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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 Torus and Its Dual, Part II

After I published the last post, which I did not originally intend to have two parts, this comment was left by one of my blog’s followers. My answer is also shown.

torus talk

A torus can be viewed as a flexible rectangle rolled into a donut shape, and I had used 24 small rectangles by 24 small rectangles as the settings for Stella 4 for the torus, and its dual, in the last post — which, due to the nature of that program, are actually rendered as toroidal polyhedra. To investigate my new question, I increased 24×24 to 90×90, and these three images show the results. The first shows a 90×90 torus, the second shows its dual, and the third shows the compound of the two.

Torus90.gif

 

Torus90dual

Torus90dualcompound

When I compare these images to those in the previous post, it is clear that these figures are approaching a limit as n, in the expression “nxn rectangle,” increases. What’s more, I recognize the dual now, of the true torus, at the limit, as n approaches infinity — it’s a cone. It’s not a finite-volume cone, but the infinite-volume cone one obtains by rotating a line around an axis which intersects that line. This figure, not a finite-volume cone, is the cone used to define the conic sections: the circle, ellipse, parabola, and hyperbola.

What’s more, I smell calculus afoot here. I do not yet know enough calculus.

“Learn a lot more about calculus” is definitely on my agenda for the coming Summer, for several reasons, not the least of which is that I plainly need it to make more headway in my understanding of geometry. 

[Note: Stella 4d, the program used to make these images, may be found at http://www.software3d.com/Stella.php.]

A Torus and Its Dual, Part I

Torus

The torus is a familiar figure to many, so I chose a quick rotational period (5 seconds) for it. The dual of a torus — and I don’t know what else to call it — is not as familiar, so, for it, I extended the rotational period to 12 seconds.

Torus dual

By viewing the compound of the torus and its dual, one can see the the dual is the larger of the two, by far:

Torus dual with torus

I used Stella 4d to make these images. It’s a program you can buy, or try for free, at this website: http://www.software3d.com/Stella.php.

My Possible Encounter with Ron Paul

photo

So, while driving on an Arkansas highway, I had an encounter with Ron Paul.

Maybe.

What I know with certainty is that I saw a vehicle with a license plate that stated, “RONPAUL.”

I was unable to catch up with this vehicle to check to see if it was being driven by, um, the Ron Paul, and this is due to Ron Paul’s/the driver’s libertarian principles.

And, by “libertarian principles,” I mean that this guy, whoever he was, was driving just as fast as he wanted to.

# # #

[Photo from Ron Paul’s Wikipedia page.]

 

Ten Commandments of the Universe, Set to Music

IDL TIFF file

Because there’s nothing wrong with mixing a little Rolling Stones with your physics, that’s why. 

Beginning the Fractiles-7 Refrigerator Experiment

To begin this experiment, I first purchased two refrigerator-sized Fractiles-7 sets (available at http://fractiles.com/), and then, early on a Sunday, quietly arranged these rhombus-shaped magnets on the refrigerator in our apartment (population: 4, which includes two math teachers and two teenagers), using a very simple pattern.

160207_0000

Here’s a close-up of the center. There are 32 each, of three types of rhombus., in this double-set, for a total of 96 rhombic magnets, all with the same edge length.

160207_0001

The number of possible arrangements of these rhombi is far greater than the population of Earth.

The next step of the experiment is simple. I wait, and see what happens.

It should be noted that there is a limit on how long I can wait before my inner mathematical drives compel me to play with these magnets more, myself — but I do not yet know the extent of that limit.

Open Octahedral Lattice of Cubes and Rhombicosidodecahedra

This pattern could be continued, indefinitely, into space.

Augmented Rhombicosidodeca

Here is a second view, in rainbow color mode, and with all the squares hidden.

Augmented Rhombicosidodeca rcm

[These images were created with Stella 4d, software you may buy — or try for free — right here.]

An Absurd “Explanation” for Creativity in Humans (a short short story)

orbiting-dual-planet

[Note: I’m currently taking a class focusing on creativity, and this was first written as an assignment for that class.]

The reason some people are creative, while others are not, is simple: we’re hybrids of pure human DNA, plus that of extraterrestrials of two types, who visited earth and interbred with the natives between 60,000 and 50,000 years ago. The first type of alien came from a planet called Itaumiped, and their hybrid descendants inherited the creative abilities of the Itaumipedeans. The other type, for similar reasons, carries DNA from the inhabitants of the nearby world Almausoped — and those from Almausoped tend be be rather bright, but also very imitatitve. For example, the creation of the comic book character Spider-Man was clearly the work of an Itaumipedean/human hybrid, but the work of grinding out numerous Spider-Man derivatives in comic books (the Scarlet Spider, Spider-Girl, that Spider-Man from an alternate universe with six arms, Spider-Gwen, Spider-Woman, Spider-Ham, etc.) was performed by imitative Almausopedean/human hybrids.

