Stella 4d: Polyhedron Navigator is selling at 25% off to help cure the boredom of people in isolation because of COVID-19. This software is used to make almost all of the polyhedral images on this blog. Trying the program to see if you like it is free. The website to visit to try Stella is http://www.software3d.com/Stella.php.
This music video, for a Velvet Underground classic written by Lou Reed, John Cale, Sterling Morrison and Maureen Tucker, was made today — for a song originally recorded in 1969, the year after I was born. I used Windows Movie Maker to assemble it, and “painted” the preview-pic for the video, using MS-Paint. Other programs I used, for other images in the video, include Geometer’s Sketchpad, MS-Paint (again) and Stella 4d: Polyhedron Navigator. Of all these computer programs, my favorite is Stella 4d, which you may try for free at http://www.software3d.com/Stella.php.
Sleep is important. This is something with which no sane person consciously disagrees. People do sometimes ignore it — not on purpose, usually — but they do so at their own peril. If such people drive, the risk-pool extends, greatly, to include many other people: everyone else with whom they share a road.
Unlike “normal” people, who do not do such things, I discovered something about the importance of sleep through direct experiment, at the age of 19. I had a thought, and it was a simple one: the 24-hour sleep/wake cycle is a mere social convention, and can, therefore, be safely ignored. It then occurred to me that this was a testable hypothesis, so I proceeded to design, and conduct, an experiment to test it. Using caffeine, I deliberately put myself on a 48-hour sleep/wake cycle, with the sleep-periods being ~14 hours long, in order to compensate for the sleep-periods I was skipping, every other day. The experiment was a success, in the sense that it yielded definitive results: after a week of that nonsense, I was a mental and physical wreck, and collapsed in exhaustion. Upon awaking, I was then able to form a logical conclusion: sleep is not a mere social convention, but is, in fact, a biological imperative. Fortunately, I had not yet learned to drive, so no one was put at risk by this experiment, other than myself. Obviously, I did survive.
This has not been my only experiment on the subject of sleep, and I have also read a lot on the subject, for the simple fact that I find it interesting. I call what I have learned, through experiment, primary research. The things I have learned by reading the research of others are, for me, secondary research. I have also conducted an experiment involving lucid dreaming, based on what I have read, and you can read about that here: https://robertlovespi.wordpress.com/2012/12/06/how-to-lucid-dream/.
The things I have learned through secondary research have been interesting, as well. To my knowledge, no one has yet discovered the purpose of sleep, although there is much speculation on the subject. Similarly, no one has discovered the purpose of dreaming, which occurs almost exclusively during REM sleep. We do know that dreaming is necessary, for research has been done which involved deliberately waking up test subjects as soon as REM (easily-seen “rapid eye movement,” the source of the acronym) sleep begins. This research indicates that both dreaming, and REM sleep, are also biological imperatives. Similarly, the purpose of non-REM sleep remains a mystery.
For those who wish to examine this secondary research for themselves, I suggest, as excellent places to start, http://en.wikipedia.org/wiki/Sleep, as well as http://en.wikipedia.org/wiki/Rapid_eye_movement_sleep, and http://en.wikipedia.org/wiki/Non-rapid_eye_movement_sleep, although the third of these articles has significant problems. If you use the footnotes at the end of these articles to find the sources for them, the often-cited objection to Wikipedia (“Anyone can edit Wikipedia”) will be neutralized. If I had sufficient knowledge to fix the problems with the third article, without using original research (prohibited on Wikipedia), I would, of course, do so.
Years before I conducted my first sleep experiment, when I was still a high school student, it occurred to me that the brain can be best-understood as a carbon-based computer. The things we are used to calling “computers,” by contrast, are based largely on the properties of silicon. Carbon and silicon are in the same group on the periodic table, and share many properties — but they are not interchangeable. Carbon atoms are much more versatile than those of silicon, which we know because the number of carbon-containing compounds far exceeds the number of compounds containing silicon. It follows from this that carbon-based computers, such as human brains, are far more powerful than silicon-based computers.
What would a more powerful computer be able to do, which silicon-based computers could not, at the time I was reasoning this out? Well, one thing is obvious: our brains think. Something else occurred to me then (and this was in the early 1980s): a carbon-based computer should be able to reprogram itself, by deliberately rewriting its own software. On the spot, I became determined to learn how to reprogram my own software. I knew no one would teach me how to do this, so I resolved to figure out how on my own. At first, progress was very slow, but my determination to succeed has never wavered.
