omenti

on approaching problems

Over the weekend I had a chat with a student from twenty five years ago.  He's a computer science professor these days and, while catching up, he thanked me for the note I gave him on problem solving.  He wasn't the first to say the same thing.  

As a freshman I discovered George Pólya's notes on problem solving. Pretty basic stuff, but extremely useful in situations where rigor is expected.  I made a version for myself and updated it over the years.  While this version mostly applies to math and physics, it has some general themes that can be useful in many other areas.  Feel free to steal modify it to your needs if you like.

 

Understand the problem

First. You have to understand the problem.

What is the unknown? What are the data? What is the condition?

Is it possible to satisfy the condition? Is the condition sufficient to determine the unknown? Or is it insufficient? Or redundant? Or contradictory?

Draw a figure. Introduce suitable notation.

Separate the various parts of the condition. Can you write them down?

 

Devise a plan

Second. Find the connection between the data and the unknown. You may be obliged to consider auxiliary problems if an immediate connection cannot be found. You should obtain eventually a plan of the solution.

Have you seen it before? Or have you seen the same problem in a slightly different form?

Do you know a related problem? Do you know a theorem that could be useful?

Look at the unknown! And try to think of a familiar problem having the same or a similar unknown.

Here is a problem related to yours and solved before. Could you use it? Could you use its result? Could you use its method? Should you introduce some auxiliary element in order to make its use possible?

Could you restate the problem? Could you restate it still differently? Go back to definitions.

If you cannot solve the proposed problem try to solve first some related problem. Could you imagine a more accessible related problem? A more general problem? A more special problem? An analogous problem? Could you solve a part of the problem? Keep only a part of the condition, drop the other part; how far is the unknown then determined, how can it vary? Could you derive something useful from the data? Could you think of other data appropriate to determine the unknown? Could you change the unknown or data, or both if necessary, so that the new unknown and the new data are nearer to each other?

Did you use all the data? Did you use the whole condition? Have you taken into account all essential notions involved in the problem?

Consult with others!

Set it aside and work on something else if it is still intractable.  It may be that it has to sit in your mind for a long time.  It's surprising how often this happens with a path forward suddenly appearing in the shower or on a walk!

 

Carry out the plan

Third. Carry out your plan.

Carrying out your plan of the solution, check each step. Can you see clearly that the step is correct? Can you prove that it is correct?

Some problems are intractable!  Be prepared to move on after you've spent what you consider enough time.

 

Consider what you've done

Fourth. Examine the solution obtained.

Can you check the result? Can you check the argument?

Can you derive the solution differently? Can you see it at a glance?

Can you use the result, or the method, for some other problem?

 

 

 

rogue waves

 It's worthwhile just sitting on the beach sometimes.

 

I used to take weekend bike rides to the South Beach as part of a cycling club when I lived in Pasadena.  Mostly wasted, we had a variety of ways to spend a a couple of hours on the beach as we recovered enough for the trip home.   Some played frisbee, some played in the water, some read, I mostly watched.  There were so many wonderful questions!  What determines the frequency and height of the waves?  What determined where they "broke"?  Why can surfers move so quickly (it turns out to be the fin)?  Why do beach volleyballs sometimes corkscrew and suddenly drop?  How do the gulls "wave surf"? Suddenly fluid mechanics became interesting.   Mostly I've done other things since, but every now and again an interesting problem crops up and the techniques have been useful in areas that aren't obvious on first glance. 

 

Recently I've come across a number of papers on rogue waves. There are a couple of primary mechanisms that describe these beasts and an understanding is emerging.  There's also a good deal of experimental work. Some day soon we'll have coarse warnings (try not to use this patch of water for the next eight hours).  Localized predictions are much more challenging, but why work on something that isn't?

 

Steve Strogatz posted an excellent non-technical podcast chatting with Ton van Den Bremer - one of the leaders in the field.  Here's a link along with a transcript (I recommend the audio as there's content in their voices).

 

 

 

 

beyond the equal sign

 

 

 

About ten years ago I devoted a good deal of time to learning a bit of category theory - an area of abstract mathematics that's become important to physics.  It deals with relationships rather than objects, abstractions and analogies, context, sameness, equivalence rather than equality, how you see patterns, and much more.  It can impact thinking in many areas beyond mathematics. It provides a rigorous framework that gives choice in how problems are approached and take you beyond  putting things in boxes that don't always make sense.

 

It's very technical, but Eugenia Cheng has written an excellent non-major freshman level book with the goal of getting art students to think about the world differently.  I suspect it will work for many other people - particularly those who have to deal with complexity and seeing curious patterns that are otherwise missed.  The Joy of Abstraction is one of those rare books that can change how you see the world.  I'd recommend getting a physical version and doing the exercises. (I don't buy books from Amazon and recommend you support local bookstores)

 

Steve Strogatz chatted with her on this episode of his Joy of Wh(y) podcast.  Listen even if the book doesn't strike you as interesting!

