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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.

On John Lennon’s 1971 Song, and Song Title, “How Do You Sleep?”

This is one of the most vicious, cutting songs ever written. Here is a video for it, so you can hear it for yourself, and see the things to which the song refers.

It was, of course, written targeting one specific person, but it isn’t difficult to listen to it, and think of other people to whom this question could — and perhaps should — be asked. In other words, there is no reason, in today’s world, to limit this song to merely being an “attack song” against Paul McCartney, although that was certainly John Lennon’s intent when he wrote it.

When anyone does horrible things — far worse than anything McCartney ever did — this song’s title makes a good question one could ask them: how do you sleep?

After thinking about it, I have come up with five possible answers.

  1. Some people lack a conscience, and so are untroubled by guilt — ever — no matter what they do.
  2. Delusional people can do horribly evil things, but be convinced that they are moral, righteous, and doing what they should be doing. If you have never encountered such behavior, consider yourself fortunate.
  3. Sometimes, people are simply ignorant of how harmful their actions have been. One cannot feel guilty over harm one does not know one caused.
  4. Even those who are tormented by their own consciences, and past actions, can still sleep with the aid of sedatives, whether over-the-counter, or prescribed. This is risky, of course, and should only be done under the care, and following the instructions, of a physician. (Yes, I am aware that illegal drugs — or legal ones that aren’t considered medicines, such as alcohol — can also be used for this purpose, but that’s even more risky, and I wish to make it perfectly clear that I strongly advise against doing that.)
  5. Narcolepsy.

If you can think of any others, please leave them in a comment.

Sixty and Sixty: A Chiral Polyhedron, as well as the Compound of It, and Its Own Reflection

60 and 60 -- chiral

This polyhedron is chiral, meaning that (unlike many well-known polyhedra) it exists in “left-handed” and “right-handed” forms — reflections of each other. These “reflections” are also called enantiomers. I call this polyhedron “sixty and sixty” because there are sixty faces which are irregular, purple quadrilaterals, as well as sixty faces which are irregular, orange pentagons.

I stumbled upon this polyhedron while playing around with Stella 4d: Polyhedron Navigator, software you can try right here. For those who research polyhedra, I know of no better tool.

To see the other enantiomer, there is a simple way — just hold a mirror in front of your computer screen, with it showing the image above, and look in the mirror!

With any chiral polyhedron, it is possible to make a compound out of the two enantiomers. Here is what the compound looks like, for this “sixty and sixty” polyhedron cannot be seen this way, so here is an image of it, also created using Stella 4d.

60 and 60 chiral --Compound of enantiomorphic pair

One of Many Faceted Rhombicosidodecahedra

Faceted Rhombicosidodeca the dual of the 32nd stellation of the strombic hexaconta

This was created by making the dual of the 32nd stellation of the strombic hexacontahedron, which is itself the dual of the rhombicosidodecahedron. This technique for finding facetings works because faceting is the reciprocal function of polyhedral stellation.

I did this using Stella 4d, which you can try for yourself, for free, at http://www.software3d.com/Stella.php.

A Rhombic Ring of Icosahedra, Leading to a Rhombic Dodecahedron Made of Icosahedra

As it turns out, eight icosahedra form this rhombic ring, by augmentation:

Rhombic ring of Icosa

Measured from the centers of these icosahedra, the long and short diagonal of this rhombus are in a (√2):1 ratio. How do I know this? Because that’s the only rhombus which can made this polyhedron, a rhombic dodecahedron, dual to the cuboctahedron.

RD of Augmented Icosa

This rhombic dodecahedral cluster of icosahedra could be extended to fill space, since the rhombic dodecahedron itself has this property, an unusual property for polyhedra. Whether space-filling or not, the number of icosahedron per rhombic-dodecahedron edge could be increased to 5, 7, 9, or any greater odd number. Why would even numbers not work? This is a consequence of the fact that opposite faces of an icosahedron are inverted, relative to each other; a pair of icosahedra (or more than one pair, producing odd numbers > 1 when added to the vertex-icosahedron) must be attached to the one at a rhombic-dodecahedron-vertex to make these two inversions bring the triangular face back around to its original orientation, via an even number of half-rotations, without which this consruction of these icosahedral rhombi cannot happen.

