An Experiment Involving Augmentation of Octahedra with More Octahedra, Etc.

I’m going to start this experiment with a single octahedron, with faces in two colors, placed so that two faces which share an edge are always of different colors.

1

Next, I will augment the red faces — and only the red faces — with identical octahedra.

2

The regions with four blue, adjacent faces look as though they might hold icosahedra — but I checked, and they don’t quite fit. I will therefore continue the same process — augmenting only the red faces with more octahedra of the original type.

3

I’ve now decided that I definitely like this game, so I’ll keep playing it.

4

Immediately above, at the fourth of these images, some of the octahedra have started to overlap slightly, but I’m choosing to not be bothered by that — I’m continuing the now-established pattern, just in order to see where it takes me.

5

The regions of overlap are now far more obvious, but I’m continuing, anyway. Why? Because this is fun, that’s why! Right now, Stella 4d, the program I use to do these polyhedral manipulations, is chugging away on the next one. (This program is avilable at http://www.software3.com/Stella.php.) Ah, it’s ready — here it is!

6

Rather than repeat this process again, I now have another question: what would the convex hull of this figure look like? (A convex hull of a non-convex polyhedron is the smallest convex polyhedron which can contain a given non-convex polyhedron.) With Stella 4d, that’s easily answered.

Convex hull

I must admit this: that was nothing like what I expected — but such unexpected discoveries are a large part of what makes these polyhedral investigations with Stella 4d so much fun. And now, to close this particular polyhedral journey, I will have Stella 4d produce, for me, the dual of the convex hull shown above. (In case you aren’t familiar with duality regarding polyhedra, it describes the relationship between the octahedron, with which this post began, and the familar cube. Basically, with duals, faces and verticies are “flipped” over edges, although that is an extremely informal and imprecise way to describe the at the process.)

dual of Convex hull

And with that, my friends, I bid you good night!

My Aqua Regia Story

This is my twentieth year teaching, but only the first year when I have not taught at least one class in chemistry, and I miss it. One of my fondest memories of chemistry lab involves the one time I experimented with aqua regia — a mixture of acids which, unlike any single acid, can dissolve both gold and platinum, the “noble metals.” I had read a story of a scientist’s gold Nobel Prize being protected from the Nazis by dissolving it in aqua regia, and then recovering the gold from solution after World War II had ended. Having read about this, I wanted to try it myself, and also thought it would make an excellent lab for classroom use — if I could figure out how to recover the gold, and also learn what precautions would be needed to allow high school students to perform this experiment safely. For sensible and obvious reasons, I conducted a “trial run” without students present, but with another chemistry teacher nearby, since aqua regia, and the gases it produces when dissolving gold, are quite dangerous. Someone else has put a video on YouTube, showing aqua regia dissolving gold, so you can see something much like what I saw, simply by watching this video.

First, I obtained one-tenth of a troy ounce of gold, which cost about $80 at the time. I had read about the extreme malleability of gold, one of the softest metals, and wanted to see evidence for it for myself — so, before I prepared the the aqua regia, I used a hammer to try flattening the gold sample into a thin sheet. That didn’t work, but it didn’t take long for me to figure out why — I had accidentally bought gold coin-alloy, which is 10% copper, not pure gold. Since this alloy is far less malleable than pure gold, my attempt to flatten it had failed, but I also knew this would not pose a problem for my primary experiment — the one involving aqua regia. Also, I didn’t have another spare $80 handy, to purchase another 1/10 troy ounce of pure gold, so I proceeded to make, for the first time in my life, a small amount of aqua regia — Latin for “royal water.”

Unlike what is shown in the video above, I prepared the acid-mixture first, before adding the gold, using a slightly-different recipe:  the traditional 1:4 ratio, by volume, of concentrated nitric acid to concentrated hydrochloric acid. Both these acids look (superficially) like water, but the mixture instantly turned yellow, and started fuming, even before anything was added to it. Wearing full protective gear, I watched it for a few minutes — and then, using tongs held by gloved hands, lowered my hammer-bashed sample of gold into the fuming, yellow mixture of concentrated acids.

It worked. It was a fascinating reaction, and a lot of fun to watch. At approximately the same time that the last of my gold sample dissolved, something occurred to me:  I had failed to research how to recover the dissolved gold from the resulting solution! No problem, I thought — I can figure this out. (I am seldom accused of lacking self-confidence, even when I’m wrong.)

My first idea was to use a single-replacement reaction. Many times, I have had students extract pure silver from a solution of silver nitrate by adding a more-active metal, such as copper. The copper dissolves, replacing the silver in the silver nitrate solution, and silver powder forms, as a precipitate, on the surface of the copper. Thinking that a similar process could be used to precipitate out the gold from my gold / aqua regia mixure, I simply added come copper to the reaction beaker. The corrosive properties of my aqua regia sample had not yet been exhausted, though, and so the remaining aqua regia simply “ate” the copper. The result was a mess — I had only succeeded in turning an already-complicated problem into an even-more-complicated problem, by adding more chemicals to the mixture. More attempts to turn the gold ions back into solid gold dust, using other chemicals, followed, but all of them failed. Finally, I used a strong base, sodium hydroxide, to neutralize the still-acidic mixture, and then, disgusted by my failure to recover the gold, found a way to safely dispose of the mixture, and did so.

In retrospect, I think I know where I messed up — I should have neutralized the remaining acids in the mixture with sodium hydroxide first, before adding copper to cause the gold to precipitate out, in a no-longer-acidic solution of ions with much less hydronium present. That, I think, will work, and I do intend to try it sometime — after doing more research first, to increase my level of certainty, and also after waiting for the current price of gold to drop to less-expensive levels. Right now, after all, a tenth of a troy ounce of gold costs roughly $120, not a mere $80.

As for the lost $80, I’m not upset about that anymore. I definitely learned things while doing this, and now view the $80 spent as simply the cost of tuition for an educational experience.

On Sleep, Non-REM Sleep in Particular, and Asperger’s Syndrome

sleep brainwaves

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!