omenti

the natural language of growth

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 = e^x ..  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 e^x the gradient is e^x and the area under the curve is e^x.   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. 

 

 

some things are central

Last night I found myself reading a paper on a futuristic propulsion system.  It's been around for several years and manages to excite the tech beat websites every time.   I'll spare you details, but it makes a rather dramatic claim.  Extraordinary claims require extraordinary evidence.  This wasn't it.   It was peer reviewed, but I doubt by practicing physicists.  The reported result was very small and several sources of error that could have easily swamped the result were not investigated.  A standard technique known by any undergrad physics student was not used and there was evidence of cherry picking data.  This doesn't mean an effect wasn't there, but it is likely they didn't really see it if it was.   Their explanation of the effect was a bit on the cranky side, although you have to give them a pass on this part if everything else is in order.   People want there to be magic, but Nature has her own playbook.

 

Without invoking magic of some kind they violate the conservation of momentum ... like sitting in your car, pushing on the steering wheel and having the car move down the road. Sort of wrong at the fundamental level.   Conservation of momentum basically states the linear momentum of a system stays constant if the system isn't subjected to external forces.  This is a really big deal.

 

Aristotle  said to keep something moving you, or something, has to keep pushing.  Push a plate of pumpkin pie across a table and it comes to a stop unless you keep pushing it.  The same thing happens even with a hockey puck on the ice.  It goes a good distance, but eventually comes to a stop  It was directly observable and everyone bought it.  Well -- almost everyone.  Galileo came along and said wait... what if you imagine doing this in a perfect environment.  One without air resistance or surface friction.  In that case the plate of pie would just continue moving along in a straight line forever.  It went against common intuition, but turned out to be right.  More time goes by and Newton offers Newtonian physics.  At it's core it's very simple.  You do simple calculations and get the gist of the physics and then add in the effects of friction and air resistance.    You reduce the problem down to its core and later bring back complexity.  What we've found since then is this is how Nature works.  

Galileo did more than a few brilliant things.  The telescope was fine and fundamentally changed how we saw ourselves in the universe, but this notion of reducing a problem to an ideal situation and then understanding the rules obeyed by the individual bits in a similar way we bootstrap into understanding complex real-world systems.  Modern physics was born.  I consider this the most radical and important thought in the last thousand years ... perhaps even longer.¹  All of science and technology is based on it. 

 

One more thing...

 

The conservation of momentum has huge implications for how the Nature works - something a lot of people still have problems with.  Aristotle talked about the natural state of things.  The plate of pie was naturally at rest.  Since some items moved there had to be a mover.   Cause and effect ..  he said everything happens for a reason. He bootstrapped back to the notion of a prime mover or first cause.  But the conversation of momentum, and all of physics that follows, says the opposite.  Things move because they are.. no reason is necessary.  The relation between events is not one of cause and effect.  

 

We still talk about causes and effects all the time and there are on the surface ... but reality is heavily layered. If you follow the thread all the way down .. in the case of the moving plate of pie, or the Earth orbiting the Sun it is easy, but if you keep going to the deepest levels and get to basics: spacetime, equations of motion, quantum fields and so on .. you find no causes lurking about.

 

Perhaps the most mysterious thing about Nature is that we can observe and work out how to understand it.

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¹ This is superhigh level.  The notion of conservation of momentum slowly emerged in dribs and drabs for hundreds of years prior to Galileo's brilliant insight.  He was an undeniable genius of the first order, but no one - not even the best - study Nature in isolation.

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No recipe corner as many of us are currently over-indulging.  Instead here's something about a neat number.  I don't know if you have a favorite integer.  Some friends tend to go for 17, 42, 47, 73 (paired with 37) and 137 for curious reasons.   My taste is more catholic .. I like numbers in general.  Here's something neat about 17 I realized a long time ago...  

1/17 is the first reciprocal of a positive integer whose periodic decimal expansion contains all 10 digits. (1/17 = 0.0588235294117647 0588235294117647...) 

 42, of course, with a tip of the hat to Douglas Adams, but it is also the integer closest angle at which red light is bent into its position in a rainbow by a water droplet ..  it is the rainbow angle.

folklore has it that 47 is the most likely integer given when someone is asked to name an integer between 1 and 100.

73 and 37...  there is a rich connection which was picked up on the Big Bang Theory

 

a determined bug and it's useful if mysterious result

Let's say you find a determined bug - a bug with the singular ambition of walking in one direction.  It covers no more and no less than one centimeter every second.  You also have an unusual band a meter in circumference.  You make a mark on it and start the bug at the mark.  At the end of the first second you stretch the band so it adds another meter to its circumference, and so on...  does the bug ever make it all the way around?  

 

Most people guess that it can't - after all, the band is growing by a meter for every centimeter the bug travels.  But let's analyze what's going on.  The first insight is the band is expanding uniformly so it's expanding behind as well as in front of the bug.  In the first second the bug travels  one centimeter divided by one meter of the way around ..  one percent.  At the end of the second second it adds a centimeter, but the total distance is now two meters..  it has traveled an additional half percent of the total circumference,  at the third second it has added another third of a percent, then a fourth and so on...      If we add up the percentages we get a sum:

 

1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + ... + 1/n    basically the sum of 1/n for all integers from n = 1 to infinity if the bug walks forever...  

 

If you've had some physics or math, this is the harmonic series related to overtones of musical notes.  Now on to the sum...  does it converge to some number or is it unbounded running off to infinity?

 

Let's see if we can bound the series from above and below with series that are easier to calculate.  Here's a really clever way to answer the question first proposed by the 13^th century monk Nicolas Oresme.  He made series where each element was less than or equal to each element in the harmonic series. By asking what is the largest power of one half that is less than or equal to the number you're looking at.    

 

Consider the first term:  the largest power of 1/2 that is less than or equal to 1 is (1/2)⁰ so the first term of the series is 1

the largest power of 1/2 that is less than or equal to 1/2  is (1/2)¹ or 1/2 so the second term is 1/2

the largest power of 1/2 that is less than or equal to 1/3 is (1/2)² or 1/4 ..  the third term is 1/4

the larger power of 1/2 less than or equal to 1/4 is (1/2)² or 1/4 .. the fourth term is 1/4

the largest power of 1/2 less than or equal to 1/5 is (1/2)³ or 1/8 .. the fifth term is 1/8

the largest power of 1/2 less than or equal to 1/6 is (1/2)³ or 1/8 .. our sixth term is 1/8

  work out a few more for yourself and a pattern emerges

1 + 1/2 + 1/4 + 1/4 + 1/8 + 1/8 + 1/8 + 1/8 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + ...

gathering the 1/2s

1 + 1/2 + (1/4 + 1/4) + (1/8 + 1/8 + 1/8 + 1/8) + ...  = 1 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 ...

1 + an infinite number of 1/2s equals infinity...

 

Every term in this series is less than or equal to each corresponding term of the harmonic series.  Since it diverges, the larger harmonic series has to diverge.  Given enough time our singularly determined bug only has to go 100 percent of the way around which is less than infinity so it eventually completes the trip.  Working out how long it takes is a bit more involved, but a long time - about e¹⁰⁰ seconds or about 10⁵⁰ years..