Here’s another view of this rhombic dodecahedron, in “rainbow color” mode:

RD of Augmented Icosa RB

All images above were produced using Stella 4d, software which may be tried for free right here.

Two Versions of a Slowly Rotating Rhombic Triacontahedron, Adorned with Spectral Patterns on Each Face

Rhombic Triaconta

It took three programs to make this. First, outlines of the “double rainbow” patterns on each face were constructed using Geometer’s Sketchpad. A screenshot from that program was then pasted into MS-Paint, which was used to add color to the outline of the pattern on each face. Next, the colorized image was projected onto each face of a rhombic triacontahedron, using Stella 4d: Polyhedron Navigator — the program that put this all together, and what I used to generate the rotating .gif above. Stella is available at http://www.software3d.com/Stella.php, with a free trial download available.

Interestingly, while this polyhedron itself is not chiral, the coloring-pattern of it, shown above, is.

With only small modifications, Stella can produce a very different version:

Rhombic Triaconta

Which one do you like better?

The Great State of Memphis?

stateofmemphis

This is a real ad. Except from cropping away the rest of the webpage where I found it, I did not alter it.

When did Memphis gain statehood?

A Hypothesized Method for Washing Away Anger

washing away anger

This particular method is simple: sleep. Eight hours usually does it for me, >90% of the time the anger originated on the previous day. For others for whom this works, I expect the amount needed will vary from person to person.

Relevant medical research comes from many sources I have read, speculating on the (still unknown) complete list of the purposes of sleep, which includes (in lay terms) “washing away” junk the brain doesn’t need any longer. I am of the opinion that anger qualifies for that category.

My evidence: repeatedly observing this happening to me, hundreds of times.

Replication of experiments, and creation of new ones, to search for more evidence, is obviously needed. While this is a testable hypothesis, I certainly have not conducted a definitive test. For one thing, this lies outside the fields I have studied, formally, the most, and my sample size (one) was far too small to count for much.

An important point, in case anyone is wondering: no, I do not think this ability is limited to any one segment of the population, such as those with Asperger’s. If “Aspies,” like myself, have any advantage at all in this area, it’s limited (in my opinion) to the fact that many of us spend an unusually-high amount of time studying our own minds, and how they work. However, my hypothesis does not require that one know what the hypothesis states, which is no more than this: in a majority of the human population, the activity of sleep reduces levels of anger. Clearly, more reliable results could best be obtained by double-blind studies.

If I’m right, chronically sleep-deprived people, as a consequence, will be more likely to be angrier, on average, than is the case, overall, in the general population. This offers another avenue for testing.

Comments are welcome, especially regarding other research on this subject.

Also, please comment if you know of a good method for anger-elimination, or anger-reduction, which does not require sleep — for I may wish to try it myself.

Speculation Regarding Future Human Speciation, Part II

tree

[My first post on this subject appeared here.]

In the future, the human race, homo sapiens, will do what all living organisms do when circumstances change, as they always do, sooner or later. We will adapt, migrate, and/or die. Since this is a post about speciation, I’m mentally setting that part of my brain which worries about extinction happening relatively soon to “hopeful,” for the duration of writing this post.

For speciation to happen, two things must happen first: reproductive isolation, and the passage of a lot of time. Migration (to off-earth colonies, for example) can produce reproductive isolation, but so can other things. As for adaptation, this is done in many ways by our species, consciously, as well as unconsciously. Evolution is always happening, but it is the slowest of all human adaptive processes. It is estimated that it has taken us (and everything else) 3.85 billion years to evolve, after all, and that’s roughly the most recent third of the time since the Big Bang.

Extra-terrestrial colonies would definitely be a huge “push” toward speciation, especially colonies with low gravity, relative to that experienced by other people. As well-described in Robert Heinlein’s novel The Moon Is a Harsh Mistress, return to a high-gravity environment, after many years in a low-gravity environment, would carry significant health effects, with serious risks of sever complications, up to, and including, death. If we ever establish colonies outside the solar system, this isolation will become even more extreme, simply due to the enormous distances and travel-times involved.

