omenti

order! - a minipost

Apparently a math puzzler became something of a theme on the Internet..  a half dozen people asked me what I thought the answer was

8÷2(2+2)=?

Some came up with 1, others with 16.  Apparently there were near-religious arguments. A mathematician would tell you both are incorrect - the question posed is ambiguous and not an answerable bit of math ..  at least not generally.

 

The problem is the order in which one evaluates expressions.  It turns out the one you learned in grade school is a bit different from the one you may have had when you learned algebra. If you remembered grade school you probably get 16.. algebra people are more likely to get 1.¹  If your algebra teacher was more careful they would have said the expression is wrong - you need to add parenthesis to make it unambiguous. The exception would be if the only people you communicate with agree on representation. That can be a big if..

 

Taking it a bit further, the simple rules you learn in secondary education don't cover don't cover unary ±, which implicitly have lower precedence than exponents on one side and higher on the other!  For example

-x³ → -(x³)
x^-3 → x(^-3)

This is so well agreed upon that no one uses parenthesis in this case.  Historically juxtaposition was treated as implicit multiplication, but that led to ugliness like

1/3x →  1/3*x →  (1/3)*x
a^2x →  a^2*x →  (a²)*x

uck! 

People decided juxtaposition of numeric literals is like a unary operator ..  so the 2 in 2x is attached to the x like the - in  -x.  

 

Back to the original problem. The fact that people had different interpretations and mentioned rules from school meant a better rule was required.  The problem needs to be unambiguous to anyone who might encounter it.  This is an issue for computer languages - different languages can parse the same written expression differently and come to different results.  Agreement on standards is an issue too ..  the Mars Climate Orbiter was lost as  teams at JPL and Lockheed Martin had different takes on a single critical bit of data - one used metric units, the other english..   when the retrorockets on the spacecraft were fired to go into Martian orbit the spacecraft burned up in the Martian atmosphere - it was literally lost in translation.  And this was code that passed multiple very careful reviews - code written for space probes is some of the carefully crafted software period.  Just wait for complex coordination systems and things that move with people nearby or inside.

 

That said there are some areas were common agreement among practitioners creates a local set of standards..  Einstein summation notation in physics is confusing to outsiders, but all physicists know it. And then there's my favorite - the double factorial.  In math (n!)! is not n!! ...  (note that I also have the problem of people interpreting ! as an exclamation point:-)

 

This local agreement is probably true in most fields - even descriptions of an action in a sport are local to the sport.  We all use local shorthand - it's a form of jargon, but we need to know when we can use it and when we need to be more general even it it's awkward.

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¹ In one case 8÷2(2+2) → (8÷2)*(2+2) = 4*4 =16

   in the other 8÷2(2+2) → 8÷(2*(2+2)) = 8÷(8) = 1

 

 

an ecology posed by a children's song and multidisciplinary problems

Last week I read Range: Why Generalists Triumph in a Specialized World by David Epstein.  David writes well  I've recommended The Sports Gene as another excellent read.   In Range he talks about real world messy problems that often poorly defined and open-ended.  The really interesting ones usually require a range of backgrounds.  Experts often do worse than generalists.  If you have a group of people, you want a range of expertise and backgrounds.   

 

You've probably heard of Fermi calculations.  Enrico Fermi liked to frame problems in such a way that you might be able to approximate an answer using common sense and a few bits of information.  Things like "how much water flows through the Mississippi River in a year?"  or "how many piano tuners work in New York City?"  They're extremely common in physics and are used to think about how good or bad a conjecture is.  The type of question you can get a handle on in your head or on the blackboard.

 

There are many other approaches to problems. The school where I did my undergrad work has a course that teaches multidisciplinary thinking.  Although it's a physics class, it really isn't about physics - or at least it doesn't have to be.  People get together a couple of times a week for a few hours and a problem is posed to talk about.  Students come from a variety of backgrounds.  What is common is serious motivation and curiosity.  There aren't midterms or finals, but there is a project each student proposes and works on.  They're given some funding and access to other class members, professors and grad students.  And each week they're immersed in the two hour sessions each featuring something that is best approached as a cross-disciplinary problem.  They learn to sort out the various strengths and weaknesses of the other students and the professor.

