While following my own lightly compulsive tracking of the blog’s viewer statistics and wondering why I don’t have more followers or even people getting e-mail notifications (at least I’ve broken 2,222 hits!) I ran across something curious. I can’t swear that it’s still true so I’m not going to link to it, and I don’t want to know if it’s not true. However.
Somehow, one of my tags has become Google’s top hit for the query “christiaan huygens logarithm”. Oh, the post linked to contains the words, don’t doubt that. But something must have got riotously wrong in Google’s page-ranking to put me on top, even above the Encyclopaedia Britannica‘s entry on the subject, and for that matter — rather shockingly to me — above the references for the MacTutor History of Mathematics biography of Huygens. That last is a real shocker, as their biographies, not just of Huygens but of many mathematicians, are rather good and deserving respect. The bunch of us leave Wikipedia in the dust.
I assume it to be some sort of fluke. Possibly it reflects how the link I actually find useful is never the first one in the list of what’s returned, so perhaps they’re padding the results with some technically correct but nonsense filler, and I had the luck of the draw this time. Perhaps not. (I’m only third for “drabble math comic”, and that would at least be plausible.) But I’m amused by it anyway. And I’d like to again say that the MacTutor biographies at the University of Saint Andrews are quite good overall and worth using as reference, and are also the source of my discovery that Wednesday, March 21, is the anniversary of the births of both Jean Baptiste Joseph Fourier (for whom the Fourier Series, Fourier Transform, and Fourier Analysis, all ways of turning complicated problems into easier ones, are named) and of George David Birkhoff (whose ergodic theorem is far too much to explain in a paragraph, but without which almost none of my original mathematics work would have what basis it has). I should give both subjects some discussion. I might yet make Wikipedia.
McDonnell issues a $9 million subcontract to General Electric to design and develop fuel cells for the Gemini spacecraft. The General Electric design, selected by the Manned Spacecraft Center after an analysis completed January 23, appeared to offer advantages over the competing solar cells or other fuel cells in terms of simplicity, weight, and compatibility with other Project Gemini requirements. Much of this advantage is credited to the use of ion-exchange membranes rather than gas-diffusion electrodes within the fuel cells.
Kevin Fagin’s Drabble from Sunday poses a nice bit of recreational mathematics, the sort of thing one might do for amusement: Ralph Drabble tries to figure how long he’s spent waiting at one traffic light. I want to talk about some of the mental arithmetic tricks I’d use to get through the puzzle without missing the light’s change. In the spirit of the thing I’m doing the calculations for this only in my head, though I admit checking with a calculator afterward to see if I got close.
Continue reading “How To Multiply By 365 In Your Head”
McDonnell awards a $3.2 million subcontract to Advanced Technology Laboratories, Inc, of Mountain View, California. The subcontract is for the horizon sensor system for the Gemini spacecraft. One primary and one secondary horizon sensor are to be part of the guidance and control system. The sensors are to detect and track the gradient of infrared radiation between the Earth and outer space.
McDonnell also awards a $400,000 subcontract to the Thiokol Chemical Corporation of Elkton, Maryland, for retrograde rockets. The solid-propellant retrorockets, four of which are designed to be put in the adapter section, are to start reentry or, in the event of a high-altitude suborbital abort, separate the spacecraft from the Titan II booster. It is believed that only slight modifications of a motor already in use are necessary, and that the qualification program will not need to be elaborate.
I might turn this into a regular feature. A couple more comic strips, all this week on gocomics.com, ran nice little mathematically-linked themes, and as far as I can tell I’m the only one who reads any of them so I might spread the word some.
Grant Snider’s Incidental Comics returns again with the Triangle Circus, in his strip of the 12th of March. This strip is also noteworthy for making use of “scalene”, which is also known as “that other kind of triangle” which nobody can remember the name for. (He’s had several other math-panel comic strips, and I really enjoy how full he stuffs the panels with drawings and jokes in most strips.)
Dave Blazek’s Loose Parts from the 15th of March puts up a version of the Cretan Paradox that amused me much more than I thought it would at first glance. I kept thinking back about it and grinning. (This blurs the line between mathematics and philosophy, but those lines have always been pretty blurred, particularly in the hotly disputed territory of Logic.)
Bud Fisher’s Mutt and Jeff is in reruns, of course, and shows a random scattering of strips from the 1930s and 1940s and, really, seem to show off how far we’ve advanced in efficiency in setup-and-punchline since the early 20th century. But the rerun from the 17th of March (I can’t make out the publication date, although the figures in the article probably could be used to guess at the year) does demonstrate the sort of estimating-a-value that’s good mental exercise too.