However, we will speciate, as will all other species, if we stay alive long enough, even if no more of us leave low-Earth orbit.

In my previous post on this subject, I speculated about the emergence of fully-speciated homo lunaris,  homo literaticus, homo illiteraticus, artificially-genetically-modified humans (whom I did not name then, but homo techogeneticus would do nicely), and homo cyberneticus. An increase in the number of infertile offspring between two groups, combined with a falling birthrate between them, persisting over millennia, would be an early warning signal that speciation might be happening — that’s how long this evolutionary process takes.

Given enough time, and increased reproductive isolation from non-“Aspies,” another group which could speciate is the population with Asperger’s, to form homo aspergerus, primarily because of the difficulties each group of people have communicating with the other group. However, I’ve written several posts already which seek to help Aspies and non-Aspies understand, and bridge, the communication-gap. What’s going on here? Simple: evolution is simply a process which happens. It has no ethics. It is neither good nor evil. More to the point, just because I can discern a possible future does not mean I want that possibility to become real.

In what other ways could speciation happen? A widenining gap between the rich and poor certainly seems possible as a “push” towards speciation, at least currently, which could create homo plutocraticus and homo povertus. It is easily possible to imagine this happening along with the emergence of homo cyberneticus.

Another interesting possibility would occur if humanity survived in the North and Southern hemispheres, near the poles, but extreme global warming made tropical regions uninhabitable, and the equator untraversable, or at least very rarely crossed. In this case, it is hard to imagine what direction the changes would take, for they could go in many directions. Names for such Northern and Southern humans could be homo borealis and homo australis, from our point of view. In theirs, of course, as in many of these scenarios, the likely terms each group will use for the other will translate, roughly, as “us” and “them,” unless they somehow manage to shed the primate characteristic of forming rival groups, one which long predates humanity.

How could the maximum number of new species emerge in the least time possible?  This is known: a mass extinction would be needed, one which leaves a small (but large enough) percentage of humanity alive, and fertile. This would open numerous ecological niches which we, and other extant species, would rapidly fill, by rapid speciation, into an amazing variety of forms. Again, this simply happens — that does not mean it is to be desired, and I am certainly not hoping for it to happen soon. I simply realize that it has happened before, and we have no reason to think it will not happen again.

All the Classes I Have Taught, or Am Teaching (Updated for 2024-2025)

This is my 30th year teaching. Just as a test of my memory, I’m going to try to list every class I have ever taught, or am teaching now. The italics indicate the subjects which I am most confident I can teach well, whether I am teaching them currently, or not. Bold indicates courses which are in my current teaching assignment.  As for improving the ones not in italics, I’ll work more on that . . . when I have the time. 

  1. Algebra I
  2. Algebra II
  3. Algebra III
  4. Algebra Lab
  5. American History to 1877
  6. Anatomy
  7. A.P. Biology
  8. A.P. Physics
  9. Area I Mathematics at Arkansas Governor’s School — a course focusing on polyhedra
  10. Arkansas History
  11. Biology
  12. Bridge to Algebra II, which I can’t help thinking of as “Algebra 1.5”
  13. Chemistry
  14. Chemistry I (no, I have no idea why that particular school called it that; I never found “Chemistry II” there)
  15. Civics
  16. Economics
  17. Environmental Science
  18. Formal Geometry
  19. Geography
  20. Geometry
  21. Geometry Lab
  22. Informal Geometry
  23. Integrated Chemistry
  24. Integrated Science 8
  25. PAP Algebra II
  26. PAP Physical Science
  27. Physical Science
  28. Physics
  29. Pre-AP Chemistry
  30. Psychology
  31. Religion, 9th grade (at a private, religious school)
  32. Religion, 12th grade (at a private, religious school)
  33. Study Center / Credit Recovery
  34. Study Skills (while student teaching)
  35. Summer School Transition Camp (for incoming high school students)
  36. University Studies (my only foray into teaching at the college level; basically, an “Intro to College” course, for entering freshmen)
  37. U.S. History Since 1890
  38. World History (while student teaching)
  39. World History Since 1450

X. In-school Suspension (ISS), also known as SAC, which stands for the horribly-misleading euphemism, “Student Assistance Center.” I used an “X” instead of a number because, as a student or a teacher, SAC is not a class, nor a subject. It is, rather, a non-class which one endures until the merciful ringing of the bell at the end of the school day.