 

Getting into the course is difficult. About a hundred apply and maybe six are accepted  You have to pass a couple hurdles - problems you spend about four weeks each on showing you can bring a range of thought to the table.  You must show they're capable of cross-disciplinary thinking.¹  A hurdle problem I'm familiar with was  posed as a line from a children's song:²

Mares eat oats, and does eat oaks, and little lambs eat ivy.

The question is:

Considered in an ecological sense, we have three species each of plant and animal life that interact in a finite geographical area. Supposing that we start out this system with roughly equal numbers of the six species, determine the way the system changes with time.

You have four weeks to come up with something clever and compelling.  Then on to the next hurdle.  The class has had extraordinary success over the years and is something of a legend.,

 

Very few people have a great range of background and expertise.  At least in the US we seem to value those who are broad less than those with a very narrow deep expertise.  One of my favorite thinkers was a mathematician some of you knew - Norm. He would sit and listen to ideas coming from a variety of backgrounds.  Later .. a few minutes to a week or two .. he'd stop you and say "Let me see if I understand what you were saying"  Then he'd proceed to give a much deeper and better thought-out version of what everyone had been trying to say, adding his own bit in the process.  Not always, but often enough history or art might be part of the framing.   These sessions were marked by deep play as well as chocolate coated coffee beans.  Play is the glue that allows communication across disciplines .. even if it's in your own head.

 

My own range isn't that great, but Norm and others - friends who have very different backgrounds - have changed the way I think.   I have this habit of "tithing" my time to the projects of friends.  I'm particularly interesting in areas outside of physics and math as I end up learning something and find myself in Norm's shoes at times.   This broader approach of posing and approaching questions strikes me as richer than the traditional expert deep dive as you can always add expertise where you have it when it's useful.

 

I've had a bit of serious insight on a very difficult problem that changed how I saw the world.  Insights came from a forestry program, a couple of sociologists and a professional volleyball player who has a degree in microbiology.  There are dozens of other problems where outside insight has combined with my own contribution to be a bit out of the box. And who would have thought thinking about bats would be just the right analogy for what turned out to be a neurology problem I didn't realize I needed to know about...

 

I don't know if I manage to help others.  If I do it's probably by offering a bit of thinking that might seem novel as their background is different.  At the same time I'm often lost as an outsider in their world until I start working in it for awhile.  I probably don't get very good at it, but it's the diversity that might be useful.

 

I'm very much against the hyper-specialization in STEM focused education we seem to be moving towards.  There's enough change in the world that the humanities, liberal arts and just practical skills are increasingly necessary. There has been some work on teaching cross disciplinary thinking. That strikes me as having great potential. 

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¹ As these are freshmen when they work on the hurdles it's interesting to look at their backgrounds. It's a bit of a problem as so few have taken the course, but it's been around for decades.   I won't go into that here, but it's very different from those of kids who focus on standardized tests and assigned activities.  A really interesting question is how much of the ability to acquire cross disciplinary thinking is wired in and how much is learned. I have suspicions, but that's beyond this post.

² Not to be confused with the song "Mairzy Doats and Dozy Doats" from the late 40s:-)  It's sort of a homonymic phrase. 

 

 

heat by chance

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We still have a bit of snow on the ground - slushy stuff that is partly translucent.  I tried forming someone of it into a sphere.  With clear enough ice and some polishing with your gloves you can make a spherical lens good enough to start a fire.  (from several years ago)  I didn't get there today, but a neighbor asked what I was doing and how could I stand all of that cold going to my hands.  I said it wasn't bad, but that's not what's going on.  Cold isn't moving from the snow to my hands - rather heat is moving from my hands to the snow.  There was a puzzled look.  

It turns out to be easy to understand what's going on and that discovery was part of a revolution taking place in physics at the end of the 19^th century that led directly to quantum mechanics.   It's not some fundamental law, but rather just chance .. nothing deeper than that.

Boltzman figured it out.  You can think of heat in terms of molecules vibrating in a solid or shooting around in a gas.  The more energy they have, the greater this agitation.  Consider a tank of air with a divider.  Make one side hotter than the other.  It's air molecules are zipping around faster than those on the other side.  Remove the divider and the hot and cold gases mix.  (you could do this with liquids or solids - same idea)

It is more probable that a faster moving molecule collides with a less energetic molecule than the other way around.  In the process it leaves a bit of its energy.  Energy is conserved and given enough of these collisions the extra energy in from the faster molecules tends to get equally distributed and everything eventually comes to the same temperature.