 

 

latin

Ab his via sterniture ad maiora.

 

In about 1820 the local monarch ordered Carl Friedrich Gauss to carry out a survey of the Kingdom of Hanover.   The great mathematician toiled away on the survey for about four years.  While physically difficult and stressful, it's nature gave Gauss something interesting to think about.  

 

Halfway through he realized you could chop a flat map up into small triangles and get an approximation for the topography, but it could never be perfect. The Euclidian geometry we tend to think of where parallel lines never meet and the sum of the angles of a triangle is always 180° just doesn't work.¹  He set his mind to consider the geometry of curved spaces.  It undoubtedly seemed nutty to anyone but a mathematician at the time, but he pushed on and developed the Theorema Egregium - a masterpiece of thought.  At a high level it shows you can determine the curvature of a surface with just local measurements.  You don't need to consider the space the surface is embedded in. Among other things it tells you why we hold pizza the way we do and that the sum of the angles on a sphere is greater than 180° and changes on an egg and the border of Wyoming isn't a rectangle..   Here I link to a great explanation by the wonderful Cliff Stoll.

 

Gauss realized this was fundamental, but only a beginning.  At the beginning of his paper describing the discovery was this phrase:

 Ab his via sterniture ad maiora - 'From here the path to something more important is prepared'

Indeed it was. Thirty or so years later Bernhard Riemann generalized what Gauss had started.  Its beauty was appreciated and enhanced by a few mathematicians until Albert Einstein needed a tool to describe the curvature of spacetime.  And there it was - a beautiful, but fairly obscure, mathematical result from the mid 19^th century revolutionized physics 70 years later. Our Universe is not Euclidian .. that's only an illusion, albeit a very good approximation most of the time,  dictated by where we live and our senses.  Among mathematicians Gauss is a great  GOAT candidate.²

 

We didn't have Latin in high school and I probably wouldn't have taken it anyway.  Since then I've learned a little on my own, but am very rusty. Still, there are phrases that seem more useful than their English equivalents.  Encountering one you pause and think a bit.  This is one I use encouraging people working on long term goals where much of the day to day grind to major accomplishments seem like small steps on the way to the goal. It is a reminder to keep pushing ahead.

__________

¹ Think of lines of lines of latitude on the Earth.  They intersect the equator at right angles added to 180°.  The angle they subtend at the pole is greater than 0°, so the sum of all three is greater than 180°.

² The story of a school master telling the seven year old Gauss to add the integers from 1 to 100 to occupy him for an hour appears in a summation of his life shortly after his death.  Given the problem, he thought for awhile and answered 5050.  He wrote the integers 1 through 100, folded the series and added .. so 1+100, 2+99 and so on. Each pair sums to 101 and there are 50 such pairs so 50*101 = 5050.  He had shown the sum of integers from 1 to n = n(n + 1)/2.  

I don't know if it was apocryphal, but he was doing serious math a few years later.

 

 

six is a surprising number

Ting-Chang Hsien was a dear friend when I was in grad school.  At the time China was beginning to send a few scholars to work with American counterparts and Ting-Chang came to Stony Brook for a few years to work in an area of particle physics the school was noted for.  Near the end of his stay     he decided to see America.  He flew to my parent's home in Montana and, renting a car, spent several weeks traveling through the West.  The day before he flew back to New York my parent's neighbor, Lloyd Erickson, stopped by to chat.  Extroverts by nature, Lloyd and Ting-Chang dove into conversation and homemade ice cream.  Ting-Chang mentioned as a teenager he was a student in Shanghai with an interest in music as well as math.  The school was run by missionaries - his mother was a devout Christian - and his favorite teacher told him he had real talent for math and should give up the piano.  He said "she was from Minnesota .. one of her brothers was technical and worked for a company there."  Lloyd sat up,  "I had an aunt who was a missionary-teacher in China.. she left before the war and and got married."  Ting-Chang mentioned the name of his teacher.  She was in her late 80s and still alive.  Lloyd brought out family photos.  That night old teacher and the physicist talked on the telephone for a long time. 

 

We all know it's a small world.  We find common connections with total strangers even if we don't consider ourselves well connected.  I tend to be shy and don't have that many friends, but continue to be surprised.  Of course there's the Six Degrees of Kevin Bacon.