XX. “Saturday School,” which is like ISS/SAC, but even worse, for all concerned. (I really needed the extra money at that time.)

To anyone now working on becoming a teacher: you become much more employable if you become certified in multiple certification areas, as I have. This is a two-edged sword, though, for it definitely increases the number of subjects you may be asked to teach in any given year, and that’s also the reason my list above is so long.

One other thing I definitely remember is my first year’s salary, to the cent: $16,074.00, before any deductions. You can make a living in this field, in this country . . . after you’ve been in the classroom for a few years . . . but no one should expect making it, financially, to be easy, especially for the first 5-7 years.

On Asperger’s Syndrome, Honesty, Lies, and My New, Sonic Lie-Detector

house

The TV series House MD first aired in 2004. I quickly developed a fascination with both the title character, and the show, and watched the entire run of the show, until it ended in the Spring of 2012. The following Fall, I began to seriously suspect — at age 44 — something that had occurred to me only as a possibility for a few years before that: I’ve been an “Aspie” my whole life, a hypothesis which I could tell had great explanatory power to explain my numerous peculiarities, as you can tell in this early blog-post on the subject, not long after I came to WordPress. By the time 2013 arrived, shortly before I turned 45, I had begun the necessary tests of this idea, which included, of course, discussions of the possibility with a psychiatrist, as documented in another blog-post, here.

This was a breakthrough for me, for it helped with a long-term project I begun in my teens: development of the ability to reprogram my brain’s own software, which I became aware was happening at night. I first wrote about this sleep-reprogramming here, just over a year ago, almost two months after getting married — to another math teacher. Thanks to my wife, the critical error in that sleep-related post (that I was reprogramming in the deep parts of non-REM sleep) was later discovered, for she noticed that I frequently stopping breathing while asleep. Sometimes she would have to shake me awake, urging me to resume breathing; other times she would be awakened, herself, by my own gasping for air, which it turns out I had been doing for years. At her urging, I discussed the possibility with my primary care physician, who immediately referred me to a specialist for a sleep study. This resulted in a definitive diagnosis of sleep apnea, revealing that I was only getting stage 1 and 2 sleep, but no significant quantities of stage 3 sleep, nor stage 4 sleep, nor REM sleep. This ruled out my “deep-sleep reprogramming” hypothesis.

I’ve been a teacher for over twenty years, and have more experience teaching science than any other subject. I started learning science very early in life, simply by hanging around in the science building of a university, as a little kid, where I could avoid interacting with children my own age (whom I did not understand), and communicate exclusively with professors and their college students, as described, along with a 5th-grade “sequel” involving “show and tell,” here. For these reasons, I don’t have to consciously employ the scientific method when empirical evidence shows a hypothesis to be flawed — it is automatically what I jump to. My deep-sleep hypothesis did not have to be thrown out completely, but only modified. The evidence prior to the sleep study did indicate this auto-reprogramming was happening in my sleep, but the sleep study proved that it could not be happening during deep non-REM sleep. What was not ruled out, though, was the only type of sleep I had been getting before I got the CPAP machine I use now: Stage 1 and 2 sleep. It is my current position, subject to further testing, that I sleep-reprogram in the shallow parts of non-REM sleep, most likely Stage 1, or in the “twilight” regions between wakefulness and sleep, hypnagogia (the “going to sleep” transition period), and/or hypnopompia, the “waking up” transition period. With my CPAP machine, I have now had the better part of a year to recover from the negative effects of chronic sleep deprivation. I now get the REM, stage 3, and stage 4 sleep needed for good physical and mental health. Of course, I still get Stage 1 and Stage 2 sleep, so I can still sleep-reprogram — and, for the last few months, I have been using sleep-reprogramming to seek ways to reduce the social-interaction difficulties involved with Asperger’s, while retaining the advantages.