At the time it was crazy to think that something as how heat flows might be governed by chance.  It would take years to understand his result.  In the process a very deep understanding of heat would emerge and prepare people for a radical rethinking of what the microworld was. 

In the meantime enjoy the fact that you're heating up a bit of the cool March air when you go outside.

 

 

 

a most remarkable instrument

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Let your spectrum analyzer listen to  middle C on a piano and take a look at what comes out. (guess what?  your smartphone can do it)   Most of the acoustic energy, assuming the piano is tuned to a 440 A is concentrated at a tad below 262 hertz (cycles per second), but other things are going on.  Bursts of energy at multiples of the 262 Hz fundamental  frequency paint the spectrum along with a few other bursts.  The integer multiples of the fundamental frequency are called partial harmonics and the others are inharmonic partials.  Together they form a large part of why a piano sounds like a piano - in fact why a specific piano sounds like that particular instrument - it's timbre.

The time development of a note can be very complex with the distribution of energy in the various harmonics changing as the note decays.   A really nice way to study this is with Fourier analysis.  I gave a crude non-mathematical introduction earlier, but I just came across another beautiful illustration. 

Enter Anna-Maria Hefele .  She's  a remarkable singer who has mastered the art of overtone singing.  Thanks to Richard Feynman's life-long fascination with Tuva you're probably aware of Tuvan throat singing.   Her singing is much richer.  Let's start with her voice as an instrument in this lovely version of H.I.F. Biber's Rosary Sonata 1.

 

and now a graphical look into the flexibility she has ..  simply wonderful!  Such a fine way to show off the overtones.

the olympics: spinning through the air

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In figure skating you to convert  some of the kinetic energy from your speed across the ice into air time during which you spin.  You launch yourself skyward using the teeth on front of your blade like the base of a lever (way more inefficient than pole vaulting).  Once it leaves the ice your center of mass follows a simple mostly parabolic trajectory defined by gravity.  You can't change that path, but you have another trick up your sleeves.

To start a spin you need to generate a lot of torque.  This is done by generating large forces in opposite directions against the ice.  (read powerful leg muscles!)   Now the trick.  Physics has several conservation laws.   We'll make use of conservation of angular momentum.  It's the product of the rate of spin and the moment of inertia.  The moment of inertia is basically a measure of how mass is distributed around an axis of rotation.   The further out the mass from the axis of rotation, the higher the moment of inertia.  You can't change your mass, but pull your arms in and your moment of inertia decreases and your rate of spin increases in response. 

I watched about a dozen world class skaters carefully frame by frame doing the same quadruple jump.  The longest flight time I saw was 0.66 seconds - most of the others between 0.63 and 0.65 seconds.  They were spinning a bit over nine times a second during the core part of the flight, but much slower just after takeoff and before landing.  

The 0.66 seconds of airtime resulted from increasing the height of his center of mass by about 53 centimeters.  It is difficult estimating the force of his impact when he hit the ice, but it was probably about eight or nine times his body weight -- all on one skate.

Clearly there's a lot of skill and training.  Training informed by kinesiology - a mix of anatomy, physics and sport,. It's interesting to think what body type is favored and what someone who could easily land quintuple jumps would be like physically.

The physics requirements demand a light and very thin body to optimize range of control over his moment of inertia.  Very powerful legs to get the highest possible speed and liftoff for maximum flight time as well as generating an enormous amount of torque against the ice.  The arms must be strong enough to move in quickly and there is some advantage having longer arms.. eg.. the skater's wingspan is likely to be longer than his height.   I would be surprised if his height is much more than 170 centimeters and his mass greater than about 60 kilograms.  

 

One might dream of what could be done give more air time.  Perhaps an event on the Moon with a sixth the gravity and flight times approaching four seconds.

But wait -  there's a sport for that.  Freestyle skiing.  Similar physics in that you're playing with your moment of inertia, now concentrating on all three axes rather than one, as your body moves through it's Newtonian parabola. 

 

 

the olympics: why is ice slippery?

