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It’s been a busy summer, and the large shortfall in donations last month has been demoralizing, so we’re taking a week off to rest and recuperate. The curated links section will be (mostly) silent, and behind the scenes we’ll be taking a brief break from our usual researching, writing, editing, illustrating, narrating, sound designing, coding, et cetera. We plan to return to normalcy on the 11th of September. (The word “normalcy” was not considered an acceptable alternative to “normality” until 14 May 1920, when then-presidential-candidate Warren G. Harding misused the mathematical term in a campaign speech, stating that America needed, “not nostrums, but normalcy.” He then integrated this error into his campaign slogan, “Return to Normalcy.” Also, the G in Warren G. Harding stood for “Gamaliel.”) While we are away, on 06 September 2023, Damn Interesting will be turning 18 years old. To celebrate, here are the first emojis to ever appear in the body of a Damn Interesting post: 🎂🎉🎁 If you become bored while we are away, you might try a little mobile game we’ve been working on called Wordwhile. It can be played alone, or with a friend. If you enjoy games like Scrabble and Wordle, you may find this one ENJOYABLE (75 points). Launch Wordwhile → And, as always, there are lots of ways to explore our back-catalog. View this post ▶
We’re not going to post things on Twitter X anymore. The new owner keeps doing awful stuff. If you have enjoyed our mostly-daily curated links via the aforementioned collapsing service, we invite you to bookmark our curated links page, or follow us a number of other ways. Rather than linger any longer on this tedious topic, here are some home-grown dad jokes. If there is any order in this universe, the comments section will fill with more of the same. Q: What is the flavor of a chair? Do you even know the meaning of the word ‘rhetorical?’ Don’t answer that! My friend bought an alarm clock that makes loud farting sounds in the morning. He’s in for a rude awakening. You’re right, these ARE my orthopedic shoes. I stand corrected. I want a good game of hide and seek, but skilled players are hard to find. Like tight sweaters, corporate acquisitions are hard to pull off. I was offered a job at the mirror factory. I could see myself working there. Did you hear about the farmer in Colorado raising cannabis-fed cattle? The steaks are high. Q: What is the best stocking stuffer? I used to be addicted to soap, but I’ve gotten clean. I finally worked up the courage to tell my hot female coworker how I felt. She felt the same. So we turned down the thermostat. The universal remote: This changes everything. Q: How fast are donkey trucks? It smells like death in there, and not in a good way. My dad demanded that I go fetch some water from that deep hole in the ground. He means well. Calendar makers: Your days are numbered. A: I enjoy cooking with ghee, but I don’t buy it, I make my own. I will not rest until I find a cure for my insomnia. I bought my wife a new refrigerator. I can’t wait to see her face light up when she opens it. Did you hear about the hilarious thing that happened at the mandatory meeting? I guess you had to be there. Remember that sweet grandmother on Twitter who thought that ‘lol’ meant ‘lots of love’? “Sorry to hear about your uncle passing. lol.” Yesterday, we were standing at the edge of a cliff. Since then we have taken a huge step forward. We had to cancel the big game of tag because somebody got hurt. It was touch and go there for a while. “Of course you can count on me,” said the abacus. IBS is genetic, you know. Runs in the family. My grandfather once told me, “It’s worth investing in good speakers.” That was some sound advice. Extreme camping is in tents. The solar panel company wouldn’t let me pay for the installation. They said it was all on the house. I was chopping herbs all day, and now my hands are quite fragrant. I’ve got too much thyme on my hands. A weather balloon measures about 4 feet in diameter (adjusting for inflation). A: Have you ever had a flatulence-based tea? Like a German dietitian, I tend to see the wurst in people. I don’t care for rulers. That’s where I draw the line. Why did the farmer propose to his horse? He wanted a stable relationship. I still think whiteboards are one of mankind’s most remarkable inventions. The Earth has successfully rotated around its axis. Let’s call it a day. My daughter dropped a brand new tube of toothpaste and it made a big mess. She was crestfallen. You’ve got to hand it to customs agents: Your passport. My friend tried to steal a box of lipstick for us, but she accidentally grabbed a box of glue sticks. My lips are sealed. Elevators: They take things to a whole other level. A friend gave me an expired pack of batteries. They were free of charge. Comedy: To taste a bit like a comet. A: How many times do I have to apologize? My wife said that the battery in my hearing aid needed to be replaced. That was difficult to hear. I asked the ski lift operator if I could get a free ride to the top of the mountain. He didn’t take me up on it. What makes a sentence a tongue twister? It’s hard to say. If you visit Mexico, remember to use the word “mucho.” It means a lot to them. There are more hydrogen atoms in a single molecule of water than there are stars in the solar system. To whoever discovered the number zero: Thanks for nothing. View this post ▶
