My dear friend Norm Schryer died unexpectedly this morning. This past week hasn't been a kind one.
Norm was one of those special beings you sometimes hear about and, if you're really lucky, get to work and play with. He was a Bell Laboratories mathematician - he'd probably say the old Bell Labs - The Bell Telephone Laboratories. The labs had a tendency to hire a large number of physicists and mathematicians - much larger than any other industrial lab. People were granted enormous freedom, but HR was looking for people who played well together and who might consider AT&T a problem rich environment. Given a few interesting constraints imposed by the government much of the future was invented.
Norm made numerous contributions, but more impressively they covered several fields including those being invented. He brought what appeared to be a near-reckless state of play. Norm had the energy to plow into "short term" problems for weeks or months at a time. Those of us who worked with him saw him as brilliant, generous almost to a fault, loud - very loud, and incredibly playful - think of an 8 year old wound on candy. Norm was like that almost constantly.
We had a very small working group of three with a few other players coming and going. We knew that brainstorming doesn't work (I have a great story about that - if you ever catch me with time ask about SewerNet and SantaDelivery ... perhaps my best three hour stint of working against the forces of corporate bureaucracy). What does work is when you have a small group of people that know each other very well. Diverse talents and interests are central - we had two physicists and a mathematician, but many others came in and out. People would present ideas which would be torn to shreds and new paths tried. On the fly estimates and calculations were central and we each had different approaches.
Norm would sit and listen as others opened up their ideas. Mind you not quietly. If there was something interesting he'd be up on the desk or couch whooping and hollering. (once I had an office next to his.. my number was 664 and the sign read 'neighbor to the beast'). The blackboard would fill with drawings and calculations .. all of us were old hands at this sort of thing. Everyone knew it was the idea under attack and not the person. If I tried this approach in a group I didn't know I'd probably make enemies in short order. We could move quickly. It turns out to be an astonishing fun and productive path to creative collaborative thinking ... I've even helped some companies with it. Friday afternoons were usually reserved for this - Friday from 1pm until 8 or 9 at night when we got too hungry.
Back to Norm .. his function was centrally important. He was the great synthesizer and cheerleader. I've never met anyone with such a powerful knack of synthesizing disparate ideas. He'd listen to a few things and say .. "let me see if I'm understanding this properly.." and then he'd launch into a beautiful description that was often much more interesting that what you had presented. He'd find inspiration for bringing in ideas from all over - 13th century Italy, a Norden bombsight and skip bombing from WWII, pure math, art, ... I began to see the value of bringing in ideas and people from far outside your field and to offer your services to others because they might be curious.
He helped me learn how to harness the power and joy of simple play.
This has stayed with me and helps define how I see the world. He probably had no idea he was a powerful mentor.
In two weeks time we would meet for a "ducking" to celebrate his birthday at the Peking Duck House in Chinatown. Norm would have been 75. The world seems too quiet, too sterile, too joyless tonight. I only have a few very close friends and that group is now smaller by one. One of the others suggested I howl at the moon in frustration. I may do just that ...
esad
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No recipe corner other than noting Norm didn't exactly have a healthy diet. Vegetables were "what meat eats..." Shrimp was a matrix for holding melted butter. In politics and food we were polar opposites. That didn't get in the way of a long friendship. So if you want to eat one last unhealthy thing this holiday season, Norm would probably applaud.
Instead this is the shard that managed to get us interested in each other's mind. Bail now if you don't like math. I'll just note it is semi-geekish.
I had only been with the Labs for two years when we first met. I heard he had a great PDE solver and I wanted to do some numerically solve Maxwell's equations in the near field (if that seems foreign don't worry. .. it's just trying to understand the behavior of light very close to the end of a hole much smaller than the light's wavelength. It's messy and people usually ignore it as unimportant). We were having such a fine time that one of us brought up Khinchin's Constant. It surprised me he knew it's history (it was a fairly recent result) and apparently it surprised him that I knew about it in the first place as I wasn't a mathematician. It's so sweet it needs to be mentioned.
