s Sketchpad, in 2011 — before I started this blog, which is why it has not appeared here before.

Six Random Starsplosions of Non-Convex Polyhedra with Cuboctahedral Symmetry

Compound of enantiomorphic pair

Faceted Trunc Octa dual of the compound

Small Cubicubocta dual the small hexacronic icositetrahedronUnnamed Dual

stellated disdyakis dodecahedronStellated Penta Icositetra

All of these were made using Stella 4d, a program you can try for free at http://www.software3d.com/Stella.php.

The 21st and 22nd Stellations of the Truncated Dodecahedron

Stellation of a polyhedron involves extending its faces and/or edges into space to form other polyhedra, often with a star-like appearance, which is where the words “stellation,” “stellate,” and “stellated” originate. (“Stella” is Latin for “star.”)

Since this can be done repeatedly, long stellation-series exist for many polyhedra. In the case of the truncated dodecahedron, it was the 21st and 22nd stellations which I found the most aesthetically pleasing.

Here is the 21st stellation of this polyhedron:

Trunc Dodeca 21st stellation

And here is the 22nd:

Trunc Dodeca 22nd stellation

Both of these polyhedra were created with Stella 4d, software you may try for yourself, right here.

A Zonohedron with 3540 Faces, Together with Its Dual

Zonohedrified Poly 3540 faces

Zonohedra are polyhedra made completely of faces which are zonogons. A zonogon is a polygon which:

  • Has an even number of sides,
  • Has opposite sides congruent, and
  • Has opposite sides parallel.

Parallelograms are the simplest zonogons.

Here is the dual of the zonohedron above; it has 3542 faces. Although zonohedra-duals do have distinctive appearances, they do not, as a class, have a name of their own, to the best of my knowledge. They are definitely not zonohedra, themselves.

Zonohedrified Poly 3540 faces dual with 3542 faces

Both of these polyhedra were created with Stella 4d, software you may try for yourself, right here.

One of Many Possible Facetings of the Rhombic Triacontahedron

Faceted Rhombic Triaconta

The simplest way I can explain faceting is that it takes a familiar polyhedron’s vertices, and then connects them in unusual ways, so that you obtain different edges and faces. If you take the convex hull of a faceted polyhedron, it returns you to the original polyhedron.

This was created using Stella 4d, software available (including as a free trial download) right here: http://www.software3d.com/Stella.php.

How the Homunculus War Begins: A Short Story

crime scene

Detective Bruce Kelley put out his cigarette, took a sip of his nasty-tasting, long-cold coffee, made a face of disgust, swallowed the sip of old coffee anyway, and took his time lighting a second cigarette, carefully watching the suspect every moment, as he did each of these things, without saying a word. After taking a long drag from his latest cigarette, he broke the silence. “Murdock. Mr. Peter Murdock. The crimes in which you are the prime suspect are serious, but I don’t think you’re taking this investigation seriously — at all — and I want to know why.”

“Detective, I haven’t committed any crimes. I have only been accused of killing two people, whose names I don’t even know, but as I have told you, repeatedly, I’ve never killed anyone, and never would. I’m not that kind of person.”

Kelley snorted. “Let’s talk about something else, then. Do you enjoy reading comic books, Mr. Murdock?”

“As a matter of fact, I do. Why do you ask?”

“I used to read them myself, when I was a kid. Spider-Man was my favorite, but I also liked Daredevil. And, I gotta tell you, of all the fake names I’ve ever been given by people with obviously fake IDs, ‘Peter Murdock’ has got to be the most pathetic fake name I’ve ever seen or heard . . . but maybe that’s just because we’ve both read a lot of Daredevil and Spider-Man comic books. So, here’s my next question: what’s your real name?”

“What? I don’t understand. Are you sure that isn’t my name? Hold on a second — I’ll consult my homunculus.”

“Your what?”

“You know — the little guy inside your head. The one who sees through your eyes. The dude behind the steering wheel, and at the control panel. You know, the one who decides what you do. Don’t you ever talk to yours? I wouldn’t know what to do with my life without the help of my homunculus.” He then gave the detective a huge, childlike smile.

Image source: http://commonsenseatheism.com/?p=8844

A Man, and His Homunculus

The detective said nothing, and was still processing these words when the suspect spoke again, roughly a full minute later. “My apologies, detective. My homunculus was apparently playing tricks on us both, but I don’t know why. He now tells me my real name is Reed Stark.”