In the late 17th century, natural philosopher Isaac Newton was deeply uneasy with a new scientific theory that was gaining currency in Europe: universal gravitation. In correspondence with a scientific contemporary, Newton complained that it was “an absurdity” to suppose that “one body may act upon another at a distance through a vacuum.” The scientist who proposed this preposterous theory was Isaac Newton. He first articulated the idea in his widely acclaimed magnum opus Principia, wherein he explained, “I have not yet been able to discover the cause of these properties of gravity from phenomena and I feign no hypotheses […] It is enough that gravity does really exist and acts according to the laws I have explained.” Newton proposed that celestial bodies were not the sole sources of gravity in the universe, rather all matter attracts all other matter with a force that corresponds to mass and diminishes rapidly with distance. He had been studying the motions of the six known planets–Mercury, Venus, Mars, Jupiter, Saturn, and Uranus–and by expanding upon the laws of planetary motion developed by Johannes Kepler about eight decades earlier, he arrived at an equation for gravitational force F that seemed to match decades of data: Where m1 and m2 are the masses of the objects, r is the distance between their centers of mass, and G is the gravitational constant (~0.0000000000667408). But this is only an approximation; humanity may never know the precise value because it is impossible to isolate any measuring apparatus from all of the gravity in the universe. Fellow astronomers found that Newton’s theory seemed to be accurate–universal gravitation appeared to reliably forecast the sometimes irregular motion of the planets even more closely than Kepler’s laws. In 1705, Queen Anne knighted Isaac Newton to make him Sir Isaac Newton (though this honor was due to his work in politics, not for his considerable contributions to math or science). In the century that followed, Newton’s universal gravitation performed flawlessly. Celestial bodies appeared to adhere to the elegant theory, and in scientific circles, it began to crystallize into a law of nature. But in the early 19th century, cracks began to appear. When astronomer Alexis Bouvard used Newton’s equations to carefully calculate future positions of Jupiter and Saturn, they proved spectacularly accurate. However, when he followed up in 1821 with astronomical tables for Uranus–the outermost known planet–subsequent observations revealed that the planet was crossing the sky substantially slower than projected. The fault was not in Bouvard’s math; Uranus appeared to be violating the law of universal gravitation. Newton’s theory was again called into question in 1843 by a 32-year-old assistant astronomer at the Paris Observatory, Urbain Le Verrier. Le Verrier had been following the Uranus perturbations with great interest, while also compiling a painstaking record of the orbit of Mercury–the innermost known planet. He found that Mercury also departed from projections made by universal gravitation. Was universal gravitation a flawed theory? Or might undiscovered planets lurk in extra-Uranian and intra-Mercurial space, disturbing the orbits of the known planets? Astronomers around the world scoured the skies, seeking out whatever was perturbing the solar system. The answer, it turned out, was more bizarre than they could have supposed. Continue reading ▶
An American Indian man on horseback stood outlined against a steely sky past midday on 05 October 1877. Winter was already settling into the prairies of what would soon become the state of Montana. Five white men stood in the swaying grass on the other side of the field, watching the horse move closer. Four wore blue uniforms, another in civilian attire. One of the uniformed men was tall and stout, with bright blue eyes and a large, curling mustache. He watched the proceedings with an air of self-importance. The surrender of the man on horseback might have been inevitable, sure, but it was nevertheless a nice feather in his cap. Perhaps his superiors would finally grant him that promotion after this whole affair was over. The other four men were more apprehensive. All of them were experienced in fighting American Indians on the frontier, but this opponent had been different. One man, with a full, dark beard and right arm missing below the elbow, looked at the approaching chief with grudging respect. The man had lost his arm in the American Civil War 15 years earlier, so he knew battle well. And in his opinion, the man across the field was a tactical genius, a “Red Napoleon.” Despite overwhelming odds, this Red Napoleon had wormed his way out of battle after battle, somehow always coming out on top. Continue reading ▶
More in science
The physicist Sidney Nagel delights in solving mysteries of the universe that are hiding in plain sight. The post Finding Beauty and Truth in Mundane Occurrences first appeared on Quanta Magazine
Federal enforcement of environmental laws has slowed significantly under President Trump. Read more on E360 →