 

This slow growth is logarithmic ..   Using that as a hunch take the harmonic series to some number n and then subtract then natural log(n) 

1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + ... + 1/n - ln(n) ..    or the [sum(1/n)] from 1 to n - ln(n)

as n goes to infinity we have an infinite number subtracted from an infinite number.   The amazing thing is it converges to a rather small number.  It's known as the Euler–Mascheroni constant and has been calculated to over 120 billion decimal digits .. the first few are: 0.5772156649...

 

It's called γ and turns up all over the place in physics and math.  It's in quantum electrodynamics and the standard model - calculate corrections to the mass of the Higgs or the electron and there it is.  e^γ is used to explore products of prime numbers.  It's not as famous as e, i, π, 1, 0, or ∞ .. but is one of the small list of important numbers used in our equations.   Very little is known about it ... it is suspected to be transcendental, but that hasn't been proven. It isn't known if it is irrational..  There are other important numbers - fundamental constants for example, and the deep mystery of why math works so well in describing Nature.  Way too deep for the blog, but perhaps something to frame up at some point.

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Recipe Corner

 

Thai Carrot Soup

 

Ingredients

 

° tbl vegetable oil

° 1 medium onion roughly chopped

° 1 garlic clove chopped

° 1 stalk lemongrass stalk chopped

° 1 piece of ginger grated ..  use your taste.. I did about an inch

° 1 pound of carrots peeled and chopped

° 1 six oz can of whole coconut milk

° 2-1/2 cups vegetable stock

° 1 lime zested and juiced

 

Technique

 

° heat the oil in a pan and cook garlic, onion, lemongrass and ginger for 4 or 5 minutes on medium high heat.  Add carrots and go another 5

° reserve a few tbl of coconut milk and add the rest to the carrot mix.  Simmer until carrots are soft  

° throw in a blender or use a blender stick (I like the later) 'til smooth

° stir in lime juice and zest and then serve..  

 

° drizzle with the left over coconut milk

 

how much is enough? - homer simpson and olympic swimming events

It turns out The Simpsons has little math and science jokes making their way into the show - usually as very short sequences that fans can stop and study.  In a favorite episode Homer scribbles on the blackboard leaving four items.¹    3987¹² + 4365¹² - 4472¹² immediately caught my eye ..  it smelled like they were hinting at Fermat's Last Theorem.

 

There are an infinite number of equations of the form x² + y² = z² ..  like 3² + 4² = 5². Fermat hunted in vain for x³ + y³ = z³ and later claimed xⁿ + yⁿ = zⁿ  has no solution for whole numbers n > 2.  In the margin of a book he stated there are no whole number solutions for the infinite number of equations of that form and then added Cuius rei demonstrationem mirabilem sane detexi, hanc marginis exiguitas non ca- peret - 'I have discovered a truly marvelous proof of this, which this margin is too narrow to contain'.    Nasty.    Frustrated mathematicians toiled for hundreds of years until it was finally proved in a very non-trivial fashion in the mid 1990s .. indeed it wouldn't fit in the margin of a book:)

 

I pulled out my old HP calculator and tapped in the numbers.  Sure enough it worked.  It appeared to be a counterexample .. enough to disprove the proposition.  You suspect a near miss .. something lost as a rounding error.  The next step was to go beyond the ten digit accuracy of the calculator.  I reached for the laptop and fired up Mathematica

the first suggests the first two terms equal the third, but the second shows they aren't in the twelfth digit ..  looking at the first fifteen digits gives a bit more.  For most purposes it isn't necessary to have that many digits of accuracy .. but there are times when the question requires a bit of deeper knowledge - in this case the precision of a calculator.   But swimming pools?

 

There was a bit of a kerfuffle in Rio trying to explain ties in swimming events.  Tightly spaced finishes - often to within a few hundredths of a second are not uncommon and every once and awhile you get two and even three way ties (Phelps, Le Clos, and Cseh in the 100 meter Butterfly).   Identical right down to the hundredth of a second.  That can't be right can it?  Someone must have touched first.  The media mostly butchered the explanation.  Going to thousandths of a second like speed skating and a few other events - seems to be the way to go, but closer examination turns up a flaw.  

 

It turns out swimming pools aren't rigid.  They can't be or they'd break given what they're made of.  The ground around them shifts as does the water.  Even a degree change of water temperature causes a change of size.   In a 50 meter pool the time of the fastest event has a swimmer moving at about 2.39 meters per second.  In a thousandth of a second - a millisecond - the swimmer would travel about 2.4 millimeters.  It turns out the allowable tolerance of the length of a lane is 3 centimeters - something that has to be re-established several times a day during an important event. The pool can vary by more than ten times the distance the swimmer travels in a millisecond ..  timing at that level becomes meaningless.  Records couldn't be compared with each other at that precision and even the lengths of lanes in the pool can't be guaranteed. ² 

 

Which brings up another problem.  How do you have a fair start?  A starting gun goes off and runners or swimmers start out.  The bang of a gun travels at the speed of sound - you probably know the rule of thumb of counting seconds between the flash of lightning and its thunder and multiplying by one thousand feet.  The speed of sound varies with temperature, pressure and humidity, with the standardized value of about 343 meters per second.  Sound travels a about meter in three milliseconds.  Imagine a starting gun at the side of a track.  The path from the gun to a runner can vary by up to fifteen meters in some situations - that's 45 milliseconds .. over four hundredths of a second and a big deal in some of the shorter events.  

 

You might get around that by flashing a light, but the problem is we're wired in such a way that we respond to sound faster than to light - and by a lot ... like 50 milliseconds.  The solution is to put a speaker under each starting block and connect it to an electronic gun.  But another interesting question comes up.  How do you judge a misstart?  That one works out nicely.  Elite runners have a reaction to the starting gun of about 145 milliseconds and remarkably similar to each other.  A misstart is judged at something shorter than what reaction time would allow .. usually 100 milliseconds.   Some runners have very slow starts - reaction time and just getting the muscles going .. Usain Bolt is one of the slowest, so go figure... 

 

And for fun consider a problem I've given to students...  we all know refrigeration is inefficient and that opening the door of a refrigerator is a way to heat a room.  What would it take to detect the effect of a refrigerator in an average house?  What would it take to detect the impact of a single refrigerator on the power grid?  The first case isn't too difficult .. the second case demands some very specialized knowledge.

 

A thread connects these examples.  If you want to make an accurate measurement you need to understand something about what you're measuring and how you're measuring it.   The measuring tool, what you're measuring, its environment and even how they interact over time.  There are limits and when you bump into them it is useful to realize something is amiss so you know if you need to fix it or not - or if the situation needs a complete rethinking.

 

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¹ I suspect this was aimed at physicists as the two physics jokes are a bit deeper than the math jokes.    two math and two physics.  The first has terms you'd see in particle physics - and suggest they predict the mass of the Higgs boson (this was fifteen years before it was confirmed).  The terms may look normal, but the equation don't make sense.  If you plug in numbers it does give about 775 GeV which isn't terribly far from the real figure of 125 GeV. (admission - I did just that immediately)  The third, Ω(t0) > 1,  suggests the universe will implode ..  when things in the room start rushing in on each other Homer replaces the > with a <  which causes a major explosions.  The fourth is a topology joke.  A circle and a triangle are the same, technically homeomorphic, in topology.  Consider a circle made of a perfectly pliable rubber.  You could easily stretch it into a square, pentagon and almost anything else without holes (that's a bit of a stretch, but you get the idea:-)  An object with a hole in it, a doughnut for example, can be stretched into a coffee cup - one hole that goes all the way through is the distinguishing characteristic.  A doughnut is not homeomorphic with a sphere.  Homer suggests that taking a bite out of a doughnut leaves it a doughnut, but you're removed the hole and destroyed its doughtnutness in the process.