Evidence for the creativity of the residents of Itaumiped comes from the name of the planet itself: a rather clever anagram of the letters in the sentence “I made it up.” Evidence that those from Almausoped are imitative, rather than creative, may be found in the fact that their planet’s name is an anagram for “Also made up,” something clearly borrowed, then slightly altered, from Itaumiped and its creative residents. This raises another question, though: how could the names of planets given to them by ancient civilizations come from anagrams of modern English sentences?

The search for evidence is now ongoing for the obvious explanation: what really happened clearly involved not just space travel, but time travel as well.

# # #

[Note: This double planet/polyhedra image of Itaumiped and Almausoped first appeared here, and was created using Stella 4d, software you can try for free at this website.]

Bicycles Don’t Need Telepathy: A Short Short Story

bicycle

Since I hate being told what to do, and people kept telling me I had to learn to drive, and get a driver’s license, and so on, I refused, getting around by foot or bicycle, instead . . . or even hitchhiking . . . until age 20. It wasn’t unusual, therefore, as a teenager, for me to bounce out of the house, ready to jump on my blue ten-speed, and take off for my high-school weekend-job. What was unusual was to find, in mid-bounce, a different ten-speed, this one red, just where I had left my blue ten-speed the night before.

“What the #@&%? Did someone sneak into the carport and PAINT my bicycle?”

No, fool. I’m your bike now. Your old bike’s gone, man.

Clearly, it seemed, some all-too-human prankster was at work. I circled the house. I looked on the roof. I checked the storage building, and under the house. I found no one. Apparently, what seemed to be a bicycle talking to me was, well, exactly that.

You’re gonna be late for work, dummy. I told you, your bike got stolen — by my old, ungrateful human. He left me here. Let’s go!

“Why?”

Because you’re going to be late, dumb&%#! Are you trying to get fired?

Annoyed that this replacement-bicycle had a point, I jumped on the seat and took off, dodging the numerous potholes which have been considered as unofficial Arkansas State Monuments, and which I got to see more clearly than all those (ugh) car drivers out there.

Hey, jerk, my best friend is a car!

“A telepathic bicycle? You’re actually reading my mind? Someone had access to a telepathic bicycle, and abandoned that kind of power for my crappy old ten-speed? You’ve gotta be kidding!”

We were passing a female jogger at that moment. My mind was on keeping control of the bicycle, in order to stay out of traffic, while carefully not hitting her. My new “telebike,” however, had a different agenda, and “mind-shouted” at the jogger — with me able to hear it.

Hey! WOO! Great #@#! When I come back, ya wanna get together and $#@%#& the old %$#%@ &*%$#@$?

The understandably-irate jogger thought that was me, not the bike, threw an ink pen at me, and hit me in the head. “Drop dead, &%$#$#@!”

Clearly, having a telepathic bicycle just wasn’t worth the trouble involved.

After a turn (to get out of direct view of the offended jogger), I knew this street would go over a short bridge over a very deep drainage ditch, not far from here — one lined with large boulders, placed there by the city. By the time I got there, I was heading for the drainage ditch at top speed, and jumped off into some soft grass just before this crazy red ten-speed went airborne, screaming in my mind until crunching into the boulders at the bottom of the ditch, at which point “psycho-bike” fell silent. I, however, had a bit more to say, so I dusted myself off, and walked to the short bridge.

Looking over, into the drainage ditch, I surveyed the bicycle-wreckage, and said, “I’m gonna be out $100 or so to replace my old bike with another bike that doesn’t talk, and I’ll be late for work today, too . . . but at least I’ve gotten rid of my primary problem today, and that’s YOU!”

With that said, I walked to work. I was late, but did not get fired, and I never saw either the red, nor the blue, bicycle again.

###

[Note: the central event here — a bike being stolen, and replaced by the thief with another one — actually happened to me. So is the fact, and the reasons given, that I did not start driving until age twenty. Most of the rest of this is pure fiction. This was written as an assignment for a class I am taking which focuses on creativity, with the writing prompt being to “carry on a conversation with [my] childhood bicycle,” and I decided to post it here, as well.]