One night, and I suspect it was last night, I used sleep-reprogramming to prepare myself to learn to do something I have not been able to do before: detect lies, and employ certain specific, related social skills, using alternative methods than those used by most non-Aspies. The program required watching House MD to activate it, however, as I discovered while watching it today, after a couple of episodes, while resting last night, before going to sleep — the first time I have watched the show in months. The changes that happened while I watched a single episode (season one, episode four, “Maternity”) were both rapid and dramatic; it was quite clear to me that an already-prepared unconscious subroutine was being activated.

Before this lie-detecting subroutine was activated, these had been long-term statements which accurately described me:

  • I was rarely able to detect sarcasm in others — but could dish it out in large quantities.
  • Although I sometimes made jokes (linguistic wordplay; made-up words, etc.), or even pulled pranks, I rarely recognized the same behavior for what it was when it was turned around, and aimed in my direction, by others, leading to numerous serious misunderstandings, in both childhood and adulthood.
  • I could not “read” other people’s emotional states, and, for most of my life, would not even acknowledge that the ability to do so was a useful skill. Most people do this by analyzing such things as facial expressions and body language; analysis of sound is, for most people, secondary to this.
  • I found it nearly impossible to detect dishonesty, and, while I can lie, I learned (as a teenager) that I could not do so effectively. If I try to lie, I figured out decades ago, there is a very good chance that another person will know I am lying. Under these conditions, having lies detected (and then experiencing the consequences of this) led me to consciously make the choice to be honest.
  • I have often behaved in ways which unintentionally offended, angered, or irritated people, simply due to my lack of awareness of any emotions which are not my own, and, 99+% of the time, I was not even aware that I had done so, until later, when informed that I had. My usual reaction has been surprise and confusion, for my emotions are not like those of non-Aspies. When younger, I spent absurd amounts of time stuck in intensely angry states, for reasons connected to another condition I have, PTSD (post-traumatic stress disorder). I recognize anger; early exposure to far too much of it was involved in creating my PTSD in the first place. However, as I have recently learned, anger is not binary — most people experience a whole range of “shades of grey” related to, but less intense than, white-hot fury, such as simply being annoyed, or irritated. It was only recently that I became aware that other people experience such intermediate states between “angry” and “not angry,” although I know of no other person who made such a “discovery” in their 40s . . . and, without help from my wife, I might never have made it at all.

There is an important subgroup of the general population who form an exception to one of the common-among-Aspies characteristics above: not reading facial expressions or body language well. This group is the blind, and, to a lesser degree, the visually impaired. I wear glasses at work, and while driving, for mild-to-moderate myopia, but do not need them to function indoors, especially in familiar settings, such as home. Lately, I have simply stopped wearing my glasses while home. However, I certainly use my vision a lot, and think visually; the sheer volume of geometry (with apologies for the pun) on this blog is evidence enough of that. I have more familiarity with blindness than most sighted people, though, for three reasons: (1) I have read comic books since I was a kid (sometimes while hanging out in some science lab, although I was more likely to be found playing with a some dangerous chemical when very young), and my favorite title for most of my life has been Daredevil, the source of the lower-left part of the image above, (2) I have had close friends who were blind, and (3) I learned to read and write Braille as a college undergraduate, so that I could communicate with these friends by mail, and have not forgotten this skill, although it has lost speed — but I made the Braille lettering in the image above, which simply reads “everybody lies,” a line made famous by House MD.

