omenti

couplings and friction

a quick minipost

 

Beginning physics students learn about simple coupled systems.  Things like pendulums with a weak spring between them and the like.  Next you look at somewhat more complex systems like a group of pendulums.  Put them on a solid surface and start them in different oscillations and they do their pendulum thing unaffected by their neighbors.  Now add a little communication by putting them on a floor that can transmit a bit of vibration, something interesting happens.

 

The coupling of systems is common in nature covering a very wide range of complexity inducing our brain.  Many of these systems are mathematically fascinating.  Turn up the amount of coupling and they behave as you might expect - at least to a point.  Then, as the comments of the system impose themselves on each other,  there's a dramatic change.  Sometimes these are good from our point of view and sometimes they're disasters.  The point of inflection is often marked by a perturbation - lightning strikes, hurricanes, pandemics, big lies.   The couplings are too strong to allow for anything but a change to a new state.

 

One feature of our society is an increased coupling of systems.  Social media and other targeted messaging, GPS, the Internet of Things, globalization, just in time everything, religion, democracy....  More systems are being connected and the amount of communication is increasing at a terrific rate. These couplings are often introduced to decrease cost and the "friction" parts of the system have with each other.   Many are  too complex to understand well, but the basic math of over coupled systems is clear - these systems can be very unstable. 

 

Building alternate pathways and redundancy into these systems is an answer, but that costs money and clever design.  Another alternative is reducing the coupling of certain important subsystems.  It's something I find myself paying more careful attention to.  Highly coupled or low-friction systems are usually brittle at some point.  I dislike the term "black swan" as real black swans (the bird) are not uncommon in nature.  But we continue to build serious and increasingly effective black swam bait...

 

 

 

things aren't always what they seem

I was asked why I didn't do a Pi day post. Sadly I didn't have any pie to celebrate, but  I'll try and make up for it a bit late.

 

π is a wonderful number. Deeply related to so much as well as being irrational and transcendental. But it turns out I’ve never needed more than five or six digits. Randall Munroe (xkcd) nails it:

 

 

 

My sophomore quantum mechanics professor liked to add a few math problems to homework assignments. Nothing deep, but unusual to encourage critical thinking. On the 14th of march he gave a pop quiz. We had twenty minutes to calculate the probability that the ratio of two random numbers x and y, where each is between 0 and 1, is even.

 

It turned out this was his Pi day fun. (he was Swedish where Pi day would be the 22nd of July, but he used the American date convention). He didn’t grade it - either you saw it immediately or 20 minutes wasn’t long enough.  He served apple pie afterwards. Apple, of course, for Newton although I’ve though cherry more appropriate as Einstein loved cherry pie and the 14th of March was his birthday.

 

Here's how I approached it:

 

 

I’m summing the areas for regions for ratios closest to even integers for all even integers by measuring triangles. The area for the region closest to 0 is 1/4, 2 is (1/3 - 1/5), 4 is (1/7 - 1/9) ... there’s a clear pattern. The pattern is almost the expansion of the arctangent of 1 which happens to be π/4.   Adding 1 gets us there.   The probability the ratio is even is 1/4 + 1 - π/4 or (5 - π)/4.  

 

A more important point is things aren't always what they seem.  Not only does π turn up, but our intuition about the answer was wrong.  The component for 0 isn't symmetrical and that leads to a 46.46% percent probability rather than 50%.    

 

Obvious first guess are often wrong and can lead to enormous problems.  We're bad at probability and risk analysis.  Not exactly a good thing given technology.

 

bananas, bagels, pizza and gauss

minipost

I watched by the sidelines as a few people managed got in a verbal tangle about what makes proper pizza.  As the religious wars raged I remembered pizza illustrates a deep feature of mathematics - Gaussian curvature and the Theorema Egregium (remarkable theorem). It turns out you can get a good feel of it with a ball, banana, bagel and a slice of pizza.  

The always enthusiastic Cliff Stoll demonstrates:

 

Math, properly taught, is beautiful and not that hard ...  And go out and get some pizza.  I won't judge.

 

a shot echoed in the dark

   Our guide, the Director of Development for The Cleveland Orchestra, noted how difficult it was to find matching marble for the lavatories.   There were only two of us and we wanted to set up in the main hall, but impressions must be made and little was being spared in the big three year remodeling job. 

