This 110-faced polyhedron has, in addition to the eight regular dodecagons, six rectangles, and 96 triangles. I made it using Stella 4d, a program you can try for free, as a demo version, at http://www.software3d.com/Stella.php. I wish I could remember how I made it!
Fortunately, I have many friends who are more knowledgeable than I, when it comes to mathematics. Perhaps one of them will be able to solve this mystery.
I’m spending this odd, COVID-19-dominated Holy Saturday inside, watching Jesus Christ Superstar. This was actually one of my primary information-sources regarding Christianity before it occurred to me to read the New Testament for myself, in my early twenties. If you’d like to watch it with me, here’s a link to the 1973 film-version, as a YouTube playlist.
The sky bursting full of rapid and illuminated clouds, rushing bright blue against an indigo background, made me feel I was looking up at the planet Neptune, stretching from one horizon to the other. I went inside, to get my phone, to snap a picture, but, when I got back out, the eighth planet above had been replaced — by a stormy-but-normal third-planet sky. I came back inside with no images, except in memory.
(Image source: NASA / JPL / Voyager 2 / this website.)
Someone nudged my shoulder, stirring me from deep sleep. “Wake up, grandpa,” said an unfamiliar voice. Grandpa? Who’s that? I opened my eyes to see a young woman, dressed in black, looking back at me. Her face was brown, and her eyes looked like deep pools of water.
She smiled. Nothing in twenty-plus years of teaching could have prepared me for this, I thought. I looked around, trying to find my cell phone, without success. Nothing here was like anything I’d seen before. Small lights, like fireflies, circled us in the darkness.
“I know it’s confusing to be called ‘grandpa,'” she said, answering a question I had not yet had the chance to ask. “This is, well, complicated.” Her voice sounded excited, even though she was speaking softly. She reminded me of teachers new to the profession, positively bursting with new ideas, and looking forward, enthusiastically, to the new school year ahead.
“It would have to be complicated,” I mumbled. Sleep was fading as I rubbed my eyes, trying to see where I was. A light came on, but it was unclear where the lightbulbs were. We were alone, inside a blue and white cube. The cube slowly moved, but its direction kept changing. “What am I doing here? Where’s my wife? Where am I, and who are you?”
“So many questions! I expected that, though. I will explain what I can.”
“That’s good, because . . . .”
“Please don’t interrupt,” she said. I stopped talking, but did not stop thinking. It appeared to be time to listen, not talk. “Thank you,” my alleged granddaughter continued. “In order, here are the answers to your questions. First, you are here for an important conversation. Second, your wife is peacefully sleeping. Third and fourth, you’re in my time-travel cube, and my name is Xiahong Al-Nasr. Technically, you’re my great-great-great-great-great-grandfather, but . . . .” I raised my hand to ask a question, as if I were in class myself. She shook her head, and continued, “. . . I’ve always thought of you as, simply, ‘grandpa.’ It’s a time-saver. May I continue explaining why we are here, or can your question wait?”
I thought fast. What should I say next? There was only one logical response. “I’ll listen,” I replied, and put my hand back down.
“You’re about to go back to school,” she said, “and you’re the teacher. It’s important that you understand why you are doing what you do, this year, above all others.” This reminded me of advice I’d heard before, but this time I was listening as if I were hearing for the first time.
This woman’s name, Xiaohong Al-Nasr, combined a Chinese given name with an Arabic surname. I hoped she would explain how that had happened.
“You’re wondering about my name,” she said. I swallowed, and nodded. My mouth was too dry to speak. “I’m from the 23rd Century,” she continued. “Nearly everyone where I work and learn, including me, has DNA from every continent on Earth. I’ve also got a little from off-world colonies, but I’m 100% human, just as you are. I was given my name by all of my parents.” She paused. Her gaze was locked to my own. “I’ve been authorized to tell you that much, but I have to be careful about revealing more, to prevent altering the timestream. Do you believe me?”
“If you know anything about me, you know that I teach science, as well as other subjects.” It was a relief to finally have my turn to speak. My alleged descendant, Xiaohong, was listening to me now. Finally! “You’ve either studied me, somehow, or you’re reading my mind, or it’s something else even more complicated, but you seem to know what I know. You must know, then, that scientists are trained to be skeptical. Everything has to have evidence to support it. In science, there is no higher authority than experiment.”
“I understand that, grandpa. We knew you would need evidence, so I do have a gift for you. It’s a t-shirt. You like t-shirts, after all.” Xiaohong smiled, and removed a small capsule from her pocket, no larger than a quarter. She opened it, and — somehow — pulled a full-size t-shirt from that impossibly small place.
