Lanas-own-blog - My Personal Space.

Charon from a different angle

via reddit

How to stabilize a wobbly table?

You are in a restaurant and you find that your table wobbles. What do you do?  Most people either put up with it, or they attempt to correct the problem by pushing a folded table napkin under one of the legs. But mathematicians can go one better. A couple of years ago, four mathematicians published a research paper in which they proved that if you rotate the table about its center, you will always find an orientation where the table is perfectly stable.

This problem - as a math problem - has been around since the 1960s, when a British mathematician called Roger Fenn first formulated it. In 1973, the famous math columnist Martin Gardner wrote about the problem in his Scientific American column, presenting a short, clever, intuitive argument to show how rotation will always stop the wobble. Here is that argument.

This only works for a table with equal legs, where the wobble is caused by an uneven floor. However uneven the floor, a table will always rest on at least three legs, even if one leg is in the air. Suppose the four corners are labeled A, B, C, D going clockwise round the table, and that leg A is in the air. If the floor were made of, say, sand, and you were to push down on legs A and B, leaving C and D fixed, then you could bring A into contact with the floor, but leg B would now extend into the sand. Okay so far?

Here comes the clever part. Since all four legs are equal, instead of pushing down on one side of the table, you could rotate the table clockwise through 90 degrees, keeping legs B, C and D flat on the ground, so that it ends up in the same position as when you pushed it down, except it would now be leg A that is pushed into the sand and legs B, C, and D are all resting on the floor. Since leg A begins in the air and ends up beneath the surface, while legs B, C, and D remain flat on the floor, at some point in the rotation leg A must have first come into contact with the ground. When it does, you have eliminated the wobble.The result follows from the Intermediate Value Theorem (Proof).

For more - VIDEO: Fix a Wobbly Table (with Math) by Numberphile.

Consider an interval I = [a, b] in the real numbers ℝ and a continuous function f : I → ℝ. Then, Version I. if u is a number between f(a) and f(b), f(a) < u < f(b)   (or  f(a) > u > f(b) ), then there is a c ∈ (a, b) such that f(c) = u.

This argument seems convincing, but making it mathematically precise turned out to be fairly hard. In fact, it took over 30 years to figure it out. The solution, presented in the paper Mathematical Table Turning Revisited, by Bill Baritompa, Rainer Loewen, Burkard Polster, and Marty Ross, is available online at Mathematical table turning revisited 19, Nov 2005, http://Arxiv.org/abs/math/0511490

The result follows from the Intermediate Value Theorem.  But getting it to work proved much harder than some other equally cute, real-world applications of the IVT, such as the fact that at any moment in time, there is always at least one location on the earth’s surface where the temperature is exactly the same as at the location diametrically opposite on the other side of the globe. As the authors of the 2005 solution paper observe, “for arbitrary continuous ground functions, it appears just about impossible to turn [the] intuitive argument into a rigorous proof. In particular, it seems very difficult to suitably model the rotating action, so that the vertical distance of the hovering vertices depends continuously upon the rotation angle, and such that we can always be sure to finish in the end position.” The new proof works provided the ground never tilts more than 35 degrees. (If it did, your wine glass would probably fall over and the pasta would slide off your plate, so in practice this is not much of a limitation.) Is the theorem any use? Or is it one of those cases where the result might be unimportant but the math used to solve it has other, important applications?…… “I have to say that, other than the importance of the IVT itself, I can’t see any application other than fixing wobbly tables. Though I guess it does demonstrate that mathematicians do know their tables”-  Mathematician Keith Devlin.

[SOURCE - MAA.org, K. Devlin, Feb. 2007]

[PDF] On the stability of four legged tables, A. Martin, 15 Aug. 2006:  Proving that a perfect square table with four legs , place on continuous irregular ground with a local slope of at most 14.4 degrees and later 35 degrees, can be put into equilibrium on the ground by a “rotation” of less than 90 degrees. And Discussing the case of non-square tables and make the conjecture that equilibrium can be found if the four feet lie on a circle.

Also, I think we can add an actual argument: “The table would be stability (not wobble) if their four legs contact the ground - not necessarily that they have lie on the same flat surface ”, then everything will be easier to approach the problem that the authors wrote.

Merging clusters in 30 Doradus.