 

Cleveland's Severance Hall is a remarkable place.  A odd mash-up of Art Deco and Egyptian Revival architecture, it was finished during the Great Depression and became home to The Cleveland Orchestra.  Like many symphony halls the acoustics were bad boarding on terrible.  Remodeling a few decades later partly fixed the problem, but in addition to rendering the pipe organ unusable,  was architecturally "wrong."  The work underway when we visited would address all of those issues and make the space one of the best concert halls in the world. 

 

Our company was doing quite a bit with digital music at the time.  There was a relationship with the Rock and Roll Hall of Fame and another with the Oberlin Conservatory.  Two of us who were doing much of the Oberlin work developed a relationship with The Cleveland Orchestra and Telarc Records on the side.¹  At the same time work on sound field reproduction was a serious research topic.

 

The idea behind sound field reproduction sounds simple enough. If you've ever been around live music - or even in a recording studio - you've probably noticed your home equipment can't begin to reproduce the acoustic experience.  The problem stems from the fact that the wavelengths of the sounds we hear range over three orders of magnitude and much of it is similar in size to musicians, audience members, seats, curtains, the room itself, etc etc.  These sounds are reflected, refracted and absorbed.  Moving anything causes a change.  It's a difficult problem that the music industry and Hollywood would love to see solved.  Plus it's an interesting problem!

 

The approach we were using had a microphone array on a tripod.  One up, one down and an odd number between them in the same plane as the floor.  Severance was undergoing acoustic tuning and we were going to set up the array to record some music to compare it with other techniques.

 

We were left alone in the open hall. It was very quiet so I did the obvious thing and clapped my hands once.  There was a lovely echo coming from everywhere at slightly different times.   If we recorded an even better "clap" - an impulse function - we could mathematically play with it and create a signature of the room.²  Armed with this we could have some fun.  We needed to create a good enough impulse.

 

Severance Hall is on the grounds of Case Western Reserve University - a lovely area with good restaurants and a lot of art and music.  (if you ever stay in the area get a room in Glidden House - fantastic little hotel).  I figured the school probably had an athletic department and quickly found a track coach.  I was able to talk him into lending me a starter's pistol.

 

Later that night in the darkened hall a shot rang out - only to be captured digitally.  Then a few more just to make sure there was enough.

 

Back at the Labs we had a special acoustically isolated room - an acoustically neutral room (these are rare and not cheap!)   Using the same microphone array that was used in Severance Hall we recorded some live music with musicians in the area.  The music was convolved with Severance Hall's signature and then mixed down to six channels for playback in the room.  

 

It was fairly convincing..  close your eyes and it really felt like the place the shot rang out in the great hall.   Of course you can do better, but it was a great excuse to have a bit of fun.

 

__________

¹ Over the years I’d spent some time worrying about image and sound quality. At one point I thought better was - well - better. My epiphany came in the a perfectly restored 66 red Mustang. It was near Cleveland with the head recording engineer from Telarc Records and a seriously good musician from the Oberlin Conservatory. We were talking about where music reproduction might go with sound field reconstruction and dramatically more information than CDs could provide when a favorite Beatles tune came up on the 8 track (gasp! - it *was* an authentically restoration). The volume came up and the two of them were singing along to the music in pure delight. There is something transcendent about the music - even with awful reproduction in that noisy environment. Portable music players with cheap headphones would be good enough for most. As long as there's a personal touchstone.

²  take its Fourier Transform.

diving into boiling stones

The air you breathe is about 21% oxygen and 78% nitrogen by volume.  People suffering from serious Covid-19 need much higher oxygen concentrations to hopefully buy some time.  Hospitals need to keep local stocks of liquid oxygen or high pressure oxygen tanks filled to provide treatment.   Patients can be treated at home with a portable oxygen supply and lower the burden on the hospital.  Often this means oxygen concentrators.

 

Around 1800 a mineralogist noticed heating stilbite produced steam. The class of materials was called zeolites from the Greek for "to boil" is zeo and lite is a mineralogy shorthand for lithos or stone.  There are a few hundred such materials each with a similar internal structure or framework.  The framework creates tiny pores - pores that can absorb substances and even be used as molecular sieves.  Wikipedia shows a structure for a common zeolite called mordenite.  It's based on a silicate (SiO₄) building block - I show it as this framework is common to zeolites and there's a nice public domain image.