For those who love math there are other gems scattered through the shows.

In one a screen comes up in the Springfield stadium:

Tonight's Attendance:

a. 8128

b. 8208

c. 8191

d. no way to tell 

(a perfect number, a narcissistic number and an Mersenne prime)

at one point the spine of one of Lisa's books reads e^iπ + 1 = 0

great stuff... to see this on tv is astonishing.

Futurama has some cultural and writing overlap and tends to leave techie trailings..  

 

² There are other problems with swimming that get a bit complex and pool dependent, but suffice it to say some lanes are better than others and the faster swimmers usually are given the best. 

 

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Recipe Corner

 

Grilled Sweet Potatoes

 

Ingredients

 

° 1 tbl cumin seeds

° about 2 pounds or a kilo of sweet potatoes cut into inchish wedges

° 2 tbl olive oil 

° 1 tbl maple syrup

° salt and pepper

dip

° 1/4 cup tahini

° 1 tbl maple syrup

° 1 tbl lemon juice

° 2 tbl water

° fresh cilantro

 

Technique

 

° get the grill to a good medium high heat

° toast cumin seeds to light brown in a small pan while the grill is heating. Then move them to a chopping board and crush them until fine

° toss the sweet potatoes and everything else up to dip

° make a half dozen aluminum foil packets with two large pieces of foil each (cut it off roughly square) brush the inside with a bit of olive oil

° lay the wedges in a line in each packet and fold tightly

° grill till tender .. about 15 minutes for me, but my grill isn't great.  check in a packet and look for the beginnings of charring.  Flip the packets a couple of times during the process

° remove and carefully open .. let them cool.

° whisk the dip ingredients and drizzle..

° garnish with cilantro 

 

 

 

like buying a parachute from a bird

About twenty years ago Mattel rolled out Teen Talk Barbie - a talking Barbie that had 270 phrases.  One of them turned out to be controversial and the doll disappeared from the shelves to be replaced by one with 269:

The remake was warranted, but sadly many Americans agreed with what she was saying and helped instill the sentiment in their children.  Americans tend to believe math is hard and that you need special gifts to do it well.  They also see it as little more than a tool.  An exceedingly unfriendly tool.  

 

A few friends mentioned Andrew Hacker's book on math education was wrong to the point of being clueless.  I had come across a few of his newspaper op-eds that were enough to keep me away, but last week Evelyn Lamb, a math postdoc at the U of Utah, published a great commentary in Slate.  

 

I've only taught in universities and can't claim to be a professional teacher.  Some of you are and have a much better sense of challenges and direction, so consider what I say nothing more than opinion from someone who went deep on the science and math track years ago.  It may seen odd, but I worry that STEM education is gets too much attention and, in its current form, is poorly tuned to the needs of most students and society.  I've commented on my belief that those who will go on into technical areas are already well served.¹   Science  and math courses need to be taught so as to engage and even inspire more of the students.  We need to move from the pre-professional track that excludes most students to something that engages most students.  We can't have people nodding their heads thinking Barbie was onto something.

 

Math is in particular need of re-work.   There is too much repetition, memorization and mechanical effort.  Math has a beauty of its own - a beauty that can be taught at many levels.  Kids can worry about puzzles, knots, various types of infinity, probability, risk analysis, and dozens of other areas without touching on trains traveling from Cleveland at 47mph.  Interdisciplinary work is obvious - probability fits in with sports, biology, history, science and many other subjects.  Here's an example that might appeal to young bike riders - the math could easily be worked out by high school students.  The fact it took a real mathematician to work on it is commentary on how many interesting questions there are rather than the problem being difficult.

 

Beyond the useful there is the beauty of the subject.  It really needs to be considered on a level with art and music and not just a tool.  There are simple ways that can bring flashes of insight ..  You may have seen the construction attributed to Archimedes that seems to indicate the circumference of a circle is 4 rather than π.  Of course this is wrong, but understanding why and gets you into a class of beautiful objects.  I remember being enthralled with them when I discovered them in an article by Martin Gardner in Scientific American.  Here Vi Hart encourages students to dip their toes in the subject:

 

 

Unfortunately a good deal of the STEM curriculum was written by technical types who work towards the preprofessional pipelines.  One is reminded of buying parachutes designed by birds - maybe for those who will grind it through to become experts, but everyone else gets killed along the way.   We've seen this type of panic over job relevant skills before.²  The notion of relevant STEM is fine, but it needs to engage and inspire most students on one hand and isn't watered down like Hacker suggests on the other.   Students need to realize there are many dots to connect and having played with dot connection a bit first seems right and math is a big fundamental dot that has connections, both obvious and subtle, nearly everywhere.

 

Evelyn, the young mathematician who took on Andrew Hacker, came to her passion as an undergraduate and is fascinated by the mathematics of sewing (among other things).  It turns out to be deep .. she notes:

When you get down to it, sewing is applied geometry. You are using flat pieces of fabric to approximate the curvature of a complicated surface. Seamstresses and other sewists don’t get much credit for their research in applied geometry, perhaps because traditionally "feminine” activities are not assumed to be very mathematical. Of course, those of us who sew or make crocheted hyperbolic planes or Klein quartic curves know better!

Barbie indeed

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¹ There is a good deal of underemployment in many of the sciences - far too many Ph.D.s, often brilliant, for each position.  

² So many stories here.  One of the funniest was a group of universities trying to mandate computer literacy by mandating APL classes for the entire freshman class.  This was in the mid 70s and fortunately crashed and burned before the end of the first semester.  A friend who happens to be one of the great computer scientists of the world has taught introductory CS courses and notes kids who had a high school background often come with bad habits that are difficult to unlearn.  The kids who do best are those who had a bit of logic and probability in high school. Also music students did very well.

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Recipe corner

 

Spring has finally arrived!  Here's a quick linquine

 

Spring Asparagus Linguine

 

Ingredients

 

° 12 oz linguine 

° 1/3 c olive oil

° 1/4 cup of nuts - pine nuts are traditional, I used pecan bits

° 4 garlic cloves sliced

° 2 pounds of asparagus, trimmed and cut into inchish pieces

° salt and pepper

° 2-3 oz shaved parmesan (omit if vegan)

 

Technique

 

° cook the pasta according to directions on the box, drain and return to pot

° heat the olive oil in a skillet over medium heat. Add the nuts and garlic and cook for a minute or two - don't overcook.  Add the asparagus and cook 'til tender .. 2 or 3 minutes. 

° add the asparagus+nuts to the linguine with some salt and pepper and toss.  Sprinkle with the cheese if you like and serve

 

 

Technique

egg illusions

Even though I'm not very good at it I find art a very important part of the playful part of math and science.  I've bemoaned the narrow-minded focus on STEM in schools enough, so here's a delightful example.

 

Before film it was discovered we have a persistence of vision - the mind stitches together smooth motion by combing something like 25 still images per second.¹  In the early 1800s the zoetrope was invented.  A drum with image frames painted on the inside would be spun.  It had a series of slits around the periphery that you would look through to watch the short animation.  These became toys of the Victorian era leading to more sophisticated design.  