Detective Kelley put his unfinished cigarette out, right into the cup of swill which, two hours ago, had been half-way decent coffee. The cigarette hissed, as the lit end hit the liquid. “Reed . . . Stark. Of course he did. Are you quite sure about that name, though? Why not ask him if he meant to say ‘Tony Richards,’ instead? I thought we had already established that we’ve read the same &%$#ing comic books, you lunatic!”

“No, he says he’s sure it’s Reed Stark. Absolutely certain. If you can’t trust your own homunculus, detective, well, then, whom can you trust?”

“Well, ‘Mr. X,’ you’ve already been thoroughly searched, so there’s no point in asking for ID that says your name is Reed Stark. Look, Mr. Stark, Murdock, Richards, Stan Lee, or whatever your name is, I think we’re done here — at least for now.”

The detective picked up the handset-part of the old-style, land-line phone on the desk between them, in this small room which was so barren that the suspect had, until the detective picked up part of it, and talked into it, thought of this shiny black telephone as the room’s sole piece of ornamentation. Speaking into the phone, Kelley said, “Yeah, have the guard take Mr. ‘Murdock,’ here, back to his cell. We’re not getting anywhere with this, and there’s no point in my wasting any more of my time questioning him.”

“Um, detective, that’s not ‘Murdock.’ I told you, my name is Stark. Reed Stark. I don’t know why my homunculus –”

“Shut up. Just — just shut up.”

The suspect did not argue, and the guard arrived quickly, removed the restraints holding the suspect to his chair, but did not take off his handcuffs. As the guard left with the suspect, Kelley spoke once more. “Guard — don’t take him back to the same cell. Put this man in isolation.”

“Yes, sir.” They left. The door closed behind them. Detective Kelley picked up the phone again.

“This is about the murder suspect with the obviously-fake ID — the one who claimed his name was Peter Murdock, and had poor-quality fake ID to match. Change the name on his intake records to ‘John Doe.’ Run his fingerprints again, just in case the system missed a match last time. Also, get in touch with the psychiatric unit at the state hospital. Tell them we’ve got a dangerous head case coming their way, and to be ready to receive him — and warn them that he is the prime suspect in a double homicide — which he claims not to remember . . . . What was that? Yes, actually, I do believe him — the part about him not remembering killing those two men in that blind alley, anyway — that much I believe. This guy thinks there’s a little man in his head who tells him what to do — a ‘homunculus,’ he called it. I’m about to go google that word right now. No, I’ve never heard that term, but I know a psych case when I see one, and this guy’s definitely nuts.”

Once locked in his isolation-chamber, the suspect asked his homunculus what to do next, but heard no answer — so, instead of just asking silently, he asked again, aloud. To that, he heard a response, sort of: snoring. His homunculus was, apparently, asleep.

“I guess I’m supposed to sleep, too, then,” he said aloud, to no one in particular, and with that, he stretched out flat, right on the floor (ignoring the small, hard bed which was attached to the far wall), and instantly fell asleep.

Soon, the suspect was dreaming. He heard the voice of his homunculus, and it sounded oddly, subtly different than normal — and familiar, also. He finally recognized it, for he had heard it in multiple superhero movies: it was the voice of Stan Lee.

Trans-dimensional portal now open for transfer of both powers and equipment. Begin transfer immediately. Matt Murdock, code-named “Daredevil,” with two billy clubs — complete. Peter Parker, code-named “Spider-Man,” with web-shooters, and extra cartridges of web-fluid — complete. Reed Richards, code-named “Mr. Fantastic,” with unstable molecules for uniform-purposes — complete. Tony Stark, code-named “Iron Man,” with latest version of Extremis-augmented, body-integrated bio-armor — complete. Files containing knowledge regarding the use of new abilities, and equipment, now being transferred through trans-dimensional portal — complete. Begin extraction and installation of all files . . . wait . . . processing . . . processing . . . complete. Peter Reed Murdock Stark, it’s time to wake up now. You have important work to do.

The suspect woke up. He spent a few minutes using his newly-enhanced hearing to listen to conversations going on in every room in the building, until he was certain no one was nearby, except for those locked behind bars. He smelled for the distinctive smells of gun oil, ammunition, and gunpowder — and again, found none that were too close. His homunculus then told him, silently “This may seem a little strange, at least the first time,” and then flipped the newly-installed ‘armor’ switch, at which point the suspect saw his body and jail-inmate-uniform quickly covered with red-and-gold armor familiar to many fans of comic books, around the globe, although those same fans would have found the presence of billy clubs and web-shooters unusual, attached to Iron Man armor. “The unstable molecules will let the armor stretch with the rest of your body, so don’t bend the bars to escape — just stretch between them, instead, for that will make much less noise.”