 

 

Now back to oxygen concentrators.  Rather than getting bogged down in their operation just note regular air is passed through a molecular sieve made out of a different zeolite that traps nitrogen and allows oxygen to pass.  Relatively inexpensive machines for home use can produce about 10 liters per minute of enriched air that is 80 to 90 percent pure oxygen.   These are regularly used by people who need oxygen therapy and have kept a lot of people out of the hospital during the pandemic.

 

A wonderful invention, but there's something extremely beautiful about zeolite.

 

Chemists and physicists have their own way of looking at its repeating framework - the one called the secondary building unit.  Mathematicians have their own interesting way.  To get there a bit of a digression is warranted. 

 

Take a deck of cards.  Assuming it's well shuffled (say a dozen shuffles or so) I guarantee the order of the cards has appeared only once and will never appear again.  The reason comes from the fact there are an enormous number of possible combinations of cards: 52!  (52*51*50 ...  *3*2*1) or something over 8 *10⁶⁷ unique orderings!  One way of thinking about it a mathematician might consider each possibility a point in a 52 dimension space.  It's a bit abstract as we are only used to three physical dimensions, so here's a way to think about these configuration spaces.

 

A mathematician looks at a cube and says "that's a three dimension cube..."  A square is a 2 dimension cube, a line is a 1 dimension cube and a point is a cube of 0 dimensions.  She can go in the other direction too..  4 dimension cubes and up.   OK - now back to the cards.  Take three cards labeled 1, 2 and 3.  There are six (3! = 6) possible orderings:    {(1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), (3,2,1)}  Make yourself a 3d grid with an x, y and z axis and plot the six points.²  Now connect them up - pretend you have a rubber band to snap around them.³ 

 

Sweet!  You get a hexagon.  Shine a light on it at the right angle and it will cast a hexagon shadow.  The combinations of the three cards are linked to a shape!  

 

Now take four cards labeled 1,2,3 and 4.  There are 4! = 24 possible ways to order them.  Plotting them on a 4 dimension grid doesn't work for our spatially limited brains, but if you do the same trick of connecting them up you get a 3 dimensioned gem-like shape .. the exact shape of zeolite's framework (the secondary building unit).  If we lived in a place with four physical dimensions the shape of that object would cast a three dimensioned shadow..  

 

Thinking about the structure of an important molecular sieve with four playing cards..  math can be a curious playground.

 

__________

¹ The repeating unit is fairly complex: (Ca, Na₂, K₂)Al₂Si₁₀O₂₄·7H₂O

² you can also do it with two 2d grids..  say x,y and z,y and see the pattern  

³ the convex hull

medical tests and probabilities

a mini post

 

There are many references in the news to how good viruses tests are and what conclusions you can draw from them.  Unfortunately some are wrong and I haven't seen any easy to understand explanations of how to figure these things out.  It's all conditional probability. The standard approach is to use Bayes' Theorem.  Unfortunately most people have never heard of it or vaguely recall a formula with P's and |'s they once memorized without knowing what it really did  

 

Here's an easy way to think about it in the form of an example.  Imagine a population, say a city, where 1% of the people have a viral disease (I'll call the disease 'virus').  There's a test with a 90% chance of returning a correct result.  So if you have the disease there's a 90% chance the test returns a positive, and if you don't have the test there's a 90% chance the test returns a negative.   So what's the probability that someone who has just been tested has the disease?

 

For simplicity imagine an unbiased group of 1000 people in the city.  Let's screen them and see what happens. Ten people will have the disease, 990 will be disease-free.Of the ten people with the disease the test will correctly identify nine of them and incorrectly identify one.   Looking at the people who do not have the disease 90% of 990 or 891 will correctly test negative, but 10% of the 990 or 99 will incorrectly test positive.

 

 

 

 

So the probability a person who tests positive actually has the disease is the number who test positive and have the disease divided by the total number who test positive:  9/108 or about about an 8% chance.

The example has nothing to do with the specifics of CV-19, but maybe will give you a way to think about how to sort out what the real tests are saying when an accuracy is published.   Going deeper really requires tools like Bayes' theorem, but hopefully this is clear.

