Friday, March 30, 1962

Martin-Baltimore has submitted to the Air Force Space Systems Division the document Description of the Launch Vehicle for the Gemini Spacecraft. This defines the concept and the philosophy for each proposed subsystem as well as laid out the design for the Gemini launch vehicle.


Thursday, March 29, 1962

McDonnell awards an $18 million subcontract to the Saint Petersburg, Florida, Aeronautical Division of Minneapolis-Honeywell. This subcontract is to provide the Inertial Maneuvering Unit of the Gemini spacecraft.
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How To Forget The Area Of A Trapezoid

When I lecture I like to improvise. I prepare notes, of course, the more detailed the more precise I need to be, but my performing instincts are most satisfied when I just go in front of the class with some key points to hit and maybe a few key lines worked out ahead of time. But I did recently make an iconic mistake, repeating the mathematics instructor’s equivalent of the lawyer asking in court a question without already knowing what the answer will be. Improvisation has to be carefully prepared.

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Wednesday, March 28, 1962

McDonnell awards a subcontract worth $2.5 million to the Collins Radio Company of Cedar Rapids, Iowa. The subcontract is to provide the voice communications system for the Gemini spacecraft.
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I Know The Square Root Of Five, Too

Now I want to do a little more complicated problem of showing two numbers are equal because the difference between them is so tiny. It struck me that if I wanted to do that, I’d have to do some setup to even start. What I really meant to do was to show that some number was equal to the square root of five. I picked the square root of five because I had it burned into my memory from a children’s book that knowing the first few digits of an irrational number would be sufficient to immobilize the mind-controlled population of an all-powerful computer dictator, and I’ve kept it in mind just in case ever since. I’m also glad to know on double-checking that I remembered the first couple digits of the square root of five well (2.236). I’m shakier on the square root of seven (2.something) so if it’s a more advanced computer we’re up against I’m in trouble.

Still, most square roots would do. It’s a neat little property of the whole numbers that the square roots of them are either whole numbers themselves — the square root of 4 is 2, the square root of 169 is 13, the square root of 4,153,444 is not worth thinking about — or else they’re irrational numbers, going on without ending and without repetition. Most people who’d read a mathematics blog on purpose have heard about how the irrationality of the square root of 2 was proven in ancient days, and maybe heard the story of how the Pythagoreans murdered the person who let slip the horrifying secret that there were irrational numbers and they represented real things that might be of interest, and a few are even aware we don’t really know with certainty that the story’s actually true. (At this point, I suspect it’s too strong a claim to say we know anything about the Pythagoreans for certain, but I haven’t looked closely. Maybe matters are not quite that dismal.) Whether true or not the legend of the Pythagoreans turning to murder is a fine way to get an algebra class’s attention. I just fear that what the students take away from it is, “if you learn any of this math stuff a cabal of mathematicians will murder you” and they stay oblivious for reasonable self-protection.

But anyone who’s understood a proof that the square root of two is irrational is perfectly able to show that the square root of three is irrational as well, or the square root of five, or any other such desired number. The proof that way runs just about the same route, but takes longer to get there.

Similarly, if you have a rational number that comes to an end, such as 0.49, then the square root either is a rational number that comes to an end, in this case 0.7; or else it never comes to an and and never repeats. That’s easy to prove, if you have that idea about the square roots of whole numbers. The square root of 4.9, for example, is not a rational number, although I can’t promise anything for its ability to halt world-spanning computers.

Disputed Cards

In one of my classes we’ve plunged into probability, which is a fun subject because suddenly there’s no complicated calculations to do — at worst, there’s long ones — but you have to be extremely careful about what calculations you do, so all that apparent simplicity gets turned into conceptual difficulty. So it’s brought to mind something a student in an earlier term told me about.

I’d given as a problem one of the standard rote probability puzzles: the chance of picking three red cards in a row from a well-shuffled and full deck. The chance of doing this depends a bit on whether you put the just-picked card back in the deck and reshuffle or not, but in either case, it’s pretty close to about one in eight. Multiple students got this exactly right, glad to say, but one spun it out into an anecdote.

The student was pretty enthusiastic about the course topics and while hanging out with a sibling mentioned this as a problem, and the solution. The sibling, however, didn’t believe it, and insisted that since there are an equal number of red and black cards there should be a one in two chance of drawing three red cards in a row out. The two disputed the subject for the whole weekend, and my student apparently rather appreciated having something novel to argue about.

I’m always delighted when a student is interested enough in a problem to mention it to anyone else, and probability puzzles often give things with real-world models simple enough to catch the imagination. But I was (and still am) surprised the question could last a whole weekend. Freaky things can happen in small sample size, but I’d be willing to bet that trying it out with a deck of cards a couple times would at least provide convincing evidence that the chance of three reds in a row wasn’t one in two.

Possibly my student wasn’t communicating the problem well; one in two would be about right for the chance of picking a third red card, after all, regardless of what the first two were. Or perhaps they didn’t have a deck of cards to try it out. I couldn’t reliably lay my hand on a deck of cards until a few weeks ago, when I bought a deck so I could demonstrate problems in class.

Also, a standard-size deck of cards is far too small for a class demonstration. I need to find a magic store and get an oversized deck of cards. I have the same problem with the dice I picked up, but there I should be able to find a giant pair of dice in an auto parts store. They’ll be fuzzy, but should express the idea of dice well enough for that.

Proving A Number Is Not 1

I want to do some more tricky examples of using this ε idea, where I show two numbers have to be the same because the difference between them is smaller than every positive number. Before I do, I want to put out a problem where we can show two numbers are not the same, since I think that makes it easier to see why the proof works where it does. It’s easy to get hypnotized by the form of an argument, and to not notice that the result doesn’t actually hold, particularly if all you see are repetitions of proofs where things work out and don’t see cases of the proof being invalid.

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Friday, March 23, 1962

Air Force Space Systems Division has published the Development Plan for the Gemini Launch Vehicle System. Using experience drawn from the Titan II and the Mercury development programs it is estimated the development of the launch vehicle will require a budget of $164.4 million. This includes a contingency fund of 50 percent to cover cost increases and unforeseen changes.

Wednesday, March 21, 1962

McDonnell awards a $4.475 million subcontract to the Western Military Division of Motorola, Inc, of Scottsdale, Arizona. Western Military Division is to design and build the Digital Command System for the Gemini spacecraft. This is to receive in digital format commands from ground stations, to decode them, and to send the commands to the appropriate spacecraft systems. Two types of commands are anticipated: real-time commands for spacecraft functions, and stored program commands to update data on the spacecraft’s digital computer. The Digital Command System is to consist of a receiver/decoder package and three relay packages.

Air Force Space Systems Division awards a letter contract to the Aerojet-General Corporation of Azusa, California. This is to research, develop, and procure fifteen propulsion systems for the Gemini launch vehicle, and also for the design and development of related ground equipment. Aerojet was authorized to work on the engines on February 14th. Final engine delivery is scheduled by April 1965.

My Wholly Undeserved Odd Triumph

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.

Tuesday, March 20, 1962

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.

How To Multiply By 365 In Your Head

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.

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Monday, March 19, 1962

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.

Some More Comic Strips

I might turn this into a regular feature. A couple more comic strips, all this week on, 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.

Saturday, March 17, 1962

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.

Friday, March 16, 1962

The most recent Titan II launch, October 2003.  I have lost the original source from which this picture came and would welcome correct credit information.
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.

What Numbers Equal Zero?

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?”

Thursday, March 15, 1962

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.

Wednesday, March 14, 1962

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.

Introducing a Very Small Number

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.