If you take the geometric mean of the partial quotients of the simple continued fraction of a simple real number, the limit as n → ∞ the result is independent of the number (except for the set of measure 0) and is about 2.685452... It isn't known if it is irrational, let alone transcendental. It's enough to invoke wonder and that was enough to start a long friendship.
(here it is more formally - you can't do this easily in html, so I'll use something else and paste an image)
A ninth grader asked me what e was all about. She had seen it looking through her brother's calculus book. I took out my iPhone and opened a calculator (rpn of course;-) and rambled on more or less like this...
Imagine you're going to deposit a dollar in an extremely generous bank that gives 100% interest a year. At the end of the year you have your original dollar plus a dollar interest. Now imagine a slightly different rate ... 50% interest, but now every six months.. At the end of six months you have $1.50 and at the end of the year that increases another 50% giving you $2.25 ... not bad. Now you're interested so you inquire about the monthly rate. You see they offer 8-1/3% interest - a twelfth interest - compounded monthly. So at the end of the first month you have $(1 + 1/12) or $1.083.. One the second month it is (1 + 1/12)*(1+1/12) = (1 + 1/12)2 .. a bit over $1.17 .. You see the pattern. At year's end your total is $(1 +1/12)12 . $2.613 .. things are getting better.
They don't offer daily interest, but you figure it out anyway and add it to the little table you've been making
(1 + 1) = 2
(1 + 1/2)2 = 2.25
(1 + 1/12)12 = 2.613
(1 + 1/365) 365 = 2.7146
You're doing better, but the rate of increase is slowing down. What about every hour - ( 1 + 1/8760)8760 ? Your calculator shows $2.7181. A slight improvement. You could try every second, every microsecond, every femtosecond... but what if the bank compounded every instant?
What is (1 + 1/n)n if n is infinity?
Euler worked it out. It is the irrational constant e. He didn't name it after himself, but it's a convenient way to remember who did it . There are a variety ways to calculate it.. he showed 1 + 1/2 + 1/6 + 1/24 + ... + 1/n!He managed to work it out eighteen digits 2.71828182845904523536028...
Some of the important numbers are buried in geometry .. π is the ratio of the circumference of a circle to it's diameter, √2 is the hypotenuse of right triangle with side lengths of one. The Greeks loved geometry and found rich pickings. e, on the other hand, is not geometrical. Rather it is fundamentally associated with growth.
Try plotting y = ex .. Notice y =1 at x =0 and e at x =e . If you look a little deeper the gradient of the curve at e turns out to be e and the area under the curve up to e is, you guessed it, e. In fact for ex the gradient is ex and the area under the curve is ex. Remarkable - this is the only function with the value, gradient and area under the curve is the same at every point. This property makes it the natural language of calculus - the math of rate of change, areas, etc. So if you study anything involving area, or growth .. finance, physics, biology... e has a deep and fundamental importance. It makes the math much easier. If you force yourself to avoid it the math gets complicated in the most ugly way imaginable.
You can use it for geekspeak... A friend would indicate something is huge by saying "e to the something" brrr ... it's e to the cold outside. Or "Trump is e to the narcissistic"...
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Recipe Corner
How to roast potatoes. Some tricks. Chop them into golf ball or smaller sized chunks and parboil them in salted water for about ten minutes. Drain in a colander and shake them up to rough up the edges a little. Toss in some olive oil (I used about three tablespoons per kilogram of potatoes) and roast at 425°F or so for about 40 minutes turning a few times as you go. The real trick is the parboil and roughing step.
Here's a seasonal recipe with another trick
Ingredients
° about four pounds potatoes ( I like red potatoes . use what you like to roast)
° two oranges
° bunch of sage
° 6 tbl olive oil
° 8 cloves of garlic
Technique
º oven to 425, chop potatoes to golf ball size chunks
° parboil for 10 min in salted water, drain and rough up a bit
° peel long strips of orange peel (use a spread peeler!) and pick the sage leaves
° pour the oil on a large roasting tray over a couple of range elements or burners on low heat and add the sage and garlic and orange peel. Let them fry for about a half minute .. now add the potatoes and toss everything with tongs. Pop in the oven and broil for 40 to 50 minutes get them crip and golden.