I note that where Mutt divides 150,000,000 into 700,000,000 I would instead have divided the 150 million into 750,000,000, because that’s a much easier problem, and he just wanted an estimate anyway. It would get to the estimate of ten cents a week later in the word balloon more easily that way, too. But making estimates and approximations are in part an art. But I don’t think of anything that gives me 2/3ds of a cent as an intermediate value on the way to what I want as being a good approximation.
There’s nothing fresh from Bill Whitehead’s Free Range, though I’m still reading just in case.
McDonnell awarded a $5.5 million subcontract to AiResearch. AiResearch is to provide the reactant supply for Gemini spacecraft fuel cells.
The fuel cells, which are designed to provide power and water, are to store hydrogen and oxygen in two double-walled, vacuum-insulated, spherical containers in the Gemini spacecraft’s adaptor section, jettisoned at the start of the reentry procedure.
The Air Force successfully launched a Titan II intercontinental ballistic missile at 18:09 Greenwich Time. This suborbital flight, the first full-scale test of the vehicle to also be Project Gemini’s booster, from Launch Complex 16, flew five thousand miles out over the Atlantic Ocean and reached an apogee of about eight hundred miles. This vehicle was serial number N-2.
Launch Complex 16 has since its inauguration on December 12, 1959, been used for six launches of the Titan I, three of them successful.
North American awarded a $225,000 subcontract to the Radioplane Division of Northrop Corporation today, as part of North American’s contract to design and develop emergency parachute recovery systems and test vehicles for the Paraglider Development Program.
McDonnell contracted with Vidya, Inc, of Palo Alto, California, today to test new ablation materials for the Gemini heat shields.
I want to give some examples of showing numbers are equal by showing the difference between them is ε. It’s a fairly abstruse idea but when it works amazing things become possible.
The easy example, although one that produces strong resistance, is showing that the number 1 is equal to the number 0.9999…. But here I have to say what I mean by that second number. It’s obvious to me that I mean a number formed by putting a decimal point up, and then filling in a ‘9’ to every digit past the decimal, repeating forever and ever without end. That’s a description so easy to grasp it looks obvious. I can give a more precise, less intuitively obvious, description, though, which makes it easier to prove what I’m going to be claiming.
Continue reading “What Numbers Equal Zero?”
The Air Force Space Systems Division contracted today with the Aerospace Corporation of El Segundo, California, for general systems engineering and technical direction of the development of the Titan II booster used for Project Gemini. Aerospace itself had established a Gemini Launch Vehicle Program in January, and Space Systems Division issued a Technical Operating Plan for this support on February 18.
The Gemini Project Office reiterated its intention that Project Mercury hardware and subcontractors are to be used for Gemini. Using different equipment or subcontractors requires justification for each item.
The Gemini Project Office made a major decision about seat ejection. It is to be initiated manually, with both seats ejected simultaneously in case either ejection system is energized. The seat ejection is to be useful as a way to escape an emergency while on the launchpad, during the initial phase of powered flight (to an altitude of about 60,000 feet), or on reentry following a failure of the paraglider landing system.
The escape system is to include a hatch actuation system, opening the hatches before ejection; a rocket catapult to shoot both seats away from the spacecraft; and parachutes for the astronauts following their separation from the seat. The system is also to provide for survival equipment for the astronauts to use after landing.
The design is to allow for an automatic initiator in case this later becomes a requirement.
In other news the Manned Spacecraft Center issued its second analysis of the Gemini program schedule. This is the first to consider launch vehicles as well as the spacecraft. (The earlier analysis, of just Gemini operations, was published January 5.) Analysis of the Agena vehicles is limited, as their procurement began only with a request the Manned Spacecraft Center sent to Marshall Space Flight Center on January 31 for the eleven Atlas-Agena rendezvous targets believed needed.
The Gemini program is projected to use a number of test articles for engineering development, correcting a problem which had delayed the Mercury Program at times. The first, unmanned, qualification test is projected for late July or early August 1963. The second, manned, flight is now planned for late October or early November 1963. The first Agena flight is projected for late April or early May 1964. The remaining flights in the program are to be at roughly two-month intervals from then until the middle of 1965.
Last time I talked mathematics I introduced the idea of using some little tolerated difference between quantities. This tolerated difference has an immediately obvious and useful real-world interpretation: if we measure two things and they differ by less than that amount, we’d say they’re equal, or close enough to equal for whatever it is we’re doing. And it has great use in the nice exact proofs of some sophisticated mathematical concepts, most of which I think I can get to without introducing equations, which will make everyone happy. Readers like reading things that don’t have equations (folklore has it that every equation, other than E = mc2, cuts book sales in half, although I don’t remember seeing that long-established folklore before Stephen Hawking claimed it in A Brief History Of Time, which sold a hundred million billion trillion copies). Writers like not putting in equations because web standards have evolved so that there’s not only no good ways of putting in equations, but there aren’t even ways that rate as only lousy. But we can make do.