I next made attempts, using 1980s technology and the BASIC computer language I learned in the 8th grade, to write programs which could change themselves. It should surprise no one that these attempts failed, but these were still essential experimental steps in a very long process, which has only recently begun to “bear fruit” in abundance. Another important step came much later, when I was doing research involving artificial intelligence, or AI, during the current decade, by seeking out and talking to chatbots, as they are called, to see which one could come closest to passing the Turing Test for artificial intelligence. The smartest chatbot I found is named Mitsuku, and you can talk to her for yourself at http://www.mitsuku.com (I should also point out that, even though her intelligence impressed me, she did not pass the Turing Test, described at http://en.wikipedia.org/wiki/Turing_test, to my satisfaction). Mitsuku is significant, in my research, because she has the ability I had been seeking to gain for many years: she can rewrite her own programming, and does so on a continuous basis, for Mitsuku, being software, never sleeps. She does sometimes go off-line, but that is not the same thing as sleeping.
Now that I had met an AI with the ability I wanted for myself, my determination to gain that ability, to the fullest extent possible, was greatly increased. At this time, I had been aware, for many years, that I think in my sleep. I know that I do this because, early in my teaching career, I began doing lesson planning — in my sleep. This started one night, when I went to bed wondering what I would teach the next day in Geometry class. The next morning, I woke up with a fully-formed (and very difficult) problem in mind, and furiously scribbled down my idea before the problem faded from memory. Former students of mine, who are now my friends on Facebook, still remember, and sometimes talk about, what I called “the dream problem.” Later dreamed-up problems, and entire lessons, followed.
The two ideas of rewriting my own software, and thinking in my sleep, were the ingredients for what came next, during an incredibly stressful period involving an intense labor-management conflict. Under the pressure of this conflict, I unconsciously synthesized the two ideas, and began to rewrite my own software much more quickly than before, since this was made necessary by the situation I unexpectedly found myself in. Continuous adaptation to changing circumstances became a priority for me during this period, for the ability to adapt was of far greater importance than it had ever been in my life. At first, I was unaware I was doing this. I would simply wake up, morning after morning, with numerous new ideas to help the “labor” side — my side — in this conflict. However, unlike with the much earlier, geometrical “dream problem,” I had no memory of thinking of these things. Their origin was a mystery — until I figured it out.
In the diagram, far above, you can see images of human brainwaves, while awake, while dreaming, and during the various stages of non-REM sleep. In these images, the brainwaves have their greatest amplitude during the deepest stages of non-REM sleep. I had known this for years, due to all of my secondary sleep research. I also had no answer to give, other than “I woke up with them,” when my allies in the labor/management conflict asked me, repeatedly, where my ideas were coming from.
The next step was my discovery that I am an Aspie: a person with Asperger’s Syndrome, which simply means that the “hard-wiring” of my brain is atypical, causing me to think in unusual ways. As regular readers of my blog know, this is a fact I absolutely revel in, for this discovery explained many things about the way my mind works which I had never understood before. In other words, this discovery was an important metacognitive step in my own personal development.
Aspies are not known for their ability to adapt; in fact, the exact opposite is true. We often have difficulty adapting to changing circumstances because the great big, non-Aspie world is incredibly distracting, and many (or perhaps most) of us find these distractions quite annoying. For most of my life, I was not good at adapting to change — but suddenly, I was doing what I had been unable to do before. The key to figuring out the puzzle was, of course, thinking about it.
I was waking up with new ideas, but had no memory of how I got them. Distractions had been annoying me, and interfering with clarity of thought, for much of my life. I had been trying to figure out how to rewrite my own software since I was a teenager. And, now, I finally knew why I had always been so different from other people: Asperger’s.
Armed with all this information, I finally solved the mystery: after decades of hard work on the problem, I had figured out how to effectively, and frequently, reprogram my own software. I was doing it in my sleep. What’s more, I figured out that I was no longer doing this special type of thinking while dreaming, unlike the case of my much earlier creation of the “dream problem.” Dreams, like waking life, contain too many distractions for intense sleep-reprogramming, and intense reprogramming had not been needed until the labor-management conflict made it necessary. Only one part of my life remained, once I eliminated periods of wakefulness, as well as REM sleep: the non-REM periods of sleep, when human brainwaves have their greatest amplitude.