 

Armed with this technique perhaps this otherwise well-written piece in the NY Times will make sense.  Also note this is for an unbiased initial sample.  If you're screening on existing conditions before testing (which is common outside of Iceland), additional complications appear. But at least this should give you an idea.

 

 

 

 

a dive into pizza pt.1

Sarah and I were talking and the subject of pizza came up.  It turns out she's a big fan - particularly of Hawaiian pizza.  Rather than getting mired in the cultural war of pizza types, let's go for the basics - the mathematics of pizza.  You'll only need a piece of paper, a ball (Sarah recommends a volleyball), and a marker. 

 

Take a piece of paper and lay it flat on something.  It's flat with zero curvature everywhere.  if you draw a triangle on it and sum the angles, you get 180°.  A circle will have a circumference of 2π times the radius - 2πr - and an area of πr². All is well. Now fold it along an axis - say the long one if it's a rectangular piece.  You've introduced some curvature.   Now grab it by a corner and let it flop. Again there's curvature, but notice which one is stronger and save that thought.

 

Here's where the math comes in.  Exploring what curvature means, Carl Friedrich Gauss wondered if there was anything intrinsic about the curvature of a surface.  The sheet of paper - what is the relation between its flat shape and when you fold it?  His approach was to consider the paths you can take along the surface.    Imagine you're tiny and walking on the sheet of paper.  You can take any path you want.  When it's folded state there are a lot of paths that are concave (he called these negative curvature) and others, along the axis of the fold, with no curvature.  Gauss defined the curvature at any point on the surface as the product of the two most extreme curvatures running through that point.  In this case the red one that runs along the axis has zero curvature and the most curved green path has some negative value.  The total curvature for that point is zero times a negative number or zero.  The beautiful part is the curvature for all points on the paper are zero if it's flat or folded in such a way that doesn't stretch or tear it.  

 

Now try a ball.  The surface is convex everywhere and the curvature is always positive in any direction.  The Gaussian curvature for any point - the product of the curvature of the two extreme paths - is always a positive number times positive number or positive. (for now just consider the sign of the curvature - the actual measure is a bit beyond this post).  This is why you can't perfectly wrap a sheet of paper - or pizza - around a ball.  The paper always has zero curvature and the ball is always positive.   This also explains why can't make an accurate flat map of the Earth and have to rely on projections with distortions.

 

Many surfaces are more complicated. For fun explore points on bagel or bananas, but for now back to the pizza.  If you let it flop it still has zero curvature, but no strength.  Putting a fold down an axis gives it rigidity.  You find this throughout nature and engineering. An empty Coke can should hold your weight if you put it on the ground upright and carefully stand on it.   Lay it on its side and will crush flat under your weight. The fold in a surface of zero curvature gives a lot of rigidity in one direction.   Next go somewhere that is easy to clean and try to break an egg by squeezing it in your hand.  The positive curvature everywhere makes it surprisingly strong.  If the shell has an imperfection or if you tap it with the edge of a knife or fork while squeezing, you'll have a mess. The positive curvature was distorted and the surface failed.

 

Assuming you didn't make a mess, go back to the ball.  Draw an equator and then mark a North pole. Drawing an arc at a right angle to the equator, you'll always intersect the North pole - it's a line of longitude.  Next pick another point on the equator and make another arc at a right angle.  You've made a spherical triangle. Sum up the angles.  If you started with the points close together the sum will be just over 180° and if you go all the way around it will be nearly 540°. Our old rules of trig don't work on this non-flat surface.¹  A neat thing you can do is measure the shape of the Universe in a way that is similar to making these trigonometric measurements.  So far it looks extremely flat and that's a deep result about the structure and evolution of the Universe.

 

So get some pizza to celebrate the Remarkable Theorem its explanation of why we hold pizza the way we do. 

__________

¹ If you draw and measure some circles on the sphere you'll notice the circumference is less than 2πr and the area is less than πr².  If you try it on a surface with negative curvature - a saddle shape or the inner part of a bagel for example - the circumference is greater than 2πr and the area is greater than πr².

 

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.

__________

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

 

 

is the drug test fair and the prosecutor's fallacy

a minipost

Some sports have doping issues and there is drug testing in many of these.  There's always an issue of how good the tests are, but there's also an issue of how fair they are and how good they have to be to be fair.   The best way to illustrate this is with an example.