I took the t-shirt from my descendant. Touching it was, well, real! I turned it over. It said “Go Bears!” on the back. Even if I believed her, though, I knew I would need more than just a t-shirt to convince anyone else. After all, time travel to the past was considered impossible by every scientist I had studied. Quickly, I did the arithmetic, using the year on the shirt. “That’s the year I would turn 300 years old, if I could live that long!” I was now catching Xiaohong’s excitement. “Clearly, Arthur C. Clarke’s Three Laws apply here, as does the Sagan Standard, Feynman’s First Principle, the grandfather paradox, and — and — and — the entire scientific method!”
“You’re absolutely correct, and it will be important for your students to understand all those things as well.” She was right; these are all things I talked about in science class, every year. This year, though, I can try to explain them differently, or perhaps have my students research them, and then have the students explain them to my class. Correction: my classes. My students. All of them.
Something fell into place in my mind at that moment, and I finally understood what was going on. It wasn’t my own accomplishments that had brought my descendant back in time to visit me, but the unknown creations of a student of mine — from the school year about to begin. Xiaohong smiled.
“You’ve figured it out, haven’t you?” She was asking a question, and, this time, I had the answer.
“Yes. You came back through time to refocus my attention to my own true purpose in the classroom. My job is to help my students learn to do great things. It’s not about me. It’s about them!” Xiaohong’s smile grew larger. I continued. “This school year is critical. This is true of all school years, in fact. Each year is both important, and urgent. In every school, and for every student, we must always do our best to learn — together.”
Xiaohong extended her hand, and received a firm handshake from me. “Now that you know the truth, grandpa, our work here is finished. You’ll wake up in the morning, in bed with your sleeping wife, and after that, you’ll find your t-shirt, in the dryer, at home. I have to go, though; I’m needed back in the 23rd Century. After all, I have my own classes to teach, quite soon, at our Time Travel Academy, where I got your t-shirt. Goodbye, and have a great school year! I know I will, as I continue my training to become a teacher myself.”
“I will do that,” I replied. “Thank you so much! As for this evidence you’ve given me, I know how I’ll handle that. I will let the students evaluate it, with help from me, on an ‘as needed’ basis.”
“Exactly,” Xiaohong said, and then she spoke to the ceiling of her time travel cube. “Send us both back to where we were — now.” A humming sound started, then became louder. The lights began to dim. After a few minutes, everything faded to darkness, and silence, once more.
When I awoke, home again, I checked the dryer, and found it — my t-shirt from the future — waiting for me. This school year will be amazing!
We have found compelling evidence for the existence of several sub-surface oceans in various places in our solar system. The most well-known of these bodies of liquid water is under the ice crust of Europa, a moon of Jupiter, with others located elsewhere. These oceans are logical places to look for signs of past or present extraterrestrial life. However, we have yet to obtain a sample of any of these oceans for analysis. It is time for that to change, but not without taking precautions to avoid damaging any such life, should it exist.
Europa (source: NASA)
What follows is my idea, freely available for anyone who wishes to use it, to safely obtain and analyze such samples. These ice-tunneling probes could be ejected from a larger lander, or simply dropped directly onto the surface from orbit. This would be far less expensive than any sort of manned interplanetary exploration. Exposure to vacuum and radiation, in space, would thoroughly sterilize the entire apparatus before it even lands, protecting anything which might be alive in the ocean underneath from contamination by organisms from Earth.
In this cross-sectional diagram, the light blue area represents the ice crust of Europa, or another solar-system body like that moon. The ice-tunneling lander is shown in red, orange, black, yellow, and green. The dark blue area is the vertical tunnel created by the probe, shown shortly after tunneling begins. As the probe descends, the dome shown in gray caps the tunnel, and stays on the surface, having been previously stored, folded up, in the green section of the egg-shaped probe. The gray section is designed as a geodesic dome, with holes of adjustable size to allow heat to escape into space. An extendable, data-carrying tether connects the egg-shaped tunneling module to the surface dome. Solar-energy panels and radio transmitters and receivers stay at the surface, attached to the gray dome.
The computers necessary to operate the entire probe are in the yellow section. The black section that extends outward, slightly, from the body of the tunneler would contain mechanisms to obtain samples of water for analysis. The orange section is where actual samples are stored and analyzed.
The red part of the tunneler is weighted, so that gravity forces it to stay at the bottom. It is designed to heat up enough to melt the ice underneath it, allowing the entire “egg” to descend, attached to its tether. Water above the tunneling probe re-freezes, sealing the tunnel so that potentially-damaging holes are not left in the ice crust of Europa. The heating units in the red section can be turned on and off as needed, to slow, hasten, or stop the probe’s descent through the crust.