 

Stitching moving still frames into an illusion of motion led to something of the inverse.   A blurry vibrating guitar string viewed using a bright light that flashes with a regular frequency appears to stop.  The stroboscope has a wide range of application - you've probably used one in a high school science class and electronic versions that only display the repeating parts of a signal are fundamental to electrical engineering.

 

You can write on the surface of a spinning object and by picking the right frequency of a flashing strobe light motion will appear.  Now what if you could do this on a complex surface?

 

The EggBot is a delightful little piece of tech.  A computer controlled printer that draws on spherical and roughly spherical shapes. Balls and, well, even eggs.  Jiri Zemanek at the Czech Technical University in Prague generates interesting patterns in Matlab which was used to drive his Eggbot.  Then he spins the eggs and adjusts the frequency of a strobe light.  Art, math, science and engineering and a bit of time for your imagination to take flight.

 

__________

¹ At least most of us think persistence of vision is what allows us to view motion in film.  It turns out to be more complicated and a combination of persistence of vision and the phi phenomenon.  Persistence of vision is why you don't see the black spaces that come before each frame in a film.  The phi phenomenon is the apparent motion of a sequence of still images.  

 

23 1407

A few years ago I was walking through the parking lot of JPL in Pasadena when a license plate caught my eye.  It took me a few seconds to figure it out, but it had to be a vanity plate and one I wouldn't mind having.  I pointed and exclaimed something like 'how brilliant' to a rather perplexed friend.  Just a few days ago someone set a rather amazing Internet ad.   It had an easter egg that grabbed my eye for the same reason, although this was much more explicit.  There is a connection between the license plate and the ad.  I'll get to that, but first I need to talk about some playful tools that physics and astrophysics types use.  

 

Any sort of creative work needs a playful component. Its how you develop your intuition.  I'm sure many fields have their own power tools for play and you may want to think about yours as mostly they are so natural that they become part of how we think.  You need to try and discard a large number of ideas without fear of failure. Some of these can be rather silly.  Often you build toy models in your head or at the blackboard.  Typically they aren't practical ..  if someone were to ask how many cattle are required to provide shoes for New York City you might say to yourself 'consider a spherical cow'..  Physics problems tend to exclude much of the real world so you can focus on just the bit of interest.  Mental or blackboard math rules.  You aren't worried about exact answer.  I love the power of computer simulations but anything you have to type into would just slow you down and break the flow during the play stage.  You want to create little visual models that seem like an extension of your mind so you can dance with them.  People develop a certain affinity for numbers and you carry around a few rough rules of thumb.  Things like π² ≈ 10 to about 1.3% .. when you're only worried about 10% errors the arithmetic is easy.  Some bits of Nature are stored in this fashion:

 

° there are π × 10⁷ seconds in a year to better than a half percent

° the speed of light in a vacuum is close to 3 × 10⁸ meters/second.  An interesting coincidence is the original definition of a meter was 1/10,000,000 of the distance from the pole to a point on the equator. Distances can be measured in the time light takes to travel the distance.  We're used to light days and light years, but light goes about a foot in a billionth of a second or a nanosecond. A very tall friend sometimes gives her height as six and two thirds nanoseconds .. about the time light takes to traverse her length.¹ 

° a light foot is sort of practical when thinking about connecting parts of a computer.  If modules are separated by 10 feet, a delay of at least 10 nanoseconds is introduced.  This gets interesting when you're using a lot of fiber optics or physical wire as the speed of light is slower.²  The speed of light in the fiber is about 2/3s that of in a vacuum or air.  A signal will traverse 1000 km in about 5 thousandths of a second, while a radio wave going through the air only requires  3.3 thousandths of a second.  A very long time for a computer.  The difference was large enough to justify the construction of a specialized microwave network between Chicago and New York City to give some traders an advantage based on the difference between the index of refraction of air and optical fiber. 

 ° the acceleration caused by Earth's gravity is denoted by g and is roughly 10 meters per second².  It varies depending where you are on the surface, but unless you want to weigh a pound less or more, the rough figure works for quick calculations.  

° the diameter of the Sun divided by the height of a person is roughly the same as the height of a person divided by the diameter of a hydrogen atom.

° it takes about 500 seconds for light to travel from the surface of the Sun to the Earth

° photosynthesis in crops and trees is usually under 0.5% efficient

° a cow requires about 10 times as much energy as the plants it eats to produce the same amount of energy 

° ...

 

There are hundreds of little relations like this and a few curious numbers too.  One of the most famous is the fine structure constant which describes the strength of electromagnetic interactions between charged particles.  It is close to 1/137 and appears in enough calculations that every physicist knows it.  If you want to flag one down in a stream of people (airports for example), just write 137 or 1/137 on a piece of paper and hold it in view.  I've experimentally found this to be remarkably effective.

 

Some of the relationships aren't directly connected with Nature, but are just fun and sometimes are even useful.  There are too many to list, but here are two..

 

° e is the base of the natural logarithms and is of fundamental importance to economics and science.   It happens to be irrational and transcendental, but the first few digits are easy to remember:  2.7 1828 1828 45 90 45  (1828 repeats twice, and the angles in a right angled isosceles triangle)

° of course there is Hardy–Ramanujan number.  The story goes that G. H. Hardy was visiting his friend and colleague Srinivasa Ramanujan in the hospital. He had ridden in cab number 1729 and remarked that it was such a boring number.  Ramanujan counted, pointing out that it was the smallest number you can express by cubes in two ways:  1³ + 12³ and 9³ + 10³.  This has slipped into popular culture and has appeared in both the Simpsons and Futurama in the form of easter eggs.  But then again some of the principals of those shows are mathematicians. 

 

And number on a taxi brings us back to the number on a vanity plate.  My eye was probably caught by the initial 23, which happens to be my favorite prime, but  it looked suspicious - something close to 20 plus π ...  I remembered e^π - π is just a bit less than 20.  The number on the plate had to be an approximation of e^π.  The connection with the ad is perhaps the most beautiful relation in mathematics : ³

 

e^iπ + 1 = 0

 

While e^π isn't the real thing, it is suggestive enough to bring a smile.  Of course there is the possibility the owner just considered e^π cool without thinking of e^iπ, but this was the JPL parking lot.  

 

If only vanity plates allowed superscripts...

 

 

__________

¹ again an approximation.  If you want to do this more exactly the speed is about 0.984 ft/ns .. 

² the speed of light in a material is c/n, where n is the index of refraction.  n is close to 1.0 for air, but about 1.5 for optical fiber

In physics experiments signals are delayed with respect others to make logic gates work properly.  You develop a sense that 8 inches or 20 centimeters is about a nanosecond in coax.

³ It links the two most important irrational numerical constants e and π, number i which is the base of the imaginary numbers which gives us complex numbers, the additive identity 0 and the multiplicative identity 1.  It also answers the question "how many mathematicians does it take to change a light bulb?"  Then answer, of course, is -e^iπ...   