The Spider-to-Human Ratio

spider to human ratio

Since I like spiders, I was pleased to read a rough estimate of 21 quadrillion for the world’s population of spiders (source: here).

The website http://www.worldometers.info/world-population/ gives the current human population as ~7.4 billion. Dividing the estimated spider population by the estimated human population yields Earth’s estimated spider-to-human ratio: 2.8 million.

Yes, your share approaches three million spiders. At least they are good at taking care of themselves!

[Source of the image of the spider above, an adult male phidippus audax: https://www.flickr.com/photos/opoterser/2989573241]

Constructing “Nightday” — An Experiment Involving Sleep

Waning-Gibbous-Moon

The last workweek having left me rather tired, I went to bed early Friday, after work, and then, having slept all I could sleep, I then woke back up quite early Saturday morning, before sunrise, and couldn’t get back to sleep. I was tired all day Saturday, but not too weary to think. What I thought was simple: as tired as I am, it sure would be nice to have a three-day weekend this weekend. Next, I thought, yeah, this would be nice, but that won’t make it happen. Finally, I realized that I actually could, perhaps, come up with some hopefully-clever and effective way to get the three-day weekend I want . . . and, having had an idea to do exactly that, I’m trying it right now.

I’ve tested the 24-hour sleep/wake cycle before, trying to find ways to lengthen that period of time. (Ever wanted more hours in the day? Well, I actually tried to make that happen, once, but the results were less than successful.) This time, however, I’m not trying to get extra hours in a day, but an extra day in the weekend — by simply using shorter “days,” and thus making “room,” temporally, to add an extra sleep-period and wake-period into the weekend. So, Friday, I fell asleep around 5:00 pm, and did so without the prescribed medication (which includes sedatives) which I usually take at bedtime, since it wasn’t that late yet . . . so I simply fell asleep because I was tired.

Without the sedatives I am used to taking, of course, I didn’t stay asleep anything like a full eight hours, and instead “popped” back awake at around 9 pm, which was less than an hour ago, as I write this. Rather than sedating and returning to sleep, however, I took the other medication I am prescribed for this time of the day (such as that needed to regulate blood pressure), and then made my “morning” coffee, which I am enjoying now . . . to begin the extra “day” I’m attempting to add to this weekend, between Saturday and Sunday. My hypothesis is that I can deliberately alter my sleep/wake cycle in such a way that I have three (shorter) sleep/wake cycles in two calendar days, thus giving myself a three-day weekend, of a sort, and enjoy the benefits of a three-day weekend as a result. If, come Monday, I feel like I’ve had a three-day weekend — in that I feel unusually well-rested — I will consider this experiment to “create” a working illusion of a three-day weekend, without any actual extra time, to be a success (subject to the opinion of my doctors, to whom I will describe all of this).

I plan to stay awake until roughly dawn on Sunday, and then go to sleep until, well, whenever I wake up. I’ll then have a shortened post-sleep Sunday wakefulness-period, go to sleep at a reasonable hour Sunday night, and get a good, full night’s sleep then, before going to work on Monday.

Right now, therefore, I’m having the middle “day” of what feels, subjectively, like a three-day weekend, and having it at night, between what seems, now, like it was yesterday (the shortened Saturday), and what I anticipate as my shortened Sunday, after I sleep again, tomorrow. Since it’s easier to talk about this extra “day” I’m having tonight if I give it a name, I’m doing so: I’m calling it “Nightday.”

Some readers may object that I’ve merely come up with an overly-convoluted way to analyze a four-hour Saturday-afternoon nap. I’ll concede that they do have a point . . . but if my calling this “Nightday,” and telling myself that I’m enjoying a three-day weekend, actually turns out to help me feel and act more rested next week, then I’ll take those benefits and run with them, regardless of what any critics tell me (unless, of course, my doctors are among those critics). If this experiment has only beneficial results, and passes medical review, then I’ll likely use more Nightdays to get additional three-day weekends in the future, whenever I need, or simply want, them.

~~~

Important disclaimer: nothing in this blog-post should be taken as any form of medical advice, for I am not medically trained. I have taken the precaution of discussing my practice of occasionally inventing and conducting experiments such as this with my own physicians, and will continue to do so. No one should attempt to replicate this experiment without first consulting their own physician(s).

[Image credit: the photo of the Moon shown above was found here — https://photographylife.com/moon-waning-gibbous. It isn’t identical in appearance to the current waning gibbous Moon, having been photographed quite some time earlier, but it is close.]