The tolerated difference is usually written as ε, the Greek lower-case e, at least if we are working on calculus or analysis at least, and it’s typically taken to mean some small number. The use seems to go back to Augustin-Louis Cauchy, who lived from 1789 to 1857, who paired it with the symbol δ to talk about small quantities. He seems to have meant δ the Greek lowercase d, to be a small number representing a difference, and ε as a small number representing an error, and the symbols have been with us ever since.
Cauchy’s an interesting person, although it seems sometimes that every mathematician who lived in France anytime around the Revolution and the era of Napoleon was interesting. He was certainly prolific: the MacTutor biography credits him with 789 published papers, and they covered a wide swath of mathematics: solid geometry, polygonal numbers, waves, inelastic shocks, astronomy, differential equations, matrices, and a powerful tool called the Fourier transform. This is why mathematics majors spend about two years running across all sorts of new things named after Cauchy — the Cauchy-Schwarz inequality, Cauchy sequences, Cauchy convergence, Cauchy-Reimann equations, Cauchy-Kovalevskaya existence, Cauchy integrals, and more — until they almost get interested enough to look up something about who he was. For a while Cauchy was tutor to the grandson of France’s King Charles X, but apparently Cauchy had a tendency to get annoyed and start screaming at the uninterested prince. He has two lunar features (a crater and an escarpment) named for him, indicating, I suppose, that Charles X wasn’t asked for a reference.
Marshall Space Flight Center delivers to the Gemini Project Office a procurement schedule for Agena target vehicles. The Air Force Space Systems Division is to contract with Lockheed for 11 target vehicles. Space Systems Division is to put the Gemini Agena target vehicle program under the Ranger Launch Directorate.
Marshall expects that the delivery of a main engine qualified for multiple restarts will be in 50 weeks. This is an improvement in development time: the main engine is no longer considered the pacing item in the schedule for Agena development.
It’s as far from my workplace to home as it is from my workplace to my sister-in-law’s home. That’s a fair coincidence, but nobody thinks it’s precisely true. I don’t think it’s exactly true myself, but let me try to make it a little interesting. I’d be surprised if it were the same number of miles from work to either home. I’d be shocked if it were the same number of miles down to the tenth of the mile. To be precisely the same distance, down to the n-th decimal point, would be just impossibly unlikely. But I’d still make the claim, and most people would accept it, and everyone knows what the claim is supposed to mean and why it’s true. What I mean, and what I imagine anyone hearing the claim takes me to mean, is that the difference between these two quantities, the distance from work to home and the distance from work to my sister-in-law’s home, is smaller than some tolerable margin for error.
That’s a good definition of equality between two things in the practical world. It applies mathematically as well. A good number of proofs, particularly the ones that go into proving calculus works, amount to showing that there is some number in which we are interested, and there is some number which we are actually able to calculate, and the difference between those two numbers is less than some tolerated difference. If we’re just looking for an approximate answer, that’s about where we stop. If we want to do prove something rigorously and exactly, then we use a slightly different trick.
Instead of proving that the difference is smaller than some tolerated error — say, that the distance to these two homes is the same plus or minus two miles, or that these two cups of soda have the same amount of drink plus or minus a half-ounce, or so — what we do is prove that we can pick some arbitrary small tolerated difference, and find that the number we want and the number we can calculate must be smaller than that tolerated difference. But that tolerated difference might be any positive number. We weren’t given it up front. If the difference is smaller than any positive number, then, we can, at least in imagination, make sure the difference is smaller than every positive number, however tiny. The conclusion, then, is that if the difference between what-we-want and what-we-have is smaller than every positive number, then the difference must be zero. The two quantities have to be equal.
That probably read fairly smoothly. It’s worth going over and thinking about closely because, at least in my experience, that’s one of the spots where calculus and analysis gets really confusing. It’s going to deserve some examples.
In a ceremony at the Pentagon, John Glenn was presented Astronaut Wings. He is the third person so honored.
The Manned Spacecraft Center directed North American to design and develop an emergency parachute system for flight test vehicles. These vehicles, both half-size and full-size, are required for Phase II-A of the Paraglider Development Program. The Manned Spacecraft Center authorized North American to subcontract the emergency recovery system to the Radioplane Division of the Northrop Corporation.
The Marshall Space Flight Center composed a procurement schedule for the Agena target vehicles, to be delivered to the Gemini Project Office.
The Gemini Project Office accepted McDonnell’s preliminary design for the Gemini main undercarriage for use in land landings. It authorized McDonnell to proceed with the detail design. Dynamic model testing of the undercarriage should begin around April 1.
