Both of these versions of the stella octangula, or compound of two tetrahedra, were made with Stella 4d, software available at http://www.software3d.com/Stella.php.
Two Colorings of a Hollow Stella Octangula
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Author Archives: RobertLovesPi
A Polyhedral Caltrop
Created using Stella 4d, available at http://www.software3d.com/Stella.php.
A Great Dodecahedron, Augmented with Twelve Icosidodecahedra, and Its Dual
Each face of a great dodecahedron is a regular pentagon, and each of those pentagonal faces has an icosidodecahedron attached to it, in the figure above. The dual of this figure appears below.
Both images were created with Stella 4d, software available at www.software3d.com/Stella.php.
My Australia Story
I once got into a huge argument, as a 7th grade student, in a “talented and gifted” section of Social Studies. The issue: how many countries are there in the continent of Australia?
The assignment was to choose a continent, and draw a map of it on a full-size posterboard. I had worked for hours on this map, only to get it back, ruined, for the teacher had taken a red ball-point pen, slashed through my line “state and territorial boundaries” in my map’s key, and had written, as a correction, “not states — COUNTRIES.” She also docked points from my grade, but that was a minor issue, to me, compared to her ruining my map. She could have, at least, written her incorrect comment on the back of my map!
When I confronted her about her mistake, she maintained that the political divisions you see above are independent countries. In my opinion, “Northern Territory,” especially, doesn’t sound particularly sovereign, and I said so, but she may not have understood the definition of “sovereign,” for that did not work. Confronted with this absurd situation, I proceeded to grab the “Q” volume of a nearby encyclopedia, and began reading the article about Queensland, loudly enough for the entire class to hear: “Queensland: one of the states of Australia….” I freely admit that, at the time, my goal was to embarrass and humiliate her right out of the teaching profession — for the benefit of her present and future students. I’ve changed my approach, a lot, since then.
A huge brouhaha ensued, and we ended up taking each other to the assistant principal’s office: her, to report a disruptive and defiant student; and me, to report an incompetent teacher, who, in my view, at that age, should have been fired on the spot. Dealing with this situation was probably one of the stranger, and more difficult, situations of that assistant principal’s career, for he knew that Australia is both a single country, and a continent — but he could not, for political reasons I did not yet understand, agree with me in front of this teacher. As for me, I was simply incredulous that someone could be a certified social studies teacher, and not know this basic fact about world geography. The whole scenario, to me, was surreal.
The assistant principal handled it well. To the teacher, he said, “You can go back to class — I’ll handle Robert.” He then “handled” me, after she left, in the only way that could have possibly worked: with an apology, and a polite request to do my best to endure her ignorance until the upcoming end of the year. I respect honesty, was being given a request, not an order, and he had conceded that I was correct. I therefore chose to cooperate — with his polite request.
If he had not taken this approach, I likely would have added him to the list I had, at the time, of people (a mixture of administrators and teachers) whom I was trying to drive out of the education profession, for the benefit of all — but he did the right thing, thus earning my respect.
As for the teacher, I survived the rest of her class, brain intact, and assume she is now retired, this being well over thirty years ago. I’m now in my twentieth year as a teacher, myself, and am pleased to report that average teacher quality has dramatically improved since this fiasco happened. (I wish I could say the same about average administrator quality, but there are, at least, a few competent people working in that field, as well.) During my years of teaching, I haven’t encountered a single teacher who lacked this basic bit of knowledge about world geography. In fact, I count, among my colleagues, many of the smartest people I know.
I am glad, however, that I don’t have to call the teacher in this story a colleague. I simply cannot respect willful, stubborn ignorance, especially in the face of evidence that one is wrong. When one of my students catches me making a mistake, I do the right thing: I thank them, make certain everyone understands the correction, and then we move on with the lesson. That’s what this 7th grade teacher of mine should have done, as well.
On Coming Out of the Closet
I have a strong belief that asking nosy questions is an incredibly rude thing to do, and I find it particularly appalling when person A asks person B to identify his or her sexual orientation. What could be more personal, and private, than what consenting adults do in their own bedrooms? For this reason, I don’t ask questions on this subject, nor do I tell (when asked). To ask would be rude, and to answer such a nosy question would be to encourage rudeness. I don’t wish to do either.
There is exactly one situation I can think of where person A needs such information from person B, and that is if person A wants to have sex with person B. Even then, though, are there not far better ways to begin a flirtatious conversation?