Credit: NASA, ESA and E Sabbi

Hundreds of you sent in questions for my live conversation with three astronauts and NASA’s chief scientist on Tuesday. Thanks! The most common question was: “What happens when you get your period in space?”

I didn’t end up asking this question because

a) the question itself has a lot of historical baggage b) the answer is pretty boring

But because people seemed genuinely curious, I decided to answer it here.

First, a bit of history…

In the early days of space flight, menstruation was part of the argument that women shouldn’t become astronauts.

Some claimed (1) that menstruation would effect a woman’s ability, and blamed several plane crashes on menstruating women. Studies in the 1940s (2,3) showed this was not the case. Female pilots weren’t impaired by their periods. But the idea wouldn’t die. In 1964, researchers from the Women in Space Program (4) still suggested (without evidence) that putting “a temperamental psychophysiologic human” (i.e. a hormonal woman) together with a “complicated machine” was a bad idea.

Others raised concerns about hypothetical health risks. They feared that microgravity might increase the incidence of “retrograde menstruation.” Blood might flow up the fallopian tubes into the abdomen, causing pain and other health problems. No one actually did any experiments to see if this really would be a problem, so there wasn’t any data to support or refute these fears.

Advocates for women in space argued that there had been a lot of unknowns when humans first went to space, but they sent men up anyway. Rhea Seddon, one of the first six women astronauts at NASA, recalled during an interview:

We said, “How about we just consider it a non-problem until it becomes a problem? If anybody gets sick in space you can bring us home. Then we’ll deal with it as a problem, but let’s consider it a non-problem.”

Just to give you a sense of the culture surrounding female astronauts back then, here’s an excerpt of a 1971 NASA report about potential psychological problems in space. Researchers Nick Kanas and William Fedderson suggest there might be a place for women in space:

The question of direct sexual release on a long-duration space mission must be considered. Practical considerations (such as weight and expense) preclude men taking their wives on the first space flights. It is possible that a woman, qualified from a scientific viewpoint, might be persuaded to donate her time and energies for the sake of improving crew morale; however, such a situation might create interpersonal tensions far more dynamic than the sexual tensions it would release.

Kanas, now an emeritus professor of psychology at UCSF, told me this was tongue-in-cheek — part of a larger discussion about the problem of sexual desire in space (5). Still, it’s surprising this language was included in an official NASA memorandum. Even advocates for women in space were caught up in this kind of talk. In a 1975 report for the RAND corporation, Glenda Callanen argues that women have the strength and intelligence to become astronauts. But here’s how she begins the report’s conclusion:

It seems inevitable that women are to be essential participants in space flight. Even if they were only to take on the less scientific parts of the space mission, or if they wished only to help “colonize” distant planets, their basic skills must still prepare them to perform countless new tasks.

In a culture where these statements were unremarkable, it’s easy to imagine that questions about menstruation weren’t purely motivated by scientific curiosity.

In 1983, 22 years after Alan Shepard became the first American to go to space, Sally Ride left earth’s atmosphere. She told an interviewer:

I remember the engineers trying to decide how many tampons should fly on a one-week flight; they asked, “Is 100 the right number?” “No. That would not be the right number.”

So what DOES happen when you get your period in space?

The same thing that happens on Earth! In the last three decades years of female space flight, periods in space have been normal — no menstrual problems in microgravity.

Notes:

RE Whitehead, MD. “Notes from the Department of Commerce: Women Pilots.” The Journal of Aviation Medicine 5 (Mar-Dec 1934):48.

RS Holtz, MD. “Should Women Fly During the Menstrual Period?” The Journal of Aviation Medicine 12 (Sept 1941):302.

J Cochrane. “Final Report on Women Pilot Program.” 38.

JR Betson and RR Secrest. “Prospective women astronauts selection program.” American Journal of Obstetrics and Gynecology 88 (1964): 421–423.

Kanas and Fedderson’s 1971 report went on to conclude: “Information regarding women during periods of stress is scanty. This lack, plus previously mentioned problems, will make it difficult for a woman to be a member of the first long-duration space missions. However, it is just as unlikely to think that women cannot adapt to space. Initial exploration parties are historically composed of men, for various cultural and social reasons. Once space exploration by men has been successfully accomplished, then women will follow. In preparation for this, more information should be compiled regarding the physiology and psychology of women under stressful situations.”

NGC 3324.

Credit: NASA, ESA and the Hubble Heritage Team (STScI/AURA)