McDonnell subcontracted to the Minneapolis-Honeywell Regulator Company of Minneapolis to provide the attitude control and maneuvering electronics system. This is to provide the circuitry linking astronaut controls to the attitude and maneuvering controls and the reaction control system. The contract is for $6.5 million.
Now here’s another great tool Chiaroscuro did, in figuring out what number raised to the fifth power would be 1/6000. Besides trying out a variety of numbers which were judged to be a little bit low or a little bit high, he eventually stopped.
Wisely, too. The number he really wanted was the fifth root of 1/6000, and while there is one, it’s not a rational number. It goes on forever without repeating and without falling into any obvious patterns. But neither he nor anyone else is really interested in any but the first couple of these digits. We’d wanted to know whether this number was close to 0.25, and it’s closer to 0.17 instead. What the tenth digit past the decimal was we don’t really care about. It’s fine to be close enough to the right answer.
This runs a little against the stereotype of the mathematician. To the extent that popular culture notices mathematicians at all, it’s as people who have a lot of digits past a decimal point. But a mathematician is, in practice, much more likely to be interested in saying something that’s true, even if it isn’t so very precise, and to say that the fifth root of 1/6000 is somewhere near 0.17, or better, is between 0.17 and 0.18, is certainly true. Probably — and I’m attempting here to read Chiaroscuro’s mind, as the only guidance I’ve gotten from him is the occasional confirmation about what my guesses to his calculation were — he found that 0.17 was a little low, and 0.18 was a little high, and the actual value had to be somewhere between the two. The Intermediate Value Theorem, discussed in the previous non-Gemini-Chronology entry, guarantees that between those two is an exactly correct answer. (It’s conceivable that there would be more than one, in fact, although for this problem there’s not.)
Chiaroscuro specifically judged the fifth root of 1/6000 to be 0.176, or 17.6%, and I doubt anyone would seriously argue with that claim. This is even though the actual number is a little bit less than that: it’s nearer 0.175537, but even that is only an approximation. We are putting one of those big ideas into play, subtly, when we accept saying one number is equal to another in this way.
Westinghouse Electric Corporation of Baltimore received a $6.8 million subcontract from McDonnell. Westinghouse will provide the rendezvous radar and transponder system for the Gemini craft. The transponder is to be located in the Agena target vehicle.
Harold I Johnson, head of the Spacecraft Operations Branch (Flight Crew Operations Division of the Manned Spacecraft Center), circulated a memorandum on proposed training devices. The mission simulator should be capable of replicating a complete mission profile including sight, sound, and vibration cues, and be initially identical to the spacecraft, mission control, and remote site displays.
Training for launch and re-entry is to be provided by the centrifuge at the Naval Air Development Center (Johnsville, Pennsylvania), with a gondola set up to replicate the Gemini spacecraft interior. A static article is to serve as egress trainer. A boilerplate spacecraft with paraglider wing, used in a program including helicopter drops, will provide experience in landing on dry land. A docking trainer, fitted with actual hardware, capable of motion in six degrees of freedom, is to be used for docking operations training. And other trainers would be used for prepare for specific tasks.
The first regular business meeting between the Gemini Project Office and McDonnell occurred. Subsequent meetings are scheduled for the Monday, Tuesday, and Friday of each week. The initial coordination meetings had been held February 19, and introduction meetings were held the 19th, 21st, 23rd, 27th, and 28th. The objective of these meetings is to discuss and settle differences in decision-making about the project.
Mercury Astronauts Scott Carpenter and Walter Schirra went through water-egress exercises, including practice with helicopter pickups.
However I may sulk, Chiaroscuro did show off a use of the Intermediate Value Theorem that I wanted to talk about because normally the Intermediate Value Theorem occupies a little spot around Chapter 2, Section 6 of the Intro Calculus textbook and it gets a little attention just before the class moves on to this theorem about there being some point where the slope of the derivative equals the slope of a secant line which is very testable and leaves the entire class confused.
The theorem is pretty easy to state, and looks obviously true, which is a danger sign. One bit of mathematics folklore is that the only things one should never try to prove are the false and the obvious. But it’s not hard to prove, at least based on my dim memories of the last time I went through the proof. One incarnation of the theorem, one making it look quite obvious, starts off with a function that takes as its input a real number — since we need a label for it we’ll use the traditional variable name x — and returns as output a real number, possibly a different number. And we have to also suppose that the function is continuous, which means just about what you’d expect from the meaning of “continuous” in ordinary human language. It’s a bit tricky to describe exactly, in mathematical terms, and is where students get hopelessly lost either early in Chapter 2 or early in Chapter 3 of the Intro Calculus textbook. We’ll worry about that later if at all. For us it’s enough to imagine it means you can draw a curve representing the function without having to lift your pen from the paper.
Continue reading “The Intermediacy That Was Overused”