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One of the questions on my Ph.D. physics prelim was this:

what is a ruby and why is it red? why is the friction so low on a sled?  on a plot of p verses t, plot out the phases of helium-3

We had twenty minutes to impress the readers with whatever depth we could must.  The middle question was a trick to see how foolish we were. 

 

Many smooth surfaces aren't slippery.  Ice can be slippery even if it isn't very smooth.  The normal explanations are the weight of the skater puts pressure on the ice lowering it's melting point and/or the friction of motion heats the surface melting a thin layer.  The problem is even if you use a heavy skater - say 120kg on small skates - you only get a liquid surface down to about -3.5°C.  The friction is only good for a couple of degrees for the fastest speed skaters. I've (stupidly) skated in -30°C weather.  What's up?

A century and a half ago Faraday hytophesized ice forms a thin liquid surface.  His conjectures as to why and his experiments weren't good enough.  Now it is known ice has a very thin layer of liquid water - about four or five molecules thick near freezing and down to two at -35°C where the skating begins to get tough.  The pressure and friction effects are real and contribute, but only on warm ice.

It took an exotic experimental technique to verify and measure the effect and physics behind the pheomea is on the deep side, but we have something.  And that's why, when you stick two ice cubes together and put them back in the freezer they freeze into a single body.

Those ice dancers, hockey players, speed and figure skaters - they're all gliding on a film four atoms thick.

How cool is that?

 (speed skating often warms the air above the ice, but that's for the same reason they do it for indoor cycling - wind resistance)

 

what films influenced your direction?

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A friend and I were discussing films that had a major impact on people even influencing our path in life.  She mentioned Contact - the film adaptation of Carl Sagan's book.  I've run into more than a few astronomers and astrophysicists that cite the original television series Cosmos. For me it was mostly books - even comic books and a planetarium.  

She suggested getting your comments.  It can be any kind of inspiration and any media type - including public talks.  What films or books had a big impact on your life?  Comment below or send me an email if you prefer.   I'll add mine, but will keep it short for this mini-post.

Oh ..  a friend was the science advisor for Interstellar.  Here's a very accessible talk he gave on the science behind it.  About an hour, but Kip's a good story teller and you'll find out how Carl Sagan setting him up on a semi-failed blind date turned out to be important. 

ps - One of you makes films for a living.  She probably has something interesting to say. 

 

 

 

prisencolinensinainciusol - and charlie chaplin

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I’ve been meaning to do something on protowords.  Rather than  talking about neurology here are a two short and beautiful video examples..

 

silly nonsense words that your mind tries to turn into English:

Charlie Chaplin doing something similar is nonsense  romance languages.  Of course he communicates anyway.

 

 

syzgny!

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You probably won’t notice the supermoon status of the moon for the upcoming lunar eclipse..  The orbit is elliptical so the Moon’s apparent size changes by about 14%  (the Sun changes by about 3%) .. unless you’re on very familiar terms with the Moon, you’re unlikely to notice the difference.

But the eclipse…  now that’s worth watching!  Syzygy is the wonderful term used to describe an alignment of three or more bodies.    In the case of a lunar eclipse the Earth is between the Sun and the moon.  The eclipses are long due to the size of the Earth compared to the moon and the moonlight reflected back to us has to make it’s way through the ring of atmosphere surrounding the Earth. For the same reason the sky is blue most of the light that makes it to the Moon is on the red end of the spectrum.  Local stuff in the air around the Earth as well as the air over you can  change the nature of the colors.  They’re basically reddish, but can be copper colored, rust, brownish and even brick red.  The colors can shift through the eclipse and sometimes it can even be difficult to see the moon in mid eclipse.

The naked eye is just fine, but if you have a pair of binoculars it’s even better.  If you’re interested in looking at the night sky, binoculars are usually a better bet than a telescope.  I’ll be happy to give recommendations of what to look for.

 

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after posting this a question arrived..   here it is and my answer

 

Q   Why should I get excited about the lunar eclipse?

A   You can’t overstate how exciting the Universe is!  Check this out:  there’s this huge rock orbiting around a point a thousand miles below the Earth’s surface that the Earth orbits around too.  A giant rock!  And we can see it!  And then we get exactly between it and this huge fusion reactor that can burn for ten billion years and gives us life even though it’s power density is about the same as a lizard’s.  How cool is that?