Detail of Mary Death, by Matt Tarpley, 10 February 2015.
Category Archives: Mathematics
Dreaming of the Day
Image credit: Matt Tarpley, Mary Death, 14 November 2014.
Astronomy and Human Progress
This morning, the National Radio Astronomy Observatory issued a press release, which in and of itself is hardly extraordinary. Its contents, however, are extraordinarily awesome:
Astronomers have captured the best image ever of planet formation around an infant star as part of the testing and verification process for the Atacama Large Millimeter/submillimeter Array’s (ALMA) new high-resolution capabilities.
This revolutionary new image reveals in astonishing detail the planet-forming disk surrounding HL Tau, a Sun-like star located approximately 450 light-years from Earth in the constellation Taurus.
ALMA uncovered never-before-seen features in this system, including multiple concentric rings separated by clearly defined gaps. These structures suggest that planet formation is already well underway around this remarkably young star.
“These features are almost certainly the result of young planet-like bodies that are being formed in the disk. This is surprising since HL Tau is no more than a million years old and such young stars are not expected to have large planetary bodies capable of producing the structures we see in this image,” said ALMA Deputy Director Stuartt Corder.
While this photo is not about to save a life or help a man improve his intimate relations, it occasionally occurs to us to remind that astronomy is not just about fancy photos. The human species needs astronomers.
Happy π Day
Happy π Day. Π? π? I go with π, since that’s what this is all about, anyway. right?
Well, you know, to our American friends. Never mind. Dumb joke. Predictable. Happy π Day to each and every one of you around the world and throughout the Universe.
On a related note, NASA proudly recalls the Pi Transfer:
On Jan. 19, 2007, the Cassini spacecraft took this view of Saturn and its rings—the visible documentation of a technique called a “pi transfer” completed with a Titan flyby. A pi transfer uses the gravity of Saturn’s largest moon, Titan, to alter the orbit of the Cassini spacecraft so it can gain different perspectives on Saturn and achieve a wide variety of science objectives. During a pi transfer, Cassini flies by Titan at opposite sides of its orbit about Saturn (i.e., Titan’s orbital position differs by pi radians between the two flybys) and uses Titan’s gravity to change its orbital perspective on the ringed planet.
Taking in the rings in their entirety was the focus of this particular imaging sequence. Therefore, the camera exposure times were just right to capture the dark-side of its rings, but longer than that required to properly expose the globe of sunlit Saturn. Consequently, the sunlit half of the planet is overexposed.
Yeah. They can do this. Happy π Day.
• If you haven’t discovered Summer Ash, do so. Or, consider the physics of your morning coffee, an exoplanet extravaganza from Kepler, or maybe the oldest known piece of planet Earth (and other notes).
• What do you get when you cross Bill Nye, Neil DeGrasse Tyson, and President Barack Obama? I don’t know, but it sure is smart ….
• Disordered hyperuniformity: You’ll never look at a chicken the same way again.
• Two words: blue lava.
• What does the number 915,103,765 have to do with LEGO bricks?
• Cosmic spiders? Not quite, but black widow binaries are still pretty scary.
• Every cloud has a silver lining, and really bad weather helps us find things we’ve lost.
• The question on everyone’s minds: What is a dropleton?
But Does It Work On Ducks?
“So, a fake force is a force that is not an interaction between two objects. Rather it is like duck tape (I refuse to call it duct tape because it isn’t good for ducts) on your accelerating frame.”
What? I mean, it’s an obvious question, right?
Er … never mind*. But the consideration of physics in a game show context, from 2011, is still worth a read. Today’s discussion of gravity in a space movie is just as rewarding.
* I know … I know, I know. I just needed a title. Or, okay, to be honest, it was the first one that occurred to me.
A Brief Physics Lesson: Bicycle Edition
Math? Physics? Those who enjoy brief exercises in applied speculation will certainly learn a thing or two from Rhett Allain, who took some time to consider bicycles and hills. At first blush, it seems an easy enough question: What’s the steepest gradient you could possibly ride on a road bike?
Of course, there is a difference between the simplicity of straightforward mathematics and the complexity of accounting for all the factors required.
I think there are three reasons why a slope would be too steep. For all of these cases, I am going to assume that it is a prolonged slope. This means that you can’t just build up a large speed and zoom up the slope. If this was the case, you could go straight up a wall (which you can for a short time).
Those three reasons are the limitations of human power, center of mass, and friction. If one wishes to point out, “What if you used these tires instead of those?” or, “What if you had a different gear set?” it’s all well and fine to do so, but therein lies the point about the complexity of accounting for all the factors required.
Really, the friction problem might be worse than this. The bike only uses the back wheel for moving forward, so it is the friction on the back wheel that matters. If the biker is leaning forward, the weight distribution might not be even on the two wheels. I will leave this estimation (combining the previous two limits) as an exercise for the reader.
And, of course, one is welcome to pursue such endeavors. (In truth, that might be part of the point.)
e, Not E, and Before e-
Apparently, I missed e Day. More than likely, so did you. Don’t worry, though, there will be another chance this year, though you’ll be celebrating with Europeans.
Rhett Allain didn’t miss e Day:
Why is Feb. 7 e Day? Well, in the USA we use the Middle-endian date format. So, Feb. 7 would be commonly written as 2/7/13. Guess what? The first two digits of e are 2.7. If you live in other places you might use the little-endian date format. In this case 2/7/13 would be July 2. For those people, just consider this an early post.
But don’t get distracted. (Yes, my first question was the same as yours: “Really? It’s really called ‘Middle-endian’?”
But e Day is a celebration of e, the jealous little brother of π.
Allain offers his favorite definition: “e is the number that if you raise that number to the power x, the slope of the function is the same value as the function.”
And it just goes downhill from there. Or uphill, I guess, if you look at it on a graph. Sort of. The graph at right does not show you e. Maybe you could try reading the Wikipedia entry on e, but that’s the fun thing about being a poor mathematician and clueless excuse for a scientist.
Sometimes called Euler’s number after the Swiss mathematician Leonhard Euler, e is not to be confused with γ—the Euler–Mascheroni constant, sometimes called simply Euler’s constant. The number e is also known as Napier’s constant, but Euler’s choice of this symbol is said to have been retained in his honor.
By the time you get to the part about derangements, well, yeah.
Oh, right. e = 2.71828 (and a whole bunch of numbers after that; it’s irrational and trancendental, just like π).