Earlier today I gave a talk in the graduate student seminar titled “Counting is Hard. Complex Analysis is Easy.” based in part on my recent blog post about analytic combinatorics and based in part on Varilly’s notes on Dirichlet’s Theorem, showing how to count the number of trees of a certain shape and the number of “primes of a certain shape” by doing complex analysis. While prepping for this talk, I realized there’s a very pretty geometric way to see what’s going on when counting rooted ordered ternary trees! I don’t usually write blog posts about my local seminar talks anymore, but I think this is more than worthy of an exception! For instance, I think this could serve as a great visiual example of branches and mild singularities in complex analysis. So, let’s remember the problem! We want to count the number of rooted ordered ternary trees with $n$ nodes. Call this number $t_n$. We note that every such tree is either a root node a root with one child a root with two children a root with three children where each child is recursively a rooted ordered ternary tree. If we define $T(z) = \sum_n t_n z^n$ to be the (ordinary) generating function for the $t_n$ we see from this recurrence relation that it must satisfy the functional equation That is, if $P(z,w) = -w + z + zw + zw^2 + zw^3$ then $P(z,T(z)) = 0$. If we draw (the real locus of) $P$ it looks like and the implicit function theorem says that we can find functions $f(z)$ so that $P(z,f(z)) = 0$ through any starting point $P(z_0,f_0)$ on the curve for as long as $\frac{\partial P}{\partial w} \neq 0$. In particular, we know that our $T(0) = t_0 = 0$ since there are no trees with zero vertices, so that our $T(z)$ takes $z$ to the unique part of this curve passing through the origin, for as long as this is defined. Here we’ll plot our curve $P$ in blue, and the implicit function $T(z)$ in orange. Note we have to stop at $z \approx 0.27695$ since here $\frac{\partial P}{\partial w} = 0$ so that the slope is infinite and we can’t continue. Said another way, we bend backwards and if we tried to continue we would fail the vertical line test. But since we have to stop here, we see that $0.27695$ is the radius of convergence for $T$, and is the dominant singularity! Next, can we find a good approximation for $T$ near this singularity? In the main post we used puiseux series for this, but it’s instructive to do it by hand! Note that $\frac{\partial P}{\partial z} \neq 0$ at $(0.27695, 0.65730)$ so there’s nothing stopping us from approximating $z$ as a function of $w$ at this point (then inverting it). Indeed, we can solve for $z = \frac{w}{1+w+w^2+w^3}$ and then just taylor expand this at our point of interest: Of course, we chose this point because $\frac{\mathrm{d}z}{\mathrm{d}w} = \frac{- \frac{\partial P}{\partial w}}{\frac{\partial P}{\partial z}}$ is $0$ here. So we know this $2.97 \times 10^{-8}$ is a rounding error and our leading order expansion is Here’s a graph zoomed into near the singularity. The actual graph of $P$ is shown in blue, and our approximating parabola is shown in orange: But of course if $z - 0.27695 \approx -.34680 (w - 0.6573)^2$ then we can solve for and if we want this to agree with our $w = T(z)$ branch through the origin we obviously have to choose the negative square root (since we want the lower half of this sideways parabola1). Here we draw our $T(z)$ in blue, and we draw the approximation $0.6573 - 0.8936 \sqrt{1 - \frac{z}{0.27695}}$ in orange so you can see how well they line up! But now from here we can run exactly the same argument as in the previous post! We compute $t_n = \frac{1}{2\pi i} \oint \frac{T}{z^{n+1}} \ \mathrm{d}z$, along a keyhole contour around the singular point $z=0.27695$ with a branch cut along the real axis. The brunt of this integral comes from the cutout near the singular point, where $T$ is well approximated by $0.6573 - 0.8936 \sqrt{1 - \frac{z}{0.27695}}$ so that Again, for more details about exactly what this keyhole contour is and how you estimate this integral along it, see the main post. Let’s go ahead and call it here! My actual talk was slightly longer than this, since I also sketched a proof of Dirichlet’s Theorem following Varilly’s notes on the subject, but there’s no sense in me writing that up since those notes are already so good! Plus it’s getting late and I want to go to bed, haha. As always, here’s a copy of my title and abstract. Unfortunately I don’t have any slides or recordings today, but I’m giving another talk at WiSCons in Madison next week and I should have slides for that. I’m also hoping to finish a sister blog post for that talk where I do more example computations in more detail than I could possibly do in a 20 minute talk. Lots of writing to do this week! Take care all, stay safe, and I can’t wait to talk soon ^_^ Counting is Hard. Complex Analysis is Easy. Don’t you miss being a kid, when your mom would ask you to count how many of your alphabet fridge magnets were both red and vowels? When you saw the answer was “2”, you felt such accomplishment…. But counting has only gotten harder since then, and now if you want to count how many objects satisfy some properties (say, being both red and vowels) it can be borderline impossible! In this talk we’ll show how you can solve counting problems by doing complex analysis instead. First we’ll count the number of trees of a certain shape, then (given time) we’ll count how many prime numbers are of a (different) certain shape. In the main post we used puiseux series for this, and we had to essentially guess which branch was correct. Now we see how the geometry of the situation tells us that we have to choose the negative branch of the square root! After all, this is the one that locally agrees with the rest of the graph of $T$! ↩
Are GLP-1 drugs causing excess muscle loss compared to non-pharmacological weight loss?