Entering the exam room you're asked if you're carrying anything metal one last time even as you walk through a metal detector. You lie down on the exam table and try to find a comfortable position as the QALO ring wearing technician checks the machine's status and inserts your IV. Headphones go on for ear protection and this time you hope they're working well. Now you're given a panic button and the the table slides you into the magnet's bore. If all goes well you'll be out in thirty to forty minutes and there will be a rich three dimension image of a part of your body that needed a close look. Assuming you aren't a child or claustrophobic it all takes place without anesthesia.
Magnetic Resonance Imaging is relatively new making use of several branches of physics and a computationally easy version of the Fourier Transform mentioned a few posts ago. The basic technique although it gets into quantum mechanics, isn't too difficult to grasp if you think of a classical model.
Remember playing with tops and toy gyroscopes? Start one spinning and carefully place it on the floor. It will rotate around tracing out a cone around the axis of its spin. There's an angular momentum associated with the spin which, acting with the gravitational pull on the top, causes it to precess tracing out the shape of a cone. Don't worry about the details if you haven't encountered the physics, but just keep the image of the spinning toy tracing the cone.
Protons - the tiny positively charged particles in the nucleus of every atom, act like tiny magnets. They have a magnetic moment. It makes them act like the spinning top. Turn on a magnetic field and the magnetic moment will rotate around in that cone-like motion. The property was noticed before quantum mechanics was fully developed and is called spin. Although nothing is physically spinning, the model is useful for this discussion.
Consider the protons in a piece of your body. In the absence of a magnetic field their magnetic moments are scattered about randomly cancelling each other's contributions. You don't have a net magnetism. Turn on a strong magnet and something happens - they start to precess and line up. Quantum mechanics tells us a proton only has two possible orientations - pointing with the magnetic field and pointing against it. The configuration with the north pole of the proton pointing towards the south pole of the field is a lower energy state than the one pointed the other way. Add energy to the system and more magnetic moments will move into the high energy state.
The Earth's magnetic field is a feature probably necessary for life to have evolved. Depending on your location it's strength is between 0.3 and 0.6 gauss. A tesla is a more commonly used unit and is ten thousand gauss. Refrigerator magnets are a few hundred times stronger. The MRI machine you're getting into is usually 1.5 to 3.0 tesla or T, magnets in the accelerator ring at CERN produce 8T and you can levitate a frog (not the kindest thing in the world) in a 16T field. But 3.0T, the standard high end MRI in the US, is a serious and potentially dangerous magnet.1
The magnetic field is arranged to run along the axis of the bore of the magnet - from your head to your feet. The most proton rich element is hydrogen and we have a lot of it in our body - about sixty percent of you, ten percent of your mass, is hydrogen. Hydrogen atoms are simple - just a proton for a nucleus and an electron. Now consider the precession of the atom's magnetic moments. The frequency of precession - how fast that cone shape is traced - is directly proportional to the external magnetic field.2 Double the field and the frequency doubles. At 3T we have a lot of hydrogen protons precessing at 127.74 MHz. Most of the lined up fields in your body are pointing opposite the MRI's field, called Bo, with some pointing with it. Impressive but not very useful yet.
We'll see why in a moment, but the trick is to get the spins to synch up we can create a net field that has a component transverse to Bo. In other words pointing away from the axis that runs down your body. Next the energy of each of the protons is increased so more spins are in the high energy state - namely north pole to the MRI's north field rather than the other way around. Adding the energy into those spins has to be done without boiling you alive. For that it makes sense to think about music.