Oceans in other places in the solar system might require certain adjustments to this design. For example, Ganymede, another moon of Jupiter, is far rockier than Europa. If this design were used on Ganymede, the tunneling probe would likely be stopped by sub-surface rocks. For this type of crust, the probe’s design could be modified to allow lateral movement of the tunneler, in order to go around rocks.
Ganymede (source: NASA)
On Europa, Ganymede, and elsewhere, one limitation of this design is imposed by the maximum length of the tether. We would not want to go all the way down to the subsurface oceans with the earliest of these probes, though. A better strategy would be to only tunnel part-way into the crust at first, capturing liquid samples of water before refreezing of the ice. After all, this ice in the crust could have been part of the lower, liquid ocean at some point in the past, and it should be analyzed thoroughly before heat-tunneling any deeper. The decision to make the tether long enough to go all the way through the crust, into the subsurface ocean itself, is not one to make lightly. It would be best to study what we find in molten crust-samples, first, before tunneling all the way through the protective crusts of these oceans.
I just noticed that I can elect to pay attention to my breathing, or ignore it, but one or the other keeps happening. Changing which one I focus on changes the way I think. This is interesting.
[Image found here.]
It is no secret than I am not a fan of our current president, Donald Trump. I’ve been watching him carefully, and have found exactly one point of agreement with the man: humans should colonize the planet Mars. The two of us differ, however, on the details. What follows is my set of reasons — not Trump’s — for supporting colonization of Mars.
First, we should not start with Mars. We should start, instead, by establishing a colony on Luna, our own planet’s moon. There are several reasons for this. First, as seen in this iconic 1969 photograph brought to us by NASA, we’ve been to the Moon before; it simply makes sense to start space-colonization efforts there.
At its furthest distance, the Moon is ~405,000 km away from Earth’s center, according to NASA. By contrast, at its closest approach to Earth in recent history, Mars was 55,758,006 km away from Earth. With the Moon less than 1% as far away as Mars at closest approach, Luna is the first logical place for an extraterrestrial colony. It need not be a large colony, but should at least be the size of a small town on Earth — say, 100 people or so. There are almost certainly problems we haven’t even discovered — yet — about establishing a sustainable reduced-gravity environment for human habitation; we already know about some of them, such as muscular atrophy and weakening of bones. Creating a lunar colony would demand of us that we solve these problems, before the much more challenging task of establishing a martian colony. (To find out more about such health hazards, this is a good place to start.) Once we have a few dozen people living on the Moon, we could then begin working in earnest on a martian colony, with better chances for success because of what we learned while colonizing the Moon.
An excellent reason to spend the billions of dollars it would take to colonize Mars (after the Moon) is that it is one of the best investment opportunities of the 21st Century. Space exploration has a fantastic record of sparking the development of new technologies that can help people anywhere. For example, the personal computers we take for granted today would not be nearly as advanced as they are without the enormous amount of computer research which was part of the “space race” of the 1960s. The same thing can be said for your cell phone, and numerous other inventions and discoveries. Even without a major space-colonization effort underway, we already enjoy numerous health benefits as a result of the limited exploration of space we have already undertaken. Space exploration has an excellent track record for paying off, big, in the long run.
Another reason for us to colonize Mars (after the Moon, of course) is geopolitical. The most amazing thing about the 20th Century’s Cold War is that anyone survived it. Had the United States and the Soviet Union simply decided to “nuke it out,” no one would be alive to read this, nor would I be alive to write it. We (on both sides) survived only because the USA and the USSR found alternatives to direct warfare: proxy wars (such as the one in Vietnam), chess tournaments, the Olympics, and the space race. In today’s world, we need safe ways to work out our international disagreements, just as we did then. International competition to colonize space — a new, international “space race” — would be the perfect solution to many of today’s geopolitical problems, particular if it morphs, over the years, into the sort of international cooperation which gave us the International Space Station.
Finally, there is the best reason to establish space colonies, and that is to increase the longevity of our species, as well as other forms of life on Earth. Right now, all our “eggs” are in one “basket,” at the bottom of Earth’s gravity well, which is the deepest one in the solar system, of all bodies with a visible solid surface to stand on. A 10-kilometer-wide asteroid ended the age of the dinosaurs 65 million years ago, and there will be more asteroid impacts in the future — we just don’t know when. We do know, however, that past and present human activity is causing significant environmental damage here, so we may not even need the “help” of an asteroid to wipe ourselves out. The point is, the Earth has problems. The Moon also has problems, as does the planet Mars — the two places are far from being paradises — but if people, along with our crops and animals, are located on Earth, the Moon, and Mars, we have “insurance” against a global disaster, in the form of interplanetary diversification. This would allow us to potentially repopulate the Earth, after the smoke clears, if Earth did suffer something like a major asteroid impact.