 

Consider a sport with 1000 players - roughly the size of major league baseball in the US .  Suppose steroid abuse is an issue and there's a test with 95% accuracy.  And assume somehow it's known (or strongly suspected) that about 5% of the players are abusers.  These are artificial numbers, but it's just an illustration.

 

What if the players are tested?  We expect 50 to be abusers and 950 to be clean.  The test is 95% accurate.. so 48 (rounding up from 47.5) are correctly identified.  So far so good.  Now consider the 950 clean players..  5% or 48 (rounding up again) are incorrectly labeled as abusers.  So of 1000 players there would be 96 positives and only a fifty-fifty chance of being right. Careers can be be ruined ..  the stakes are high.

 

A way to sort out these conditional probabilities was described the The Reverend Thomas Bayes in the mid 1700s in a paper called Essay Towards Solving a Problem in the Doctrine of Chances.  The good Reverend described the relationship between the probability an athlete is an abuser given a failed test and the probability an abuser fails the test.   Consider these probabilities (apologies for notation)

P(A) = probability a drug test is positive

P(B) = probability a player is an abuser

P(A|B) =  probability an abusing player tests positive

P(B|A) = probability a player has taken drugs if their test is positive

Perhaps the most important takeaway is that P(A|B) and P(B|A) are not the same!  It's a very common prosecutor trick to fool the jury into thinking they are - so common that it's called the prosecutor's fallacy.  But to the problem at hand.

 

We want to know P(B|A).  We already know P(B) or at least we have the 5% estimate.  So P(B) = 0.05.  This also gives us  P(not B)  = (1 - 0.05) = 0.95 as P(B) and P(not B) must add to 1.0. We also have 47.5 (I'll go with the fractional athlete here) of 1000 drug free athletes tested positive, so P(A|not B) = 0.0475.

 

Bayes derived a formula showing how all of these relate.  Sticking with our notation:

 P(B|A) = {P(A|B) *  P(B)} / {P(A/B) * P(B) + P(A|not B) * P(not B)}

                         = {0.95 * 0.05} / {0.95 * 0.05 + 0.0475 * 0.95}

                         = 0.513

This is a very unacceptable test!  Although 95% accuracy sounds good on the surface, it only finds a bit better than half the abusers and incorrectly labels a lot of players.  For fun try a 99% probability and a perfect 100%  to see what's happening. 

 

You can have much more complex conditionals, but this is the general idea.  You really have to think things through a bit..

 

 

it's switches all the way down

The other day I saw something that a few of you might want to consider for a kid or even yourself - the Turing Tumble Mechanical Computer.  

 

It turns out digital computers are mostly switches.  A lot of switches these days - perhaps a trillion in your smartphone. As a teenager I built a very simple relay based digital computer based on a column in an old Scientific American. Later I built an updated version that substituted transistors for the relays using some (free) components from the high school physics teacher.  

 

Playing with these beasts is an excellent introduction to understanding how logic circuits work.  You'd be surprised how many computer programmers are hazy on the subject (of course they work at a level of abstraction far above the silicon - or what old timers called the iron).  The early machines of the 30s were very important and set the stage for the dramatic developments that followed.  Here's an outstanding non-technical short history of those early days. (recommended!)

 

But back to the Turing Tumble.. It's limited, but that's central to its beauty.  The instruction book has about sixty examples to play with and an aha! just might strike leading to out-of-the-book insights.  

It's also great to play around with extremely simple silicon digital computers - particularly those that you can hook up to sensors, lights and other interfaces to the real world.   But getting down to the basics is usually a paper or YouTube exercise or, if you want to do it yourself,  a frustrating experience involving a lot of soldering and a non-trivial amount of money.   The Turning Tumble might be just the ticket.  It is play and that is the best way I know of to learn. I didn't have much of a chance to play with it and can't do a thorough review, but I'll probably end up buying one to give to a kid.  The caveat is that you have to spend some time with it to learn.  If the user doesn't find it fascinating enough to devote time to there won't be any learning.  But the same can be said about so many things.

 

The education link on their site has the following in addition to a set of resources:

 

Appropriate Age Range

Turing Tumble is for ages 8 to adult - and that is not a stretch! We find that kids 8-12 are able to get through the first 20-30 puzzles. Adults get addicted by puzzle 27, and their minds are blown by puzzle 35. Younger kids enjoy the first 10 and building their own computers.