If it seems strange that the exponential of a complex number could be equal to 1, consider what the number e is.  It is closely related to compound interest ...  e^z is the limit of (1 + z/N)^N as N goes to ∞. If you set z = iπ and do the math you get -1 + 0i or just -1.  Here's an animated gif: 

 

 

__________

Recipe Corner

 

Caramelized Carrots

 

Ingredients

 

° 3 pounds of carrots - go for the fancy colored ones if you want it to look good

° salt and freshly ground pepper

° 2 clementines or one large orange

° 1 tbl red wine vinegar

° 3 tbl butter or vegan margarine 

° half a bunch of fresh thyme

 

Technique

 

° peel the carrots. you can quarter of halve them, but if they're pretty whole carrots with a bit of the tops on are beautiful

° put carrots in a large pan and just cover with water.  Add a bit of salt and ground pepper and the clementine juice, the vinegar and butter

° bring to a boil and cook until nearly all the liquid has evaporated

° add the thyme sprigs, reduce heat to low and cook for about 5 minutes to caramelize 

 

more on education

I’ve had a number of fine surprises this year, but Piper Harron’s PhD thesis in number theory is at or near the top. Why takes a bit of explaining.

Math is core to almost everything we do, but sadly it has become abstracted for too many people. Part of the blame falls on the mathematicians. The failure rate of math undergrads in many schools is very high. You come into realizing how little you know, but often bootstrapping is too difficult. Some of its fields have become exceptionally technical. No one in their right mind would do a PhD thesis in number theory unless they were extremely brilliant as the “good” problem space has become the territory of genius. These departments have become formidable ivory towers.

So along comes Piper - a young woman with an interest in number theory who understands math for what it is - a deeply cultural and human undertaking. After all, you have to convince others that your proofs work - something that is seriously broken these days (the abc conjecture in number theory comes to mind as do any number of computational proofs).

Her thesis is *very* readable and pedagogical. A sophomore math or physics major should be able to grab enough of a hold to bootstrap themselves through with a bit of work. A senior level student should find it a delight. There aren’t many math thesis that grab you. I was sent a copy on Christmas Eve day and its style drew me in. It isn’t a terribly deep result, but it is different from conventional math papers in the sense that Minute Physics or Physics Girl are different from physics for high school students and college non-majors. I’ll hazard a guess that it will be widely read and will have a greater impact than any other math thesis this year - perhaps this decade.

 

Some of you have serious math skills, others have studied other areas with not much exposure. If you know math jump it, otherwise just read the part I quote and perhaps skim the paper for cartoons and structure … the point is not so much to read the paper as to understand what she is doing culturally.

Maybe, just maybe, math and math education is changing and becoming less elite. That is exciting.

 

download it (1.8 MB pdf)

The prologue and early parts of the introduction say it all:

 

Prologue

Respected research math is dominated by men of a certain attitude. Even allowing for individual variation, there is still a tendency towards an oppressive atmosphere, which is carefully maintained and even championed by those who find it conducive to success. As any good grad student would do, I tried to fit in, mathematically. I absorbed the atmosphere and took attitudes to heart. I was miserable, and on the verge of failure. The problem was not individuals, but a system of self-preservation that, from the outside, feels like a long string of betrayals, some big, some small, perpetrated by your only support system. When I physically removed myself from the situation, I did not know where I was or what to do. First thought: FREEDOM!!!! Second thought: but what about the others like me, who don’t do math the “right way” but could still greatly contribute to the community? I combined those two thoughts and started from zero on my thesis. What resulted was a thesis written for those who do not feel that they are encouraged to be themselves. People who, for instance, try to read a math paper and think, “Oh my goodness what on earth does any of this mean why can’t they just say what they mean????” rather than, “Ah, what lovely results!” (I can’t even pretend to know how “normal” mathematicians feel when they read math, but I know it’s not how I feel.) My thesis is, in many ways, not very serious, sometimes sarcastic, brutally honest, and very me. It is my art. It is myself. It is also as mathematically complete as I could honestly make it.

I’m unwilling to pretend that all manner of ways of thinking are equally encouraged, or that there aren’t very real issues of lack of diversity. It is not my place to make the system comfortable with itself. This may be challenging for happy mathematicians to read through; my only hope is that the challenge is accepted.

Chapter 1

Introduction

1.1 Notes to My Dear Reader(s) 1.1.1 The Layperson: Math 101

I will always be honest with you.

The hardest part about math is the level of abstraction required. We have innate logical abilities, but they are based in context. If you give people a scenario of university students drinking beverages at a bar and give them information either about the person’s age or about the person’s beverage, most people know instinctively which students’ drinks or IDs need to be checked to avoid underaged drinking (i.e., if the person’s 22 you don’t care what they’re drinking, but if the person has a vodka tonic, you need to know their age). Take the logically equivalent situation of cards with a color on one side and a number on the other. Suddenly it takes some work to figure out which cards have to be turned over to satisfy a given condition (say, all even numbers have red on the back). Just one level of abstraction and the untrained, but educated, person will have a good amount of difficulty even understanding the situation. Now try doing Number Theory.

I like to imagine abstraction (abstractly ha ha ha) as pulling the strings on a marionette. The marionette, being “real life,” is easily accessible. Everyone understands the marionette whether it’s walking or dancing or fighting. We can see it and it makes sense. But watch instead the hands of the puppeteers. Can you look at the hand movements of the puppeteers and know what the marionette is doing? A puppeteer walks up to you and says “I’m really excited about figuring out Fermat’s Last Thumb Bend!” You say, “huh?” The puppeteer responds, “Oh, well, it’s simply a matter of realizing that the main thumb joint has several properties that distinguish it from...” You’re already starting to fantasize about the Zombie Apocalypse. Imagine it gets worse. Much, much worse. Imagine that the marionettes we see are controlled by marionettoids we don’t see which are in turn controlled by pre-puppeteers which are finally controlled by actual puppeteers. NEVER HAVE A CONVERSATION WITH THESE FICTIONAL ACTUAL PUPPETEERS ABOUT THEIR WORK!

I spent years trying to fake puppeteer lingo, but I have officially given up. My goal here is to write something that I can understand and remember and talk about with my non-puppeteer friends and family, which will allow me to speak my own language to the puppeteers. To you, the lay reader, I recommend reading this introduction and then starting each subsection of laysplanations (the .1s) and reading until you hit your mathiness threshold (stopping to think or write something down is encouraged; even math you know won’t necessarily make sense at the speed at which you can read and understand non-math), then skim/skip to the next lay portion. Depending on how you feel with that, you should look at the math parts (the .2s) which will look familiar if you were able to finish the lay sections. I can’t promise they’ll make sense, but things should be vaguely readable. Maybe. The weeds (the .3s) contain extra information (some lay, some math) and calculations, more for answering questions than for reading. Enter at your own peril.

1.1.2 The Initiated

Welcome mathy friend! Depending on the extent of your initiation (and your sense of humor), this thesis may be exactly what you’ve always wanted to read! Skim the laysplanations (.1s), but if they are too math-less for you, it’s okay to only read the math sections (.2s) and just go back to the lay stuff if necessary (several things are introduced/motivated in the laysplanations, including explanations of my Formula in the .1.1s). You may also be interested in the weeds (.3s) which are appendices with things that weren’t strictly necessary to get through the proof of the Main Theorem, but were necessary personally for me to get a hold of things. The weeds aren’t to be read straight through, but you might find an explicit calculation or extra explanation there.

1.1.3 The Mathematician

Dear Professor, thank you for showing interest in my thesis! Your introduction awaits at §1.3. For results, however, you may find the fluffless arxived original [BH13] easier to read (certainly quicker!) than this thesis.

1.2 This Thesis (Problem) 101: A Mostly Layscape

Every thesis is a question and (very long) answer. My question in layspeak is: “How many” “shapes” of certain degree n “number fields” are there?