You've probably played with tuning forks. Imagine two that ring at middle C and another at the A just down from it. Bang one of the A tuning forks and the other A will start ringing. The C fork? ... crickets. Resonance is an efficient way to transfer vibrational energy. The same is true for the protons. Just "ring" them magnetically at the appropriate frequency and you can efficiently transfer enough energy to move some of the magnetic moments. If you do it right they'll all be in phase too ... all of those little vector arrows will be traveling around together. A transverse component to the magnetic field of your protons is starting to appear - order is coming to the system.
(excuse my crude drawing ... this works a lot better on a blackboard or with hands)
At some point half are pointing with Bo and half against. The components of the magnetic moments parallel Bo cancel, leaving only a transverse component rotating around your core at the precession frequency. Now there is a magnetic field pointing out of your body rotating around you at nearly 128 MHz. You're a rotating magnetic beacon. Now we have the resonance part of MRI.
If you move a magnet through a coil of wire an electric current flows through the wire. Receiving coils are placed in the MRI's bore to pick up the signal from your rotating magnetic field. It doesn't mean very much at this point, but a few more tricks and we get the imaging part of MRI.
Turn off the exciting field. The spins in the high energy state lose energy on average and move to the lower energy state and the transverse magnetic field decays. In fact it decays a different rate for different tissues - muscle is different from fat for example.3 The problem is it isn't localized. To remedy that there are an array of gradient coils. Their field is much weaker than Bo - usually a few percent of a Tesla - but they're still strong enough to change the net field along in the magnet changing the resonant frequency a bit. By sweeping the exciting frequency and moving the strength of the gradient coils you make smaller regions of the body transverse while the rest isn't contributing. You "ring" small regions of the body and then carefully listen before moving on to ring other regions. The sort of task a computer can handle.
This richly changing field appears in the from analog signal from the receiving coils. The frequency and signal strength information is transformed into position and intensity information with a Fourier Transform. Additionally the relaxation rates of the signals have a lot of rich information about tissue type - eg .. tumor or regular tissue. The individual chunks that were scanned become an array of volume pixels or voxels and we have our image. It's usually read out in slices along each of three axis of your body.
A couple of issues about the ride. It is classified as minimally invasive as it don't do the damage cutting does. It heats the tissue up a bit, but not enough to worry about. If the resonance part didn't work you'd have to be heated past the point where you boiled, so yay for resonance! It is loud - often painfully so. The gradient coils may only be a few hundredths of a T, but they have to be switched on and off quickly and the motion produces a loud banging noise. Somewhat louder than a jackhammer at very close range. Past the pain limit. Protective headphones vary - on my last ride they worked poorly. Something to ask about if you have a ride. Claustrophobia is a more issue for some. A normal full body bore machine is 60 centimeters in diameter - about two feet. The largest full bores are 70 cm - a bit under 28 inches. Can you live in a very loud tube like that? If not you're anesthetized - not recommended if you can help it. There are quieter and more open machines, but they tend to be have less powerful magnets that produce fuzzier images that may or may not be diagnostic.
Contrast agents, or " dyes", are injected at different parts in the procedure to change the relaxation time in tumors producing images with better contrast. Gadolinium is commonly used. In small doses it's safe although there are some side effects to consider (one is a chilling effect when injected. My experience has been that it's common to make a full series of images without the dye and then inject it by IV and make a second run.
The experience isn't fun, but fantastically better than the alternative. fMRIs are a different beast with their own tricks - perhaps another post. And a bit on the name. Technically it's really a form of nuclear resonance, but one of the first major companies to get involved was GE. They were sensitive to public reaction to the word nuclear and somehow MRI stuck. It's also accurate, but marketing and branding are powerful.
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1 The big MRI magnets have to be superconducting ... you can't build a large and strong enough permanent magnet and the heat produced by a non-superconducting magnet would render it useless and terribly expensive. MRIs are a major user of liquid helium .. it's precious stuff and the world supply is an issue.
A MRI magnet is strong enough that stray metal objects can fly into it at extremely high speed - lethal speeds. Oxygen tanks have flown in and killed so extreme caution is taken.