Since Moon landings ended in the 1970s, we’ve made many significant discoveries with space probes and telescopes. It’s time to start following them with manned missions, once again, that go far beyond low-Earth orbit. There’s a whole universe out there; the Moon and Mars could be our first “baby steps” to becoming a true spacefaring species.
[Later edit: Please see the first comment, below, for more material of interest added by one of my readers.]
Shortly before Winter began at 4:44 CST (USA), I wondered if it were Winter yet, and googled it. Here’s what I found. I was astonished when I looked at the current time and the time of the Winter solstice, using a Google-search, and they matched. This was a bizarre coincidence, and I thought it worth a screen-shot and a blog-post, all while in a bewildered state.
2016: a year that could never have been predicted. There’s little time before 2017 is here. May it be a better year for us all.
Never before have I deliberately tried to recall my earliest memories. This morning, however, simply to see what might happen, I tried it. In my imagination, I returned to further back than I ever had gone before, to a period before I learned to communicate. In this early period, I could visualize things, with the imagery which appeared being geometric in nature. Later, I had to learn English, as a second language, to express the mathematical ideas in my head. My first word, according to my parents, described one of the two shining round things in the sky: “Moon.” I have always preferred moonlight to sunlight, for the intensity of direct sunlight is painful to me.
At least, that’s how I remember these things; I could be wrong about the earliest parts. All I know is that the image above popped into my head, when I tried to recall my oldest accessible memory. I then made the image above, in a short period of time, using Stella 4d, Polyhedron Navigator, available to try for free at this website. (I’ve used the program for over a decade, and find it an indispensable tool for geometrical investigations, such as this recreation of what I found in this morning’s early-memory-search.)
People have been trying to figure out what intelligence is, and how it differs from person to person, for centuries. Much has been written on the subject, and some of this work has helped people. Unfortunately, much harm has been done as well. Consider, for example, the harm that has been done by those who have had such work tainted by racism, sexism, or some other form of “us and them” thinking. This model is an attempt to eliminate such extraneous factors, and focus on the essence of intelligence. It is necessary to start, therefore, with a clean slate (to the extent possible), and then try to figure out how intelligence works, which must begin with an analysis of what it is.
If two people have the same age — five years old, say — and a battery of tests have been thrown at them to see how much they know (the amount of knowledge at that age), on a wide variety of subjects, person A (represented by the blue curve) may be found to know more, at that age, than person B (represented by the red curve). At that age, one could argue that person A is smarter than person B. Young ages are found on the left side of the graph above, and the two people get older, over their lifespans, as the curves move toward the right side of the graph.
What causes person A to know more than person B, at that age? There can be numerous factors in play, but few will be determined by any conscious choices these two people made over their first five years of life. Person B, for example, might have been affected by toxic substances in utero, while person A had no such disadvantage. On the other hand, person A might simply have been encouraged by his or her parents to learn things, while person B suffered from parental neglect. At age five, schools are not yet likely to have had as much of an impact as other factors.
An important part of this model is the recognition that people change over time. Our circumstances change. Illnesses may come and go. Families move. Wars happen. Suppose that, during the next year, person B is lucky enough to get to enroll in a high-quality school, some distance from the area where these two people live. Person B, simply because he or she is human, does possess curiosity, and curiosity is the key to this model. Despite person B‘s slow start with learning, being in an environment where learning is encouraged works. This person begins to acquire knowledge at a faster rate. On the graph, this is represented by the red curve’s slope increasing. This person is now gaining knowledge at a much faster rate than before.
In the meantime, what is happening with person A? There could be many reasons why the slope of the blue curve decreases, and this decrease simply indicates that knowledge, for this person, is now being gained at a slower rate than before. It is tempting to leap to the assumption that person A is now going to a “bad” school, with teachers who, at best, encourage rote memorization, rather than actual understanding of anything. Could this explain the change in slope? Yes, it could, but so could many other factors. It is undeniable that teachers have an influence on learning, but teacher quality (however it is determined, which is no easy task) is only one factor among many. Encouraging the “blame the teacher” game is not the goal of this model; there are already plenty of others doing that.