The naive short answer is: Infinitely many! But of course, though true, that is not nearly enough information. What we will show is that the infinitely many shapes we find are actually “equidistributed” with respect to the “space of shapes.” In other words, if you think of the collection of possible shapes as being a blob (a “space”), then wherever you look in this blob, you will find shapes of number fields in equal quantity.

Equivalently, though somewhat less to my liking, a thesis is a claim and a (very long) proof. My equivalent claim in layspeak is: “Shapes” of certain degree n “number fields” become “equidistributed” when ordered by “absolute discriminant.”

In what follows I hope to do enough “laysplanations” to make the whole argument approximately readable by approximately anyone. Approximately. In addition to laysplaining and “mathsplaining,” I will also, where appropriate and not too horrifying, have some “weedsplanations” where I wade into the weeds with examples and explicit calculations, sometimes with extra laysplanations that were not strictly necessary to the main argument.

 

People often say kids are natural scientists.  It isn't true, but they have bushels of curiosity and that is a key component of science and math.  Somehow this goes away in somewhere around the 5^th grade.  It shouldn't be - perhaps it is useful to look at where STEM education came from.

 

With the cold war came a recognition that the country needed scientists, engineers and mathematicians in much greater numbers than ever before. The curriculum was changed to ensure that a pipeline was created to identify and nurture those who would ultimately end up with Ph.D.s from the best universities in the country and programs within the universities were greatly expanded.  In two decades the number of physics Ph.Ds. increased from under 200 to about 800 a year.   There was this one little problem.  What do you do with the millions of people outside the pipeline?   

 

There is a deeply rooted assumption that science literacy is good for society.  That somehow the pre and pre-pre professional courses designed by Berkeley and MIT would create a citizenry that could deal with science and society issues.  Unfortunately there isn't any empirical evidence to support the assumption.   

 

Just how do people interact with science?  About ten years ago stem cell research was a hot topic in the news.   There were diverse group.  People with parents who have Alzheimer's, certain fundamentalist Christian sect members and biotech investors were some of the groups with differing motivations driving their interests.  Global warming is another example - depending how you cut it there are at least a half dozen distinct groups (I have some nasty scars on this one). And now we need to strap in for CRISPR.   Science and the public issues are important, but the reality is we have many publics each with their own issues. 

 

Another huge issue is science is not a single monolithic subject.  Science education teaches about the scientific method.  Science doesn't work like that - reality turns out to be a different beast.  Scientists agree on hypothesis testing and many methods, varying across branches and subpecalities.  Trying to sort out the methodological differences between black-footed ferrets studies, climate modeling, and theoretical particle physics is a non-trivial task to anyone outside of science.

 

Most people don't worry about science on a regular basis but rather when something comes up and they would like to use it.  Your two year old is showing signs of autism, your tap water has so much natural gas in it that you can set it on fire, the license on the nuclear power station twenty miles away is up for renewal, you are worried about losing weight and eating better...  Text book learning isn't terribly useful and pronouncements by experts and pundits are often at odds.  What is taught involves a regurgitation of memorized "facts" and the ability to plug in some numbers into equations to answer an artificial problem set.  You have learned this thing science so you can score well on standardized tests.  Generally people have little ability, five years out of school,  to form a meaningful question and come up with a reasoned path that might be useful.   We need to teach a useful science literacy.  I was going to add an aesthetic science (and math) literacy, but I think that comes with study.

 

I was very lucky in high school.  It was a time and place where some of the teachers were allowed to create their own courses on their own.  Not every class, but a few were enough to keep me excited.  My two best teachers taught history courses.  Neither taught a bag of places or dates that we had to recite and as such that probably didn't help us on standardized tests.  One had classroom discussions, both lectured and both required papers every few weeks on a topic or two of your own choice.  They were trying to get us to understand history in context and to relate it to the present - which happened to be a very spooky time.  I still can't remember dates and places well, but I think I have a better appreciation for the flow of history than the average high school graduate - enough that it helps me think about societal issues and begin to work out how to think about politics and policy.  Again - I wouldn't claim to be great at it, but the education I had was much deeper.  Had it not been for some native curiosity and some out-of-school mentoring I probably would have thought about majoring in history. (and that may have been good)

 

The good news is people are experimenting with programs that allow students to find and interpret science in the context of real world problems as well as judge the credibility of scientific claims.  Courses like these may need to find interests of the student that can be used to apply science and this may lead to paths far from what is currently measured in standardized tests.  There are barriers to be broken, but it is exciting to contemplate a much larger population that has a bit of real science literacy rather than the current artifact of pipelining.  Some exciting work is going on: Problem-Based Learning, Place-Based Education, Science-Technology-Society approaches and probably a dozen I'm not aware of.  

 

In college we have examples like Piper's thesis.  I think physics and astronomy are in better shape, but more needs to be done.  And outside of formal education the seeds being planted with laysplaining from YouTubers like Physics Girl and Minute Physics are just.plain.wonderful!

 

and with that a Happy New Year to you!

 

__________

Recipe Corner

 

Fabulous this time of year

 

 Lemon and Oil Spaghetti

 

Ingredients

 

° 2 large wax-free lemons to zest

°  handful of flat-leaf parsley, finely chopped

° a pound of spaghetti

° 1–2 garlic cloves (depending on your love of garlic - I go for two) finely chopped

° a small dried chilli, finely chopped

° ~6 tbsp olive oil

 

Technique

 

° grate the zest from the lemon and then mix with the parsley and set aside

° bring a large pan of salted water to a boil, add  the spaghetti and cook until al dente. 

° in a large pan, gently warm the olive oil, garlic and chilli over a low flame until fragrant – do not let it burn.

° once the spaghetti is cooked,  use a sieve or tongs to lift the spaghetti and a just a little residual water into the frying pan. Stir, add the lemon and parsley, a pinch of salt and if you like a squeeze of lemon, stir again,

° plates and eat immediately!

 

the context and quality of data

Passing white light through a prism to break it into a range of colored light, Newton introduced the word spectrum into optics.  The colored apparition was packed with information contained in the light - a mixture of frequencies and intensities.  Eventually it was learned that studying these revealed important aspects of matter the light had interacted with.  In the mid 1800s physicists reignited the sleepy field of astronomy by attached spectroscopes to telescopes allowing the study of what made the starlight.  Among the discoveries was an unknown yellow line in the Sun's spectrum.  It was proposed as a possible new element later called Helium in honor of the Sun. 25 years it was discovered on Earth.  

 

The technical is powerful and commonly used in many areas of science, medicine with thousands of commercial applications. The caveat is you need to use the right technique and appropriate instrument. 

 

The heart of many types of spectrometers is simple - a well characterized and calibrated light source, something to hold a sample, a prism or diffraction grating to break up the light into a spectrum and a way to measure the intensity of the spectrum as a function of its wavelength (color, sort of).  A smartphone has enough processing power to perform serious analysis and a few companies have made low cost spectrometers for teaching purposes.  A good example is this $400 unit from Pasco.  It lacks the precision to do serious work, but its great fun and works with an iPhone or iPad.

 

Three times in the past two years people have sent descriptions and pitches of even more portable units using integrated spectrometer modules.  While they might be fun, these usually do reflected light surface spectroscopy and appear to have poor resolution.  Silly claims are frequently made - that you can determine what is in your medicine, do food safety and calculate the calories in a meal.  