As always, he did as he was told by his homunculus, without question, nor hesitation. His armored and elastic form slipped through the bars easily. Once out, though, his amplified hearing and enhanced sense of smell warned him that someone, probably a guard, was approaching. Spider-sense then confirmed imminent danger, so he jumped to the high ceiling — and stuck there. He held his breath as the guard strolled casually down the corridor. The guard was nearing the end of his shift, and was sleepy — but he would have noticed bent or broken bars to a cell. Due to his fatigue, however, he did not notice that the number of empty cells in this particular corridor was one more than it should have been — and he had no reason to look up.

Once this guard exited the far side of the corridor, the escaped prisoner crawled along the ceiling, and then noticed a ventilation shaft. It was covered by a metal grille, but the elastic powers of Reed Richards nullified that obstacle, as he simply oozed through the small holes in the grille, web-shooters and all — thanks to those unstable molecules from which the Fantastic Four’s uniforms are made. Within seconds, he was completely inside the ventilation shaft.

From inside his head, he heard the next instructions from his homunculus: “Head for the roof.” He did so, soon finding his way completely out of the ventilation system, and on the roof of a police station, in a small city in a world without superheroes. On this particular Earth, superheroes existed only as copyrighted characters in comic books, movies, television shows, and the like.

He spoke quietly: “Homunculus? What should I do now?”

“You need do nothing but accept our gratitude. Your mission is now complete.” The homunculus flipped the armor-switch back to the ‘off’ position, and the red-and-gold armor retreated from view. He then flipped another switch, and the escaped murder-suspect fell to the ground, able to see and hear, but unable to move. He heard four separate voices in his head say, “Thank you,” one after another. During his escape, the homunculus had reproduced by fission, splitting in half once, then each half splitting into two parts again — a new means of reproduction their species’ scientists had only just developed, and was only being used by experimental prototypes, such as these four particular homunculi.

A pair of small, green-skinned homunculi then crawled out of each of the murder-suspect’s ears, causing red blood to pool under his head, on the roof. “Ugh, human blood is so nasty,” one of them said. “Red is such a sickening color for blood!”

“Get used to it,” said another, “for I’m sure we will be seeing a lot more of it in the next few weeks.”

“Weeks? More like just a few days,” said a third of the green-skinned quartet. “In this universe our scientists recently discovered, no super-powered humans exist at all, except in what the humans call ‘fiction’ — stories about things which never actually happened, at least not here. The problems we have repeatedly encountered with these super-powered humans, every time we try to take over the Earth, will not exist here, with this Earth, at all. No one here will have the power to oppose us!”

The green homunculi were now marching, in a triumphant circle, around the paralyzed human. Although he could not move, their walking finally brought one of their faces into plain view, so he could see the strangely-shaped, green chin of his homunculus — and then he knew his planet was as doomed as he was, as he slowly bled out, unseen by any other human, on the rooftop of a police-headquarters building.

Secret_Invasion_Vol_1_5_Textless

A Skrull. Four tiny Skrulls, in fact. They didn’t even see the need to kill him before they activated tiny versions of Iron Man’s armor, and flew off, in four different directions . . . each to become someone else’s new homunculus, no doubt.

His last thought was relief, for he was blacking out, and knew he would not personally have to witness the destruction of his world’s civilization, and the enslavement of its people . . . but he saw no way either one could be avoided. Right now was, he thought, a much better time to die.

# # #

[Image credit — the “homunculus” image, above, was found at this website: http://commonsenseatheism.com/?p=8844, and the picture of the head of a Skrull was found at http://marvel.wikia.com/wiki/Skrulls.]

[Disclaimer: This is a work of fiction. No similarity to any living nor dead person, nor institution, is intended, nor should it be inferred. This story is not being sold for a profit, but is a work of fan-fiction available on the Internet, for anyone to read, for free. No copyright infringement involving characters owned by Marvel Comics is taking place, since this work is not being sold for a profit. If a representative of Marvel Comics requests it, this blog-post will be deleted, but it is expected that having this story available on the Internet will only increase, not decrease, that company’s sales and profits.]