There is also a fourth reason, unrelated to blindness, which is a common characteristic of those with Aspergers: periods of time when my senses become painfully over-sensitive, so that the sun, indoor lighting, etc., appear to be turned up extra-bright, everything is incredibly loud, etc. — except for my sense of smell, weakened by allergies, and exposure to chemical fumes. This used to happen during migraine headaches; now, I no longer get this state with headaches, but the amplified sensory perceptions are now so intense, at times, that this state is actually worse than a migraine headache, for at least those had pain to distract me, at least somewhat, from overwhelming sensory overload. When this happens, I blindfold myself to eliminate the visual overload, but I cannot silence the world, and you can read a description of what it sounds like, when I am in “Matt Murdock mode,” as I call it, in this blog-post. My perceptions don’t become so acute that I can hear the heartbeats of those around me (as the fictional Murdock can), and I’m fine with that; I don’t need, nor want, to have the volume turned up any further.

When House MD was airing new episodes, as I remember the show,  Dr. Gregory House would be an incredible jerk to people at least six to ten times per episode, sometimes more, and I found it hilarious. Only tonight, while watching the episode mentioned above, “Maternity,” did I realize (after nearly everyone else on the planet who has seen even one episode of this show), that House is incredibly rude at a rate far closer to six to ten times per minute. I had no idea!

This made me quite surprised. I then suddenly noticed something else: Dr. Allison Cameron — lying. Next, Chase lied. Later, Foreman lied. A patient lied, but of course they always do — “everybody lies” is a recurring theme of the show, and the consequences of lying are the show’s most-used plot-device. While thinking about this, I caught House lying. And I knew about each and every one of these lies before House revealed them, as lies, later in the episode. How was I noticing these lies I had never heard, as lies, before?

Of course, I have seen every episode of House MD, so memory is definitely a factor, but had not watched season one in years. I began to focus on this puzzle. About ten detected-lies later, I figured it out: I was hearing not an increased volume, but tiny changes in pitch, no more than a half-step on the musical scale. Lying is a risky activity, and liars know this, so it causes increased anxiety when people lie — and, due to this anxiety, I reasoned, their airways constrict, at least slightly, at the end of a lie, causing an increase in the pitch and frequency of the sound of their voices at the end of a lie. House was the ideal tool for teaching this, for it allowed me to “calibrate” my internal lie-detector, by focusing on the voices intently, while using long-stored memories of Dr. House revealing lies to help me catch what I had never caught before.

Of course, House is a TV show. This needed to be tested — with something which did not involve actors, nor writers. And so, I devised a way to test it. I asked a scientifically-minded person I know to help me test a hypothesis, by choosing a random order for a true statement and a lie, and then simply telling both to me, and then see if I could accurately identify the true statement, as well as the lie.

The pitch of this person’s voice unexpectedly went down, slightly, at the end of one of the statements, but did not change when speaking the other statement. What was going on, I wondered? I then figured it out: I had chosen a scientifically-minded person — who was trying to outsmart my experimental test! I then noticed that one statement of the two — the lie — concerned the number of items in a box, and was therefore testable. The other statement was not testable. The other person, who is quite intelligent, had taken a minute or two to formulate both the true statement, and the lie — and clearly intended me to fail the test, and then be able to prove it had failed, by simply opening the box. Using logical reasoning (an old skill) and my new, sonic lie-detector, together, I can now even detect a lie that a police polygraph might not have been able to detect!

I then realized I was detecting more subtle emotions . . . catching more jokes . . . and generally doing, by rapidly analyzing sound, what other people do by somehow “reading” body language and facial expressions.

It is a myth that Aspies can’t do such things as understanding emotions. It is not a myth that Aspies think in different ways, however. Sometimes, individual Aspies simply have to figure out their own methods for doing something which comes naturally to non-Aspies — and that’s exactly what I did.

I could, presumably, use this new frequency-sensitivity to train myself to lie convincingly, by deliberately avoiding the “tell” of a slight pitch-increase at the end of a lie. However, I choose not to do that. The first week I went without telling a single lie, in my teens, was difficult, but the second week was much easier, and the third was easier still. Not having to try to keep track of previous lies began to give me a persistent, liberated feeling that can only be enjoyed by those who are honest — and that feeling would, of course, vanish if I chose to invest time, energy, and thought into becoming a skilled liar. To do that would be to move backwards in my life, rather than forwards, and I see no reason to do that. In other words, I’m still me . . . just the latest version of me, that’s all.