2 The precession frequency is called the Larmor frequency.. it's simple something called the gyromagnetic ratio times the magnetic field. Atoms have individual gyromagnetic ratios. In a 1 T field hydrogen is 42.58 MHz so a 3T magnet has them precessing at just under 128 MHz .. a bit above the FM band on your radio.
3 There's another relaxation time, but I'd spend a few pages talking about it... for now just understand that there are two relaxation rates that are part of the signal when the energizing field is turned down.
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un-recipe corner
Many of you are into holiday cooking so I'll refrain from adding to the pile today, but rather offer a video of Merlin's Swan. watchmaker+jeweler at the extreme
and something in the omg category... amber with a dinosaur feather! The linked paper is outside the journal's paywall and is quite readable for sciency types.
Well .. maybe a snowflake or two rather than a snowman.
Years ago C.L. Strong's Amateur Scientist column in Scientific American talked about homemade ice crystals. Homemade apparatus for carefully controlling temperature and humidity so you could study their growth. These pieces were great in that they had appeal to a wide audience. Remarkably just enough to suggest how you might build something sophisticated along with the physics or chemistry of what was going on. Enough to pique your curiosity to the point where you wanted to do something. You had to "lean forward." It has been said this column excited more young people about physics and chemistry than Sputnik and the space age.
The serious apparatus described was too expensive and challenging a fourteen year old, but it mentioned the right points - namely that you want to set up a temperature gradient between a warm damp region and a cold one. Place a pin or a string .. anything very thin, between them and, at the right temperature and humidity, ice crystals will form. Strong suggested a damp cloth positioned above a chunk of dry ice to get going.
That was enough for me. My father had a serious Albertson's ice cream addiction which required the regular purchase of chocolate chip or rocky road ice cream in the three gallon cylindrical containers. I rescued an empty, cut a slit down the side and created a tube about five inches in diameter .. the chunks of dry ice from the Talcott's were six inches on a side .. I had a vertical viewing port about two inches wide that was covered with a thin flexible plastic sheet (why I didn't make the cylinder out of that entirely is beyond me ... my style probably dictated an ice cream empty) A thin piece of fish line (any respectable house in Montana had a wide variety) ran vertically from a damp sponge towards the dry ice. A sinker at the bottom keep it pointing down. It took some fiddling around .. the airflow had to be very smooth .. but within an hour I had distinct types of crystals forming at different temperature locations along the line.1
I had been thinking about suggesting this as a neat family experiment and played around with something not very elegant a few years ago. I needed a better approach but had more or less given up on it when I caught a Science Friday interview of Ken Libbrecht. Ken probably knows more about ice crystals and snowflakes than anyone else on the planet and is something of a legend. When he suggested using a 2 liter softdrink bottle I had one of those slap the hand on the forehead moments. Of course - now it's simple.
If you can find dry ice, and that's not terribly difficult (I asked Siri in Manhattan and found a half dozen locations), you're almost set. It is beautiful to watch and you can go much deeper. Those of you so inclined can set up a camera (dark background with side illumination and some magnification) and make beautiful time-lapse movies. You can vary the conditions and see different types of crystals form. You'll develop a sense of how shape depends on temperature. And the basic physics is unsolved so far for those who want to really drill down. (hint.. you'll need to build a someone more sophisticated apparatus to ask that class of question properly). You can even see, rather dramatically, that warm flows to cold rather than the other way around (cold is really heat leaving a warmer object... so you don't get cold... you lose heat)
They even made a video which is much clearer than any written description I could provide.
Really fun for the family - way better than TV, although it's a bit inefficient for making snow to build snowmen or women.
__________ 1 One of the questions I had was how to accurately measure the temperature at any point. That turned out to not be easy - at least on my budget - as I didn't know about thermocouples. I tried to figure it out with math, but didn't have enough. That became one of the drivers that brought me to partial differential equations. But in the end ground truthing with accurate measurements is better.