Perhaps person A became ill, suffered a high fever, and sustained brain damage as a result. Perhaps he or she is suddenly orphaned, therefore losing a previous, positive influence. There are many other possible factors which could explain this child’s sudden decrease of slope of the blue “learning curve” shown above; our species has shown a talent for inventing horrible things to do to, well, our species. Among the worst of the nightmare scenarios is that, while person B is learning things, at a distant school, the area where person A still lives is plunged into civil war, and/or a genocide-attempt is launched against the ethnic group which person A belongs to, as the result of nothing more than an accident of birth, and the bigotry of others. Later in life, on the graph above, the two curves intersect; beyond that point, person B knows more than person A, despite person B‘s slow start. To give credit, or blame, to either of these people for this reversal would clearly be, at best, a severely incomplete approach.
At some point, of course, some people take the initiative to begin learning things on their own, becoming autodidacts, with high-slope learning curves. In other words, some people assume personal responsibility for their own learning. Most people do not. Few would be willing to pass such judgment on a child who is five or six years old, but what about a college student? What about a high school senior? What about children who have just turned thirteen years old? For that matter, what about someone my age, which is, as of this writing, 48? It seems that, the older a person is, the more likely we are to apply this “personal responsibility for learning” idea. Especially with adults, the human tendency to apply this idea to individuals may have beneficial results. That does not, however, guarantee that this idea is actually correct.
I must stop analyzing the graph above for now, because the best person for me to examine, at this point, in detail, is not on the graph above. He is, however the person I know better than anyone else: myself. I’ve been me now for over 48 years, and have been “doing math problems for fun” (as my blog’s header-cartoon puts it) for as long as I can remember. This is unusual, but, if I’m honest, I have to admit that there are inescapable and severe limits on the degree to which I can make a valid claim that I deserve credit for any of this. I did not select my parents, nor did I ask either of them to give me stacks of books about mathematics, as well as the mathematical sciences. They simply noticed that, when still young, I was curious about certain things, and provided me with resources I could use to start learning, early, at a rapid rate . . . and then I made this a habit, for, to me, learning is fun, if (and only if) the learning is in a field I find interesting. I had absolutely nothing to do with creating this situation. My parents had the money to buy those math books; not all children are as fortunate in this respect. Later still, I had the opportunity to attend an excellent high school, with an award-winning teacher of both chemistry and physics. To put it bluntly, I lucked out. As Sam Harris, the neuroscientist, has written, “You cannot make your own luck.”
At no point in my life have I managed to learn how to create my own luck, although I have certainly tried, so I have now reached the point where I must admit that, in this respect, Sam Harris is correct. For example, I am in college, again, working on a second master’s degree, but this would not be the case without many key factors simply falling into place. I didn’t create the Internet, and my coursework is being done on-line. I did not choose to be born in a nation with federal student loan programs, and such student loans are paying my tuition. I did not create the university I am attending, nor did I place professors there whose knowledge exceeds my own, regarding many things, thus creating a situation where I can learn from them. I did not choose to have Asperger’s Syndrome, especially not in a form which has given me many advantages, given that my “special interests” lie in mathematics and the mathematical sciences, which are the primary subjects I have taught, throughout my career as a high school teacher. The fact that I wish to be honest compels me to admit that I cannot take credit for any of this — not even the fact that I wish to be honest. I simply observed that lies create bad situations, especially when they are discovered, and so I began to try to avoid the negative consequences of lying, by breaking myself of that unhelpful habit.
The best we can do, in my opinion, is try to figure out what is really going on in various situations, and discern which factors help people learn at a faster rate, then try to increase the number of people influenced by these helpful factors, rather than harmful ones. To return to the graph above, we will improve the quality of life, for everyone, if we can figure out ways to increase the slope of people’s learning-curves. That slope could be called the learning coefficient, and it is simply the degree to which a person’s knowledge is changing over time, at any given point along that person’s learning-curve. This learning coefficient can change for anyone, at any age, for numerous reasons, a few of which were already described above. Learning coefficients therefore vary from person to person, and also within each person, at different times in an individual’s lifetime. This frequently-heard term “lifelong learning” translates, on such graphs, to keeping learning coefficients high throughout our lives. The blue and red curves on the graph above change slope only early in life, but such changes can, of course, occur at other ages, as well.
It is helpful to understand what factors can affect learning coefficients. Such factors include people’s families, health, schools and teachers, curiosity, opportunities (or lack thereof), wealth and income, government laws and policies, war and/or peace, and, of course, luck, often in the form of accidents of birth. Genetic factors, also, will be placed on this list by many people. I am not comfortable with such DNA-based arguments, and am not including them on this list, for that reason, but I am also willing to admit that this may be an error on my part. This is, of course, a partial list; anyone reading this is welcome to suggest other possible factors, as comments on this post.
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