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.
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.
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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.
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.
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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
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.
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 2Pi 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 find a volleyball or some other sphere. 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.
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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.
Apparently a math puzzler became something of a theme on the Internet.. a half dozen people asked me what I thought the answer was
8÷2(2+2)=?
Some came up with 1, others with 16. Apparently there were near-religious arguments. A mathematician would tell you both are incorrect - the question posed is ambiguous and not an answerable bit of math .. at least not generally.
The problem is the order in which one evaluates expressions. It turns out the one you learned in grade school is a bit different from the one you may have had when you learned algebra. If you remembered grade school you probably get 16.. algebra people are more likely to get 1.1 If your algebra teacher was more careful they would have said the expression is wrong - you need to add parenthesis to make it unambiguous. The exception would be if the only people you communicate with agree on representation. That can be a big if..
Taking it a bit further, the simple rules you learn in secondary education don't cover don't cover unary ±, which implicitly have lower precedence than exponents on one side and higher on the other! For example
-x3 → -(x3) x-3 → x(-3)
This is so well agreed upon that no one uses parenthesis in this case. Historically juxtaposition was treated as implicit multiplication, but that led to ugliness like
1/3x → 1/3*x → (1/3)*x a2x → a2*x → (a2)*x
uck!
People decided juxtaposition of numeric literals is like a unary operator .. so the 2 in 2x is attached to the x like the - in -x.
Back to the original problem. The fact that people had different interpretations and mentioned rules from school meant a better rule was required. The problem needs to be unambiguous to anyone who might encounter it. This is an issue for computer languages - different languages can parse the same written expression differently and come to different results. Agreement on standards is an issue too .. the Mars Climate Orbiter was lost as teams at JPL and Lockheed Martin had different takes on a single critical bit of data - one used metric units, the other english.. when the retrorockets on the spacecraft were fired to go into Martian orbit the spacecraft burned up in the Martian atmosphere - it was literally lost in translation. And this was code that passed multiple very careful reviews - code written for space probes is some of the carefully crafted software period. Just wait for complex coordination systems and things that move with people nearby or inside.
That said there are some areas were common agreement among practitioners creates a local set of standards.. Einstein summation notation in physics is confusing to outsiders, but all physicists know it. And then there's my favorite - the double factorial. In math (n!)! is not n!! ... (note that I also have the problem of people interpreting ! as an exclamation point:-)
This local agreement is probably true in most fields - even descriptions of an action in a sport are local to the sport. We all use local shorthand - it's a form of jargon, but we need to know when we can use it and when we need to be more general even it it's awkward.
Some sports have doping issues and there is drug testing in many of these. There's always an issue of how good the tests are, but there's also an issue of how fair they are and how good they have to be to be fair. The best way to illustrate this is with an example.
Consider a sport with 1000 players - roughly the size of major league baseball in the US . Suppose steroid abuse is an issue and there's a test with 95% accuracy. And assume somehow it's known (or strongly suspected) that about 5% of the players are abusers. These are artificial numbers, but it's just an illustration.
What if the players are tested? We expect 50 to be abusers and 950 to be clean. The test is 95% accurate.. so 48 (rounding up from 47.5) are correctly identified. So far so good. Now consider the 950 clean players.. 5% or 48 (rounding up again) are incorrectly labeled as abusers. So of 1000 players there would be 96 positives and only a fifty-fifty chance of being right. Careers can be be ruined .. the stakes are high.
A way to sort out these conditional probabilities was described the The Reverend Thomas Bayes in the mid 1700s in a paper called Essay Towards Solving a Problem in the Doctrine of Chances. The good Reverend described the relationship between the probability an athlete is an abuser given a failed test and the probability an abuser fails the test. Consider these probabilities (apologies for notation)
P(A) = probability a drug test is positive
P(B) = probability a player is an abuser
P(A|B) = probability an abusing player tests positive
P(B|A) = probability a player has taken drugs if their test is positive
Perhaps the most important takeaway is that P(A|B) and P(B|A) are not the same! It's a very common prosecutor trick to fool the jury into thinking they are - so common that it's called the prosecutor's fallacy. But to the problem at hand.