 

Steven Strogatz does applied math at Cornell.  In the mid 1990s he and his student Duncan Watts were interested in the problem of synchronization.  How is it that some insects like beetles manage to synchronize their sounds at night?  Why is it that types of glowworms blink in unison?  Why is it .. there are many of examples of this in nature ...  They weren't trying to solve the underlying science, but rather were looking to see if there was a deeper mathematical underpinning.  Cornell has lots of beetles in the Summer.  They could make measurements. 

 

They considered two extreme cases.  One is very regular.  Think of a checker board.  Each square on a checker board filled has an immediate group of neighbors - small neighborhood cluster.  To "talk" to square on the other side there are about eight degrees of separation.  Each square is only directly connected to the nearest neighbors.   This kind of model is well-studied in physics where a force may only have a very short range. None of us have connections that limited.    Next consider complete random connections. If you know 150 people, each of them would know a similar number so the total number of people connected by two degrees of separation is about 150² - 22,500.¹  Most of us know that's wrong (except for some early models of Internet connectivity:-) 

 

Their gut feeling was the real answer would le somewhere in between.  Technically they were looking for the phase transition .. the point at which most of the beetles were suddenly in synchronization, or everyone seems to have these seemingly random connections. The small world problem.  What they found was a fairly simple mathematical description that said you only need a small bit of randomness.  In your circle of friends an outlier or two.. or knowing someone with such connections.   

 

Their short paper showed examples ranging from power plant distributions to noise making bugs and is one of the one hundred most cited papers in all of science and math.   They also noted that given connections afforded by travel these days pandemics could spread rapidly.  It's very difficult preventing that phase transition.

 

The small world problem is very beautiful.  It's one of those topics that should be taught in a course that doesn't seem to exist.  In college you get music and art appreciation.  I'll never do good art or music, but 101 level non-major courses gave me a better appreciation for those very human endeavors.  

 

Math is important enough that students in high school get either working on mindless problems they'll never see again or pre-STEM work that only a small percentage will use.  People are properly recommending and even building 'data science' courses.²  These seem like a great idea - familiarity with tools, presentation and interpretation - but I'd like to see a math appreciation path.

 

Math appreciation would have history from a number of civilizations.  Was math the first example of external memory?  Why did people use different number systems?  Why was double-entry accounting so important to Italy?  What about zero?  Imaginary numbers?  The Fibonacci sequence in nature.  Why does a slice of pizza fold the way it does?  What is infinity, are there different kinds?  How are coffee mugs and doughnuts similar? What is the fairest way to vote in a democracy? It's easy to think of dozens of lecture topics that might even inspire a few students and create links to what they finally go into.

 

Just an idle dream I guess..

__________

¹ Apologies to Robin Dunbar - he'd say the number is almost always used out of context anyway).  

² I hate the term.  Generally anything that adds 'science' to a label isn't science.

 

finding what works

 

It’s a clear winter night as I write these last words. I’ve stepped out to look at the sky. With the stars up above and the blackness of space, I can’t avoid feeling awe.

How could we, Homo sapiens, an insignificant species on an insignificant planet adrift in a middleweight galaxy, have managed to predict how space and time would tremble after two black holes collided in the vastness of the universe a billion light-years away? We knew what that wave should sound like before it got here. And, courtesy of calculus, computers, and Einstein, we were right.

That gravitational wave was the faintest whisper ever heard. That soft little wave had been headed our way from before we were primates, before we were mammals, from a time in our microbial past. When it arrived that day in 2015, because we were listening—and because we knew calculus—we understood what the soft whisper meant.

                 Steve Strogatz 

 

Chatting with a friend the subject of gravity came up.  It isn't a conventional force, but we perceive it as one.  Rather it's connected with the curvature of a four-dimensional object called space-time that we happen to live in.  The fundamental concepts aren't intuitive and the math that describes it is beyond what you might get in an engineering degree.  So how to describe it?  There are very high level books and videos that speak in terms of balls on rubber sheets and then jump to oddities like clocks running faster on mountain tops and blackholes.  I usually find them a bit fluffy and disappointing.  They're bound by assumptions about their audience, so it's not a major criticism. It's just that I was talking with one person and something a bit deeper seemed appropriate.  Just how do you find the right level and where do you go from there?

 

A few years ago Wired produced the 5 Levels series.  An expert would try to explain something about a complex subject to five people:  a child, a teenager, an undergrad majoring in the same subject, a grad student and, finally, a colleague.   My favorite is Donna Strickland on lasers - give it a watch, she's excellent:

I went through something like this just before my Ph.D. thesis defense.  I was to explain my thesis work to a group of high school students. At first I thought it would be easy, but that notion quickly evaporated.  It turned out to be one of the most difficult and embarrassing things I've done.  My advisor stepped in near the end of the talk and summed everything up beautifully.  Afterwards he told me if you understand something deeply, you can explain the gist of it to a high school student.  That's high on the list of the most important things I've learned. 