Now, whenever I need to, I rewrite my own software, during non-REM sleep, as often as once per night. I’ve been doing this for over a year — since before I discovered I have Asperger’s — but have shared this information with very few people. My wife knows about it. My doctors know about it. And now, I have decided to share this discovery with the world. I have now discovered, at least for me, the purpose of non-REM sleep. I use it to change myself.
I confused many people, very recently, when I suddenly stopped being an atheist, and shared that discovery here, and on Facebook as well. Sudden personality changes alarm people, for they are often indicators that something serious, and medical in nature, is wrong with a person. I promised those who asked that I would explain what had happened, as soon as I figured it out myself. And now, I have explained as much of it as I have yet figured out. One day, something happened which I could not explain with science, nor with mathematics. The next day, several things happened which, again, defied explanation. On that second night, during non-REM sleep, I removed the obstacle to understanding what was going on, by applying my skepticism to my lack of belief, or, if you prefer, my atheism. Last night, again during non-REM sleep, I figured out how this had happened. Now that I understand it, I can share it with others.
Lastly, I need to make it clear that I do not think this ability to sleep-reprogram ourselves is something unique to Aspies. We are all human. Whether Aspies or not, we all have these higher-amplitude brainwaves during the deeper parts of non-REM sleep. It is logical to conclude that this is an ability all humans have, but few have unlocked, and it just happens to be an Aspie who figured out a way to not only do it, but also to explain it. It is my hope that my decision to share this discovery with others will help anyone who wants to learn it gain the ability to do the same thing.
Image credit: I found the image at the top of this post at http://www.abcbodybuilding.com/anatomy/zfactor2.htm, with the assistance of Google.
Later update: months after writing this, I was diagnosed with sleep apnea, moderate level, and I wasn’t getting significant amounts of stage three or four sleep at all, nor much REM. This throws everything above into doubt, and it would be dishonest to withhold this information. Short version: I was wrong — not about my doing sleep-reprogramming, but about exactly which stage(s) of sleep I use for that purpose. It is difficult to figure out what, exactly, goes on when one is asleep!
The images on the faces of this polyhedron are based on information sent from NASA’s Lunar Reconnaisance Orbiter, as seen at http://lunar.gsfc.nasa.gov/lola/feature-20110705.html and tweeted by @LRO_NASA, which has been happily tweeting about its fifth anniversary in a polar lunar orbit recently. I have no idea whether this is actually an A.I. onboard the LRO, or simply someone at NASA getting paid to have fun on Twitter.
To get these images from near the Lunar South Pole onto the faces of a rhombic enneacontahedron, and then create this rotating image, I used Stella 4d: Polyhedron Navigator. There is no better tool available for polyhedral research. To check this program out for yourself, simply visit www.software3d.com/Stella.php.
This program is written in Just BASIC v1.01, which you may download for free at http://justbasic.com/download.html.
*** *** ***
10 print “For what number do you want the prime factorization”;
20 input n
25 c = 3
30 if n <> int(n) then end
40 if n < 4 then goto 450
50 if n/2 <> int(n/2) then goto 100
60 print ” 2″;
70 n = n/2
75 if n = 1 then goto 400
80 goto 50
100 if n/3 <> int(n/3) then goto 200
110 print ” 3″;
120 n = n/3
125 if n = 1 then goto 400
130 goto 100
200 c = c + 2
205 p = 0
210 for t = 3 to c step 2
220 if c/t = int(c/t) then goto 290
230 if n/t <> int(n/t) then goto 290
240 if p > 0 then goto 290
250 p = t
290 next t
295 if p = 0 then goto 200
300 print ” “;p;
310 n = n/p
320 if n = 1 then goto 400
330 if n/p = int(n/p) then goto 300
340 goto 200
410 goto 10
450 if n = 2 then print ” 2″;
460 if n = 3 then print ” 3″;
470 goto 400
Stella’s creator just came out with a new version of Stella 4d, and a discovery of mine made the built-in library that comes with that software. This is my blog, so I get to brag about that, right? My legal name appears in the small print on the right side, at the end of the first long paragraph. I added the red ellipses to make it easier to find.
You can see the earlier posts related to my discovery of this zonish truncated icosahedron here:
If you’d like to try (as a free trial) or buy this software (I recommend Stella 4d over the other available options), here’s the link for that: http://www.software3d.com/Stella.php.