We want to know P(B|A). We already know P(B) or at least we have the 5% estimate. So P(B) = 0.05. This also gives us P(not B) = (1 - 0.05) = 0.95 as P(B) and P(not B) must add to 1.0. We also have 47.5 (I'll go with the fractional athlete here) of 1000 drug free athletes tested positive, so P(A|not B) = 0.0475.
Bayes derived a formula showing how all of these relate. Sticking with our notation:
This is a very unacceptable test! Although 95% accuracy sounds good on the surface, it only finds a bit better than half the abusers and incorrectly labels a lot of players. For fun try a 99% probability and a perfect 100% to see what's happening.
You can have much more complex conditionals, but this is the general idea. You really have to think things through a bit..
The other day I saw something that a few of you might want to consider for a kid or even yourself - the Turing Tumble Mechanical Computer.
It turns out digital computers are mostly switches. A lot of switches these days - perhaps a trillion in your smartphone. As a teenager I built a very simple relay based digital computer based on a column in an old Scientific American. Later I built an updated version that substituted transistors for the relays using some (free) components from the high school physics teacher.
Playing with these beasts is an excellent introduction to understanding how logic circuits work. You'd be surprised how many computer programmers are hazy on the subject (of course they work at a level of abstraction far above the silicon - or what old timers called the iron). The early machines of the 30s were very important and set the stage for the dramatic developments that followed. Here's an outstanding non-technical short history of those early days. (recommended!)
But back to the Turing Tumble.. It's limited, but that's central to its beauty. The instruction book has about sixty examples to play with and an aha! just might strike leading to out-of-the-book insights.
It's also great to play around with extremely simple silicon digital computers - particularly those that you can hook up to sensors, lights and other interfaces to the real world. But getting down to the basics is usually a paper or YouTube exercise or, if you want to do it yourself, a frustrating experience involving a lot of soldering and a non-trivial amount of money. The Turning Tumble might be just the ticket. It is play and that is the best way I know of to learn. I didn't have much of a chance to play with it and can't do a thorough review, but I'll probably end up buying one to give to a kid. The caveat is that you have to spend some time with it to learn. If the user doesn't find it fascinating enough to devote time to there won't be any learning. But the same can be said about so many things.
The education link on their site has the following in addition to a set of resources:
Appropriate Age Range
Turing Tumble is for ages 8 to adult - and that is not a stretch! We find that kids 8-12 are able to get through the first 20-30 puzzles. Adults get addicted by puzzle 27, and their minds are blown by puzzle 35. Younger kids enjoy the first 10 and building their own computers.
William Rowan Hamilton had become obsessed with the idea of a three dimensional number system - sort of analogous to complex numbers being a two dimensional system. The story he later told was every morning his son would say 'papa, can you multiply triples?' to which Hamilton would sadly shake his head and say 'no, I can only add and take away'. Then, on a walk on the 16th of October, 1843, he crossed the Broom Bridge in Dublin and suddenly knew how to do it. Rather than adding a single dimension to the complex plane, he'd add two giving three imaginary dimensions and one real dimension perpendicular to them. In his excitement he carved it into the bridge with his knife: i2 = j2 = k2 = ijk = -1
He called them quaternions. They had some use in the day, but when people invented linear algebra, the technique fell by the wayside ... that is until quantum mechanics came along about eighty years later. And then computer graphics further down the road.
It turns out they're a lovely way to think about quantum mechanics and perform real calculations. Erwin Schrödinger knew enough about forgotten math to realize it was a beautiful gem. Decades later, trying to improve the speed of computer graphics, a physicist realized they could be a key to quickly calculating rotations in space -- and Tomb Raider became the first computer game with smooth graphics on using relatively primitive personal computers.
Hamilton had other accomplishments that deeply impacted physics and math, but the quaternion story has a certain romantic element. To this day physicists gather at the Broom Bridge on the 16th of October to celebrate.
The story of physics is that it's extremely important to think broadly and to keep expanding your learning. It is also is relevant to education and work across many fields today. That's what I want to get to, but first this wonderful parody of a certain song.. perhaps the best math song ever.