 

Nope - it isn't going to work like that.  You can't use out of context uncalibrated data and somehow perform cloud magic with 'algorithms' ... Success during a Kickstarter funding period does not guarantee impossible magic.  There will be a flood of interesting sensors and even serious instrumentation for smartphones and some of it will enable a density of information that is potentially transformative. Consider thousands of accurate pollution monitors per square kilometer in cities.  China is ripe for a revolt from a data informed citizenry.

 

Data, even the simplest data, has context.  What was the sample. how were the measurements made, what is the accuracy of the measuring tool, did the accuracy change over time, how was the data preprocessed, how was it postprocessed...?  

 

A few years ago I came across a detailed description of a rope transmission from Bordeaux, France that moved mechanical power from a water wheel to several businesses hundreds of feet away.  These contraptions sometimes often had mechanical efficiencies that still impress.  My answer seemed far too low.  I smugly came to the conclusion that the book must be in error.

 

The problem resurfaced recently when I was reading about the spread of the metric system in France.  Standardization of weights and measures was of vital importance to commerce as hundreds of local systems were being used.  The author noted rural Bordeaux defined a foot using a standard that was 35.7cm long - dramatically different from the now standard definition of about 30. 5cm.  I had been doing my calculations based on a bad assumption.  Using this definition a six foot tall person would be a bit over five foot one...

 

About three years ago a friend came to me with a question about a distribution of measurements.  She's quite tall and was thinking about putting together a specialized shop for tall women.  An important early question was to understand the potential market size.  While there are many factors that go into clothing size, height is a reasonable proxy to get a rough idea.  She knew there were tables showing height distributions,  but it wasn't clear to her how to navigate the information.  I had been involved in a large anthropometry survey in the nineties so one thing led to another...

 

You can make a histogram of heights - 100 people at 64 inches, 125 at 65 inches and so on. Increasing the number of people in your sample will make the distribution look smoother.  With a very large population and very small measurement increments you'll get something like this:

 

It isn't exactly the bell-shaped curve you probably expected.  The problem is it is the composite of two two curves.  Men are about five inches taller than women on average and need to be considered a separate population.   Plotting men and women separately gives two bell curves. Add them and the original distribution appears.

 

 

This type of curve is a Normal distribution - anyone with a STEM background has studied them in depth as they turn up almost everywhere.  Human height turns out to approximately follow this type of distribution making it easy to calculate the distribution of potential customer heights assuming the published curve is relevant to your population.  Measurements have been made in most countries and  are repeated at regular intervals.¹ 

 

My friend's  target market was women who stand more than six feet tall.  A small market, but she and her friends were underserved so it made sense to investigate.  Perhaps the Internet would allow her to find enough customers to justify production runs greater than a few pieces.  She chose a huge American survey and found wrote a simple program that calculated how many women were predicated to be at or about a height 

from her email:

 

Something is broken Steve  I just integrate a normal distribution with a mean of 64.3 and a standard deviation of 2.5  from a height through infinity to get the fraction at and above a height.  Here are my numbers with a little rounding.  Any ideas?!?

 

72"    1 in 966

73"    1 in 3,900

74"     1 in 19,150

75"     1 in 107,000

76"     1 in 697,200

77"     1 in 5,298,900

78"     1 in 47,022,800

This is SO WRONG!  The number of girls in college basketball and volleyball are instant disproofs!

 

To produce a clean fit it standard practice to exclude the tails of a measured distribution and tell the reader where it is valid.  It wasn't published in this case,  at least not prominently, but the fit was only good to about 5'11.  The tails on the real curve tend to be thicker, but making a large enough measurement to smooth them out is unrealistic.  It is possible to get a good enough fit for all but a tiny fraction of the population with only 100,000 or so people.

 

Now I was curious.  Would it be possible to find a better fit to the distribution?  Over a period of a few months I found a few ways to build a more accurate distribution function.  Rather than a simple normal distribution it is the sum of three separate distributions.  The tricky part was calculating the errors.

 

I won't list all of the results, but I'm 95% confident that a 6'3" woman (my friend's height) is between 1 in 16,070 and 18,070 with 1 in 16,950 being the most likely.²  1 in 17,000 is probably a good enough starting point and better than any of the major surveys.  My tall friend Colleen would be about 1 in 267,000, but the bound is much larger between 1 in 219,000 and 1 in 420,000.  Another good friend is 6'8,  the estimates are very sketchy at her level:  1 in 1.38 million with a range from 1 in .76 million to 1 in 2.79 million.  Counting athletes as a lower bound is probably better:-)

 

The final plot looks identical to the second plot as the size of the change is only 0.02% of the regular normal curve.  This is a feature as the densely populated portion of the distribution function is nearly identical to the original. To get a sense of the shape of the adjusted function,  imagine a world where my corrections are 2,500 times larger making their contribution equal to that of the normal female height distribution.

 

                         

 

 

 

The new female probability curve is red and the old uncorrected female curve is blue in the first plot and the uncorrected male curve is blue in the second.³  In this different world a quarter of the female population is six feet tall or more.  One in one hundred and thirty would be at least Colleen's height.  This  alternate world could be fodder for social commentary fiction.  Tall women certainly wouldn't have problems finding clothing and female sports would probably dominate.  Perhaps height would be unimportant socially. 

 

Continuous distribution functions derived from observed data are common and, when used correctly, powerful.  It is all too easy to miss a fundamental assumption and come to the wrong result while feeling good about it.  Separate sanity checks, like looking at the roster of college volleyball teams, need to be normal practice.  Being playful and not be easily seduced is a requirement.   

 

The clothing concept turned out to be impractical.  Quality sources for reasonable costs simply don't exist for the necessary low volumes.  It became clear that made to measure is necessary. Some is done for men's shirts in Asia and South Asia, but costs are currently high.  In addition to height, there was too much variability in sizes to make the effort practical... at least for now.  The trick is mass made-to-measure.  Eventually it will be standard practice and fit will become nothing more than a memory. Until then some segments of the apparel market are tough sledding.  Her advice for anyone dealt an usual body: learn to sew, find a good tailor and make enough money for a few quality custom pieces. Find your own style rather than following the trends.

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¹ Health organizations usually take the measurements as they are a proxy for human health.  Cheating I copy and paste from an email I sent on the subject:

We tend to pay a lot of attention to numbers without worrying about their accuracy. Height is a good example. If you measure a large number of adult males and females, their heights will distribute themselves along a bell shaped curve. Most people will be close to an average (a mean). The larger the sample the more the distribution will take the shape of a bell shaped curve called the normal distribution. Remarkably all you need to know is the average and a number that tells you something about how rapidly the curve changes shape. From this a designer will know the height distribution of potential customers and can adjust their manufacturing plans accordingly. You can do similar exercises for many measurements - shoe size for example. 

The characteristic curves vary regionally. The US is very similar to England and can be taken as the same within the accuracy of most measurements. Northern Europe is taller, with Holland being the tallest. The North of Holland is particularly tall and is sometimes treated separately. Asia and South Asia are shorter than the US and South America is much shorter. Changes in mean and standard deviation are studied over time as height is a good proxy for health within a population. A tall person is not necessarily healthier than a short person, but for a child about 20% to 25% of the variation of their final height is influenced by health and nutrition rather than genetics.  If the health of a nation changes, so will it’s average height. Holland has seen a dramatic change in the 20th century - particularly after WWII with Northern Holland being the tallest region in the World.  For adults under 35 female height is only an inch less than mean male height in the US and mean male height approaches 6'2.