At a school where I used to teach, I co-sponsored a student-initiated GSA, or Gay-Straight Alliance. On the basis of this co-sponsorship, plus my wearing a rainbow-colored wristband, many assumed that I was gay. To make a statement about how such things do not matter, I refused to ever confirm, nor deny, such rumors. On Facebook, I didn’t answer the “interested in” questions, nor did my blog reveal my sexual orientation — until now.
At this time, having considered the ramifications carefully, I’ve decided it is time to come out. I’m teaching at a new school, and am no longer co-sponsoring a GSA. I’ve kept quiet on the subject of my own sexual orientation long enough to make my point. And, now, I’m going to make another point . . . by coming out, here and now.
Yes, world, I admit it . . . I am a raving heterosexual, and one who has finally decided to come out — as a straight person. I’m not ashamed of it; this is part of who I am. There may be some who do not approve of this, but I do not have to care what they think.
In case my point is escaping you, please consider what it is like for gays, lesbians, bisexuals, and others who don’t conform to society’s norms, to do what I just did. Imagine what it would be like, for “heteros,” if social pressure favored homosexuality, rather than heterosexuality. How many “straights,” in such a world, would be willing to risk being ostracized, losing their jobs, being shunned by former friends and family, etc., simply for admitting they have a preference for the other sex, rather than the same one? If you, yourself, are also straight, and lived in a gay-dominated world, would you have the courage to come out as one of “those people,” the “heteros,” with their allegedly-“disgusting” sexual practices?
It’s ridiculous that anybody has to think about such things. “Live and let live” is not too much to ask — of anyone.
A True Story of a Young Aspie Getting in Trouble with “Show and Tell”
In elementary school, in the 5th grade, I managed to get in trouble for a “show and tell” project. As usual, getting in trouble was not my objective, but it happened anyway. This was decades before I learned I have Asperger’s, but, looking back, none of this would have happened were I not an “Aspie,” as we call ourselves.
This image, which I found here, is very much like the poster I made, by hand, and used for this project:
That was the “show” part of this “show and tell” project. For the “tell” part, I explained how nuclear chain reactions work, and then explained how nuclear bombs are made. It’s very simple: you have two slightly sub-critical masses of uranium-235 or plutonium-239, and physically bring them together, so that the total mass exceeds the critical mass. At that point: boom.
The hard part, of course, is actually obtaining the U-235 or Pu-239, for those aren’t things you can simply buy at the local hardware store. Ironically, I did know where to find both uranium and plutonium — at the very same university, about an hour away, where I’d spent far too much time conducting mostly-unsupervised experiments with both elements, along with lots of liquid mercury, before my tenth birthday. (I still suspect that all that radiation may have turned me into a mutant.) However, I also knew that the uranium and plutonium there would not have nearly enough of the correct isotope of either element, making this information irrelevant to my “show and tell” report, and so, for this reason, I did not tell them where to find the uranium and plutonium I had previously used for experiments.
I didn’t figure this out in class that day, since I’m not particularly good at “reading” emotions, facial expressions, and body language, but, apparently, I really upset, and scared, my teacher. This became apparent when she called my mother, and, later, my mother asked me to tell her what I’d done in school that day. Being excited about the “show and tell” presentation I’d given that day, I immediately told my mother all about it. When she told me the teacher had called her, concerned about me explaining to my class how to build atomic bombs, I was confused, since I didn’t understand, at all, why what I had actually said posed any problem. To explain this to my mother, I simply said, “But, Mom, I didn’t tell the class where to actually get the uranium-235 or plutonium-239! I don’t know where to find those isotopes!”
This was enough to convince my mother that I had not, in fact, done anything wrong. She called the teacher back, and simply asked if I had, or had not, included that critical bit of information: where to find the actual fissionable material needed for a nuclear bomb to work. When the teacher replied that I had not done that, my mother’s response was both sensible, and logical: “Well, then, what’s the problem?”
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Postscript, for those who might be worried about the childhood experiments I mentioned above: at around age 40, I asked a physician about my worries regarding early exposure to mercury vapor and radiation. He told me that any problems I might have, as a result of such experiments, would have already showed up by then, and that I could, therefore, stop worrying about this. Thus reassured, I did exactly that.
How Richard Feynman Saved Eastern Tennessee from Getting Nuked
I’m reading the book shown above for the second time, and am noticing many things that escaped my attention the first time through. The most shocking of these items, so far, is finding out that history’s first nuclear explosion almost occurred by accident, in Oak Ridge, Tennessee, during World War II. One person prevented this disaster, and that person was Richard Feynman, my favorite scientist in any field. If you’d like to read Feynman’s account of this, in his own words, it’s in the chapter “Los Alamos from Below,” which starts on page 107.