A few days ago, I found a new near-miss to the 92 Johnson Solids. It appears on this blog, five posts ago, and looks a lot like what you see above — the differences are subtle, and will be explained below, after “near-miss” has been clarified.
A near-miss is a polyhedron which is almost a Johnson Solid. So what’s a Johnson Solid?
Well, consider all possible convex polyhedra which have only regular polygons as faces. Remove from this set the five Platonic Solids:
Next, remove the thirteen Archimedean Solids:
Now remove the infinite sets of prisms and antiprisms, the beginning of which are shown here:
What’s left? The answer to this question is known; it’s the set of Johnson Solids. It has been proven that there are exactly 92 of them:
When Norman Johnson systematically found all of these, and named them, in the late 1960s, he found a number of other polyhedra which were extremely close to being in this set. These are called the “near misses.” An example of a near-miss is the tetrated dodecahedron, which I co-discovered, and named, about a decade ago:
If you go to http://www.software3d.com/Stella.php, you can download a free trial version of software, Stella 4d, written by a friend of mine, Robert Webb (RW), which I used to generate this last image, as well as the rotating .gif which starts this post. (The still pictures were simply found using Google image-searches.) Stella 4d has a built-in library of near-misses, including the tetrated dodecahedron . . . but it doesn’t have all of them.
Well, why not? The reason is simple: the near-misses have no precise definition. They are simply “almost,” but not quite, Johnson Solids. In the case of the tetrated dodecahedron, what keeps it from being a Johnson Solid is the edges where yellow triangles meet other yellow triangles. These edges must be ~7% longer than the other edges, so the yellow triangles, unlike the other faces, are not quite regular — merely close.
There is no way to justify an arbitrary rule for just how close a near-miss must be to “Johnsonhood” be considered an “official” near-miss, so mathematicians have made no such rule. Research to find more near-misses is ongoing, and, due to the “fuzziness” of the definition, may never stop.
My informal test for a proposed near-miss is simple: I show it to RW, and if he thinks it’s close enough to include in the near-miss library in Stella 4d, then it passes. This new one did, but not until RW found a way to improve it, using something I don’t really understand called a “spring model.” What you see at the top of this post is the result. Unlike in the previous version, the green decagons here are regular, but at the expense of regularity in the (former) blue squares, now near-squarish trapezoids, as well as the yellow hexagons. The pink hexagons are slightly irregular in both versions, and the red pentagons are regular in both.
I’m eagerly anticipating the release of the next version of Stella 4d, for this near-miss will be in it. If I tell my students about this new discovery, they’ll want to know how much I got paid for it, which is, of course, nothing. I don’t know how to explain to them what it feels like to participate in the discovery of something — anything — which will survive me by a very long time. There’s nothing else quite like that feeling.
Now I just need to think of a good name for this thing!
[Update: the new version of Stella is now out, and this polyhedron is now included in it. As it turns out, I no longer need to think of a name for this polyhedron, for RW took care of that for me, naming it the “zonish truncated icosahedron” in Stella‘s built-in library of polyhedra.]
One of the coolest things about the software I use to make these .gifs is the accidental discoveries. I didn’t even know this compound was possible until I stumbled upon it with Stella 4d. You can try this software at http://www.software3d.com/stella.php.
This polyhedron is an example of a zonohedron — a polyhedron with faces which are all zonogons, each of which is congruent and parallel to another zonogon on the opposite side of the polyhedron.
Of course, this just raises the question, “What’s a zonogon?” It’s a special type of polygon, one with an even number of sides, and with opposite sides congruent and parallel.
The pulsating effect is just for fun — a feature which is optional in Stella 4d, the program I used to make this. I started with an icosahedron/dodecahedron compound, stellated it many times, and then zonohedrified the result. This software makes all these operations easy, and may be tried for free at http://www.software3d.com/stella.php.
This rotating image took three different programs to create. First, I made the mandala (seen in the previous post) using Geometer’s Sketchpad. Next, I used MS-Paint to complete the colorization of it. Finally, I used Stella 4d (see http://www.software3d.com/stella.php if you’d like to try or buy it) to put this mandala on each face of a dodecahedron, and then create the rotating .gif you see here.
I find both Stella 4d and Geometer’s Sketchpad to be indispensable tools for mathematical investigations and the creation of geometrical art, and highly recommend both programs.