What prompted all of this was an email with a link describing the most contrarian college in America. It's laser focused on a vision of an education seemingly without much a nod to vocation ... seemingly.. I'd argue that it's too focused on a narrow piece of education, but have no doubt its graduates have learned how to learn.. they have the beginnings of a real education. Mark Roosevelt, its President, sums it up:
Education should prepare you for all of your life. It should make you a more thoughtful, reflective, self-possessed and authentic citizen, lover, partner, parent and member of the global economy.
It takes a special kind of student to thrive in a school like that. I wouldn't have survived - my interests, those that I was motivated to dive into, lay elsewhere. I'm happy it exists and that there is spectrum of schools for those who look.
My belief is a deep education with at least a few focuses creates the second type of person Claude Shannon describes:
“There are some people if you shoot one idea into the brain, you will get a half an idea out. There are other people who are beyond this point at which they produce two ideas for each idea sent in.”
The ability to get multiple ideas is core to serious play and creativity. I'd claim focusing on education rather than finding the minimally easy path helps create the neural connections that make this possible.
A worry is that many, if not most, schools and students tend more towards vocation. Focus on what is important to get a degree that will get you employed to pay off that mountain of debt. Many (not all) businesses, on the other hand, would do well to have broader employees who are still learning and growing - those who can think creatively. There is a disconnect.
I hope this focus on vocation changes. The rate of real change is great and people will keep up better if they are broader and more adaptable. When recommending colleges I tend to focus on education .. often to the consternation of those asking. Of course I may be a poor example. My education was far too narrow, but I've been lucky and curious enough to have been able to expand it a bit. The other note is you can get a wickedly broad experience at many more places than you might imagine.
Finally there's this odd emphasis on over-scheduling many pre-college kids, but I've probably written on how it kills creativity enough.
I was in the basement with the brand new Motorola 21 inch color TV and a children's telephone. The wires snaked out and up to the roof where my Dad was on the other toy phone braving the Montana Winter as we tried to get a usable signal. It was my first year of high school and the Winter Olympics would be on in a day or two. The Olympics and we had a COLORTV!
We sort of got it working by the time the broadcast began. The Olympic symbol was displayed and the story given
... the six colors [including the flag's white background] combined in this way reproduce the colours of every country without exception. The blue and yellow of Sweden, the blue and white of Greece, the tricolor flags of France, England, the United States, Germany, Belgium, Italy and Hungary, and the yellow and red of Spain are included, as are the innovative flags of Brazil and Australia, and those of ancient Japan and modern China. This, truly, is an international emblem.
It's a very clean design, but it offended my sense of mathematics.
My sister had very long hair at the time. Long enough to sit on. She was always working on various braids. The year before I was curious about knots and braids and learned that a simple three strand braid becomes very interesting if you connect the ends. Take three strands here colored red, blue and green. Wrap them in a standard braid and go through one cycle so the colors end up where they've started. (I like to think of braids as musical rounds)
Rather than repeat the process like my sister, bend them back and connect red to red, blue to blue and green to green. You'll be left with three loops that are fully linked, but if you remove one the whole works falls apart. Here's a nice image. Note that they're not really two dimensional circles as they have to pass over and under each other. Making this forces you to use three dimensions.
Technically these are a special case of brunnian links - a nontrivial link (a link is a grouping of the loops here) that becomes a set of trivial (unconnected) circles if you pull out one of the loops. Cut any loop and the whole thing can easily be pulled apart. Do it with five loops and you'd have powerful symbolism for the Olympic movement - or so I thought. I knew you can make brunnian links for any odd number of loops so it was time to draw.
I sketched a five loop version. Cut and remove any loop and it all comes apart. All I had to do was find some way of making a nice drawing. My sister is the arty one. She had all of the right kit including some high quality colored pencils. She also happened to be smart enough to not let me use them - even if I had noble intensions.
Plan B. Every kid has Crayola crayons. I selected five and, after a couple of tries, I had it.
(pretend the colors are accurate. my drawing here is pretty bad, but so was the crayon version)
The drawing went into a special international airmail envelope with an explanation of the math and symbolism. It was addressed to the Olympic organization in Lausanne.. I have yet to hear from them. Apparently crayon drawings from 15 year olds don't make the cut.
I draw quite a bit. It's an integral part of how I think. It isn't pretty and I consider it an ephemeral part of thinking and trash it as soon as it's served it's purpose. If you want meaningful design it's best to use a real designer.