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 |    Not for the faint of art. |
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Complex Numbers A complex number is expressed in the standard form a + bi, where a and b are real numbers and i is defined by i^2 = -1 (that is, i is the square root of -1). For example, 3 + 2i is a complex number. The bi term is often referred to as an imaginary number (though this may be misleading, as it is no more "imaginary" than the symbolic abstractions we know as the "real" numbers). Thus, every complex number has a real part, a, and an imaginary part, bi. Complex numbers are often represented on a graph known as the "complex plane," where the horizontal axis represents the infinity of real numbers, and the vertical axis represents the infinity of imaginary numbers. Thus, each complex number has a unique representation on the complex plane: some closer to real; others, more imaginary. If a = b, the number is equal parts real and imaginary. Very simple transformations applied to numbers in the complex plane can lead to fractal structures of enormous intricacy and astonishing beauty. ![Merit Badge in Quill Award
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| I'm linking this article from Nautilus, not to instigate any arguments about climate change, but to make a point about science. Why Our Intuition About Sea-Level Rise Is Wrong  A geologist explains that climate change is not just about a global average sea rise. The post is quite old by internet standards, having been released about 9 years ago. This is irrelevant to what I have to say about it, but I'm sure the science has advanced somewhat since 2016. Jerry Mitrovica has been overturning accepted wisdom for decades. Well, that's one thing science is for. Another thing it's for is to support accepted wisdom, but I don't think that happens as often. A solid Earth geophysicist at Harvard, he studies the internal structure and processes of the Earth, which has implications for fields from climatology to the timing of human migration and even to the search for life on other planets. I could look up what "solid Earth geophysicist" is, but it's more fun to joke. Are there also liquid Earth geophysicists and gaseous Earth geophysicists? Does his purview stop at the mantle, where the Earth's guts change from solid to plastic to liquid? If so, what's he doing talking about oceans? Most of the link is in the form of an interview. The questions part of the interview, I'll put in bold. What happens when a big glacier like the Greenland Ice Sheet melts? Three things happen. One is that you’re dumping all of this melt water into the ocean. So the mass of the entire ocean would definitely be going up if ice sheets were melting—as they are today. The second thing that happens is that this gravitational attraction that the ice sheet exerts on the surrounding water diminishes. As a consequence, water migrates away from the ice sheet. The third thing is, as the ice sheet melts, the land underneath the ice sheet pops up; it rebounds. I think, perhaps, it's only his "second thing" that would seem counterintuitive. I did a link a while back that talked about the Indian Ocean gravitational anomaly, and how the lower gravity there, however tiny, causes sea level to be lower in that region of ocean. And this is why we have science: because intuition is, very often, wrong, and it can be catastrophically wrong. Gravity has a very strong effect. So what happens when an ice sheet melts is sea level falls in the vicinity of the melting ice sheet. That is counterintuitive. The question is, how far from the ice sheet do you have to go before the effects of diminished gravity and uplifting crust are small enough that you start to raise sea level? That’s also counterintuitive. It’s 2,000 kilometers away from the ice sheet. This is another reason why we have science: to quantify these effects. I know when I was a kid and I learned about tides, I was like "Okay, I get why sea level increases on the side facing the Moon, but what's up with the bulge on the other side?" I didn't fully understand this until I learned about vectors, in college. Why is the source of the melt important? If you’re living on the U.S. east coast, or Holland, you don’t need to worry what global average sea-level rise is doing. I was in Holland a few summers ago and was trying to convince the Dutch that if the Greenland ice sheet melts, they have less to worry about than the Antarctic ice sheet melting. I gotta admit, I struggled with maintaining my acceptance of his credibility as he used Holland for the Netherlands. But I have to remember: he's a geologist, not a geographer. No one can know everything. But I do hope someone clued him in that we don't call the entire country Holland. Of course, 120,000 years ago, humans weren’t having any impact on climate. That was natural climatic variability. One could make the argument (as I have in the past) that humans are part of nature, so all the shit we do is also natural. This is technically true, but that doesn't mean it's helpful. There's a lot more related to sea level change, which is interesting but, again, doesn't really impact my point here. The money quote for me is this one, further down: So many of your results seem abstract and counterintuitive. Is that a coincidence? There are so many interesting problems in our science that you can see with your eyes. But your eyes can fool you. Richard Feynman, the great physicist, used to start his physics lectures by showing students their intuition could take them a long way. They could do things just through intuition that would get them roughly the right answer. Then he used to throw some counterintuitive examples at them. Then he said, “This is why you need physics. You need to understand when your intuition might go wrong.” I firmly am a Feynman acolyte. There are some things that you can explain, but as a scientist you’re always going to face things that are counterintuitive. Most people think of Einstein as the Platonic ideal of genius, but I'm going with Feynman. Because to see the unexpected is the reward of science. And, yes, sometimes science itself turns out to be wrong, or incomplete. But it's still better than guesswork or "common sense." |
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