omenti

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.

 

a shot echoed in the dark

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

 

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

 

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

 

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

 

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

 

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

 

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

 

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

 

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

 

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

 

__________

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

²  take its Fourier Transform.

diving into boiling stones

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

 

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

 

 

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

 

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

 

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

 

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

 

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

 

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

 

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

 

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

 

__________

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

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

³ the convex hull

medical tests and probabilities

a mini post

 

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

 

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

 

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

 

 

 

 

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

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

 

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

 

 

 

 

a dive into pizza pt.1

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

 

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

 

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

 

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

 

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

 

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

 

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

__________

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