Feynman, a physicist, was one of many civilians involved in the Manhattan Project, doing most of his work in New Mexico. At one point, though, he obtained permission to visit Oak Ridge, in order to try to solve problems which existed there. These problems were caused by the American military’s obsession with secrecy, which was caused, in turn, by the fact that it was known, correctly, that at least one spy for the Nazis was among the people working on the Manhattan Project. The military’s “solution” to this problem was to try to keep each group of civilians working for them in the dark about what the other groups of civilians were doing. Most of them had no idea that they were working to develop a bomb, let alone an atomic bomb. In Tennessee, they thought they were simply working on developing a way to separate uranium isotopes, but did not know the underlying purpose for this research.
The military men in charge knew (because the physicists in New Mexico figured it out, and told them) a little bit about the concept of critical mass. In short, “critical mass” means that if you get too much uranium-235 in one place, a runaway chain-reaction occurs, and causes a nuclear explosion. The military “brass” had relayed this information to the civilian teams working in Tennessee, by simply telling them to keep the amount of U-235 in one place below a certain, specific amount. However, they lacked enough knowledge of physics to include all the necessary details, and they deliberately withheld the purpose for their directive. Feynman, by contrast, did not share this dangerous ignorance, nor was he a fan of secrecy — and, as is well known, the concept of respecting “authority” was utterly meaningless to him.
While in Tennessee, Feynman saw a large amount of “green water,” which was actually an aqueous solution of uranium nitrate. What he knew, but those in Tennessee did not, is that water slows down neutrons, and slow neutrons are the key to setting off a chain reaction. For this reason, the critical mass for uranium-235 in water is much less than the critical mass of dry U-235, and the “green water” Feynman saw contained enough U-235 to put it dangerously close to this lower threshhold. In response to this, Feynman told anyone who would listen that they were risking blowing up everything around them.
It wasn’t easy for Feynman to get people to believe this warning, but he persisted, until he found someone in authority — a military officer, of course — who, although he didn’t understand the physics involved, was smart enough to realize that Feynman did understand the physics. He was also smart enough to carefully listen to Feynman, and decided to heed his warning. The safety protocols were modified, as were procedures regarding sharing of information. With more openness, not only was a disaster in Tennessee avoided, but progress toward developing an atomic bomb was accelerated. It turns out that people are better at solving problems . . . when they know the purpose of those problems.
Had this not happened, not only would Eastern Tennessee likely have suffered the world’s first nuclear explosion, but overall progress on the Manhattan Project would have remained slow — and the Nazis, therefore, might have developed a controlled nuclear bomb before the Americans, making it more likely that the Axis Powers would have won the war. Richard Feynman, therefore, dramatically affected the course of history — by deliberately putting his disdain for authority to good use.
My Complete List of Complaints About My New School
For the last three weeks, for the beginning of my twentieth year as a teacher, I’ve been teaching at a different high school. I am much happier, now, due to this change. This being a personal blog, it is my policy not to name my school, nor school district, here. However, I see no problem with posting my complete list of complaints about this new school. Here it is:
In Anticipation of Sam Harris’s Newest Book
Sam Harris has already written these five books, all of which I have read. I wasn’t crazy about the first one, but thoroughly enjoyed all the others. They are presented here, in order, by publication-date.
- The End of Faith (2004)
- Letter to a Christian Nation (2006)
- The Moral Landscape: How Science Can Determine Human Values (2010)
- Lying (2011)
- Free Will (2012)
This month, he has a new book coming out. It’s called Waking Up: A Guide to Spirituality Without Religion. I have already pre-ordered it on Amazon, and am looking forward to its arrival.
Here is a link to a book review of Waking Up which was published in The New York Times, yesterday: http://www.nytimes.com/2014/08/31/opinion/sunday/frank-bruni-between-godliness-and-godlessness.html?_r=1.
This link is to the page on Amazon where you can order Waking Up: http://www.amazon.com/Waking-Up-Spirituality-Without-Religion/dp/1451636016/ref=sr_1_1?ie=UTF8&qid=1409584480&sr=8-1&keywords=waking+up+sam+harris.
On Sleep, Non-REM Sleep in Particular, and Asperger’s Syndrome
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.
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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.
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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!
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