 

My friend has a BA in biochemistry and knows about differential equations so I figured I should aim for something between the vague videos and what a senior level general relativity course offers.  The more I thought about it I realized it would be best to talk about the history and use a bit of calculus and geometry.  I had the advantage of knowing her well and she can stop and ask questions or give a blank look along the way.  And afterwards she was going to talk about something where she has serious expertise. 

 

Steve Strogatz is an applied math professor at Cornell. He's written several books including the one the opening quote is taken from: Infinite Powers: How Calculus Reveals the Secrets of the Universe.   It's an excellent read that makes no assumption on your mathematical background.¹  Thinking about it gave me some hints of how I might proceed.

 

First why do I have to use math?  Why won't words work?  The laws of nature obey logic that we can make predictions from.  Math, calculus in particular, is a logical calculating tool.  In a way it's a prothesis. Math lets us take a bit of logic, write down and perform logical manipulations that far exceed what we can do in our head, and then interpret the results. The logic can be crafted to represent something about the science.   Every once and awhile the results can make predictions that can lead to new discoveries, but more often they're used to to solve an enormous range of problems.   And why calculus?  Much of the  underlying structure of nature has been successfully expressed with the corner of calculus known as differential equations.  "Why?' is a deeper question.  If one encountered an intelligent alien who understood some aspects of how the Universe worked, I suspect they'd use math.  I suspect, but it's only my suspicion, that it's deeper than an artifact of how we think about things. 

 

I spent a couple of hours writing.  How general relativity came about, a bit on the structure of space-time, what goes into the main equation and how a simple prediction could be calculated.  Then about an hour of chatting that left me with two delightful philosophical questions that will probably lead to more discussions as well as her turn to teach me something. 

 

Whatever your expertise, it can be fun to try and explain the gist and maybe even the beauty of it (those can be he same thing) to someone with a very different background.   If you're like me you'll fail at first, but eventually you'll get to a point where you can find the right grounding, the right words and perhaps the right drawings or even music (some of you are artists and musicians).

__________

¹ By all accounts he's a wonderful teacher and has become a popularizer of math.  Among other things he's created a college course at Cornell for people who think they're afraid of math and have generally put it off until their senior year. 

 

 

 

 

teaching and technological change

On Saturday my niece sent a few photos from her son's robotics competition.  While standardized components and a simple scripting language are used, there's a lot of room for learning and creativity.  Winning teams spend much of their time modifying the basic design along with moving to more sophisticated programming languages.  The basic hardware is quite expensive so many of the groups are sponsored by their schools and sometimes local businesses.  It seems like a great way for kids who like to build things and play computer games to learn a bit of engineering and programming.  My niece's son isn't interested in math or science at school, but loves to build things.  It's a path that has him learning.  Given the right advisor there could be a lot of learning.  It's also a nice example of using technology as a teaching vehicle.

 

Sometimes new technologies are seen as a way to cheat.  I've read that some teachers worry that language models like GPT-3 will give students easy access to ghost-written essays.  Of course there's still the problem of students buying essays from human, but this may be cheaper and easier to get.  One can imagine a number of ways to deal with this.  A teacher might ask something like "based on the debate Adam and Sally were having last Tuesday, how would you.."   Indexicality - pointing to something in the context in which it occurs - is something these language models can't address.  Better still may be to focus on critical thinking in the class and test on something that isn't a simple regurgitation. Essays seem like a very uncreative way to teach writing and critical thinking skills.   And there will probably be teachers who figure out how to involve language models as a tool to teach creative thinking.  That would be a leap similar to using robotics competitions to teach shop, basic engineering and programming skills.

 

I'd like to see changes in how science and math are taught in high school.  Currently there's a push to eliminate areas most students will rarely use.  I'd counter that by teaching the subjects in a way to improve critical thinking.  Pure math may be a way forward, not the "new math" that crashed and burned in the 60s, but big ideas.  It doesn't have to be about formal proofs and calculations (although tools like Mathematica and Maple can be useful experimentation and visualization tools).  I've probably belabored it before, but simple topology, infinity, the continuum, maps, abstraction, inference and model building. None of this has to be taught with rigor, but playing with them can open pathways students have never thought about.   Sure teach some of the regular math, but it doesn't have to be so repetitive. I'd add probability and risk analysis as well as personal finance. The same with science - do away with memorization and focus on the concepts and how we arrived at them.  Of course extra points for making it playful like the robotics competition.

 

But I'm not a k-12 educator so all of this may be silly.

 

And for college here's something by the wonderful Woodie Flowers of MIT on changes he'd made to university engineering curriculum and beyond.  (Woodie was one of the most innovative teachers at MIT)

 

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.