² A quick comment - the curves are probability distribution functions.  The horizontal axis is height, the the vertical axis is probability, so you get a sense of roughly what it is.  The exact probability for a point is meaningless.  The probability of someone being 73.0000001 is very nearly zero.  Just integrate the function over the range you are interested in. 

Measurement errors in the height studies and insufficient statistics are the major components that make up the errors.  A trained person using a simple stadiometer - those vertical rulers with a sliding horizontal arm - can repeatedly measure height to about a half centimeter, but some surveys are self-reported.

³ Here I have plotted real probabilities for each inch of height and have connected them with a smooth curve - a similar to the density function, but a bit different.  I'm just doing it to show there are different ways to interpret and display the data and drawing conclusions requires some understanding.

Here is the probability plot of just the correction in black scaled up to represent a full population next to a normal female plot in red.  The shape isn't simple and has a much larger mean and width than the normal curve for women's height.  Scaled down by a factor of 2,500 and added to the normal women's distribution it gives a "fat tail" that squares with other measurements.

 

 

Playing games like this make it easy to spot errors in your thinking or calculations. At some level this is just a game.  A good thing as you can spend a lot of time getting everything to work.

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Recipe Corner

 

This is modified from an old recipe from a cooking club I was part of

 

Indian Cauliflower with Several Spices

 

Ingredients

 

° 1 large head cauliflower, trimmed and separated into florets 

° 1 large yellow onion, thinly sliced

° 1 tbsp olive oil (you really should use coconut oil, but I didn't have any)

° 1 tsp mustard seeds 

° 1 tsp cumin seeds 

° 1 tsp nigella seeds (I didn't have so I skipped)

° 1 tsp fenugreek seeds 

° 1 tsp fennel seeds 

° 6 ounces tofu, stirred 'til smooth

° 1 tsp paprika

° ½ tsp cayenne

° ½ tsp turmeric

° juice of a half lemon

° kosher salt to taste

Technique

° heat the oil in a pan and add the seeds. When they dance and begin to darken, add the onions.

° sauté the onions until  translucent.

° add the cayenne, paprika and salt. 

° add the cauliflower florets and mix well.

° add the tofu, mix. Cover the pan and let the cauliflower cook until somewhat tender. Stir every few minutes

° add salt and lemon juice and mix .

 ° serve warm

 

summing things up

In high school you probably learned about the properties of infinite series.   Things like the sum of the reciprocals of the powers of 2: 1/1 + 1/2 + 1/4 + 1/8 + 1/16 + ^... converges on 2, but the sum of the reciprocals of the positive integers diverges: 1/1 + 1/2 + 1/3 + 1/4 + 1/5 + ^... , growing to infinity.  There are any number of interesting convergent series related to really interesting numbers like e and Pi, not to mention it is useful to know when something blows up to infinity.  Math just works, eh?

 

 

But wait...  

 

A friend sent a link to a video that seemed wrong.  It made a completely nonsense claim: that the sum of all positive integers was not infinity.   1 + 2 + 3 + 4 + 5 + ... + infinity.   Stranger yet was the claim it was not an integer and also negative.   The video came from real mathematicians who sounded confident.  He looked for an April first posting date, but didn't find anything suspicious.   This had to be some kind of joke.  What was the trick?

 

 

 

It turns the video is correct.  In fact you need to be able to do sums like this if you want to use the Standard Model in physics as well as string theory and a few other areas.  The same folks put out a second video with a bit more math, but still at a high school level.  If you're really curious take a look.

 

Something extremely interesting is going on.

 

Imagine math in the days when the only numbers people used were real and rational.  Some clever person came along and started using square roots.  The square root of 1, 4, 9 and 16 made sense, but 2 was a problem.  You could approximate the square root of 2, but it wasn't a rational number.  A new type of math had been discovered that was not only was in another context, but expanded the notion of what math was.

 

On the surface the square root of a negative number seems pure nonsense - so much that the operation was defined as invalid on that side of the number line. But its just math, so why not?  The square root of a -1 was defined to be a new type of number - an imaginary number.   A new number system had been invented.  It was shown to be rigorous  in the proper context with certain definitions and has enormous value in physics and all of engineering.  

 

This is the story of math.  Rather than a pristine temple where people work with pure logic using only existing truth to find new truths, it is a bubbling sea of conjecture and relation that can lead to proof.  Ultimately a rigorous path needs to be found,  but the frontiers are messy.  Change comes with rebellion.

 

Leonhard Euler was just such a rebel.   He wondered if  sums that diverged to infinity had deeper meanings.   Some, like the sum of all positive integers, had a regular structure.  How might they differ from other infinite sums?  Was there a story different from the final tabulation at the end?

 

He came up with an ingenious technique and showed the sum of positive integers was -1/12 in a fashion not too far from that of the first video.  The first time I saw it in college was electrifying.   It was like sorting through an infinite pile of dirt and rock to find a tiny diamond.  When a similar series appears in quantum electrodynamics - an area of physics that gives us the highest precision of all measurements to date - you substitute in the little diamond and, low and beware, the calculation produces a meaningful result.  The most accurate measurements humans have made agree with a theory that makes use of this type of sum.

 

Euler worked out several other series.  The sum of the squares of all positive integers turns out to be 0 and that of the cubes is -1/120.  In fact the all of the sums of even powers of the the integers is 0 and the odd powers is not.  (Impress your friends by noting the sum of x_i¹⁰ over i = 1 to infinity is 0.)   He was unable to prove this rigorously.  That had to wait about a century for Bernhard Riemann who was able to show a deep connection with something that fascinates to this day - the Riemann zeta function.

 

So what seems both wrong and crazy turns out to make perfect sense in a certain context - a context that is fundamentally related to how we know the physical world at the deepest level.  Why does Nature seem to speak math?  Now there is a deep mystery... 

 

I should stress that the equivalence here is the core  issue.  Equivalence doesn't mean equal.  This gets deep into the analytic continuation of the Reimann Zeta function at s = -1 ... which happens to correspond to the sum of the positive integers.  The R-Z function needs to be extended through analytic continuation to be valid for -1.  The videos don't stress it enough.  Doing this you get the R-Z (-1) = -1/12 and also note that the sum of R-Z for -1 looks like the sum of positive integers.  Using this redefinition physics happens to work -- that is the magic.

 

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Recipe Corner

 

I don't know where the okra is coming from these days, but some is very high quality. I threw together a sort of crispy bhindi.  Not exactly traditional, but delicious.  As usual the quantities are recollections of what I think I did

 

Crispy Okra

 

Ingredients

 

° a pound of okra with the tops trimmed and sliced lengthwise into thin strips

° 1-1/2 tsp red chili powder

° 1-1/2 tsp garam masala

° 1/2 tsp turmeric

° 2 tbl chickpea flour

° 1 tsp cornstarch

° juice of one lemon

° vegetable oil

 

Technique

 

° mix the chili powerder, garm masals, turmeric, chickpea flour and cornstarch and toss the okra in it moistening with the lemon juice 

° heat some vegetable oil in a wok to something under its flash point and fry a handful of okra at a time until golden.  Drain on paper towels and let cool.

° I tried a second frying on some which makes them even crispier -- experiment!