omenti

beyond 3d

Ask a physicist, biologist or chemist what got them interested in science and chances are they'll talk about a dramatic single moment - some refer to it as a calling.  I've written about mine on Sulphur Mountain in Banff.  Two young physicists took the time to try and explain their cosmic ray telescope.  Much of it was beyond me but the idea that something from outside our solar system slammed into the upper atmosphere and created a shower of other particles stuck.  And that these wouldn't live long enough to make it from the upper atmosphere to their telescope unless time was different for them.  I had no idea how that worked, but the fact that they could show that universal time - a master clock for the universe - didn't exist ate at me for nearly two years.  I didn't know it but I was about to become a physics student.

 

Somehow I found two correspondence mentors who taught me a little physics and more math.  Two years after Sulphur Mountain I finally understood why time ran more slowly for those speeding muons than for me.  That and what E =mc² really meant and how Einstein figured it out.  Einstein figuring it out was extraordinary, but more extraordinary is Special Relativity is simple and clear - beautiful - even if the results seem odd.  A high school student could read his paper and follow everything with just basic algebra and trigonometry.  

 

There are more than a few explanations in print and online.  To really learn it you need to puzzle it out for yourself with pencil and paper guided by a good textbook and problems.  Instead I'll talk about how he came to that flash of insight. 

 

Around the age of sixteen Einstein had been kicked out of the German school system.  He moved to Switzerland and spent a year at a school that put a premium on visualization and individualism.  There he learned about electromagnetism - light was just oscillating electric and magnetic fields that traveled at the speed of light. He found himself wondering what it would be like to run alongside a beam of light. Could he visualize what these fields looked like?

 

Maxwell's equations said that electromagnetic radiation - light - travels at 300,000,000 meters per second in a vacuum.   If he was going that fast the fields would vanish.  That didn't work.  Since Galileo physics was built on the idea of relativity.  The laws of physics didn't depend on how fast you are traveling.  All you can measure is the velocity of one object relative to another.  He had run directly into a wall.

 

For a decade Einstein kept at the problem.  Finally he went with a crazy idea (to be fair he tried a few crazy ideas).  What if the speed of light was the same for all observers no matter how fast they were moving? If you were on a train traveling at 80% the speed of light and pointed a flashlight forward, you'd measure the speed of light from the flashlight as 300,000,000 meters per second, as would a stationary observer watching from the ground as your train sped past.

 

The way out of this paradox was the person on the train and the observer on the ground experienced time differently.  Two events can happen simultaneously for one observer and at different times for another and both could be right.  Understanding exactly how that works requires learning about special relativity and you can't avoid the math and some sketches.    

 

Einstein offered the following gedankenexperiment (thought experiment):  An observer near a train track watches a speeding train go by.  Exactly as the midpoint of the train goes by two bolts of lighting strike the train - one at the front, the other at the back.   The distance from each strike to her is exactly the same and she says the lightning strikes were simultaneous. 

 

A passenger is sitting at the train's exact midpoint.   Because the train is moving the light from the pulse at the back has to travel a bit further than the pulse from the front. He says the strike at the front of the train came first.  They weren't simultaneous.  The key is it's simultaneity that's relative.  Once you understand that the algebra is simple.  The construction you are left with is space and time are intimately connected in a four dimensioned coordinate system called spacetime.  Spacetime doesn't change, but objects in the 3d world we perceive and time have different measurements for observers who are moving relative to each other.

 

I remember the amazing feeling when it came together in my mind.  It was early June around sunrise.  I got on my ten speed and rode as fast as I could.

 

 

 

 

springs, the coyote and the phrase "no, and..."

A magician, in the one that usually comes to mind, uses an unrevealed trick to make something unexpected happen.  There's skill on the part of the magician and often a sense of how people perceive things.   Not being skilled at the slight of hand, I see another kind of magic - one that doesn't on a skilled trickster with a secret.  

 

Most of you have played with that long, thickly coiled, floppy spring known as a Slinky.  They're wonderful. You can make them walk down stairs, watch waves shoot back and fort, construct squirrel-proof bird feeders, lots of things.  And you can perform magic - that second kind without skill or a secret.

 

I stand on a stool and hold one high above my head and let the bottom drop as far as it will go.  I ask them to watch paying careful attention to the bottom. 

"Three, two, one, drop!"

What they see doesn't make sense. For about a third of a second the spring's top drops like you would expect while the bottom appears to levitate in mid-air as the Slinky collapses down on itself.  When the top crashes into the floating bottom the now compessed Slinky falls to the ground like anything else.  I usually have to do this a few times and then they demand trying it themselves.    

 

Some years ago I was on a flight to California with a bad case of speaker's block.  In less than two days I'd be on stage at an animation company - a physics type talking to artists and computer science people.  Sitting there at 35,000 feet I realized what I had prepared was pretty bad.  

 

The sky was crystal clear and over the Southwest the talk I needed came to me.  I'd talk about cartoon physics - the license good animators take moving objects around in the worlds they create.  Seeing the Grand Canyoun suggested the Road Runner the Wile e Coyote.  I scribbled a half dozen obvious things and had my talk before we landed.  The next day I ran it by a few senior animators who helped me find some clips. All that was left was finding a Slinky.  Someone had one in their office.  I was set.

 

The talk started with the Slinky drop.  It was doing exactly the opposite of what Wile e does when he suddenly realizes he standing on nothing but thin air high above the canyon floor.  His ears remain fixed while his body falls.  A Slinky is the un-coyote.  Both break our idea of how Nature should work.. one is right, the other fantasy.

 

And a little epiphany a few years ago.  I took Alan Alda's short course on science communication. He uses improv acting exercises to help people carefully listen to and establish trust with each other.  A central idea of improv is "yes and.." - you accept what the other person has said and then expand on it.  It's a powerful technique and in the hands of skilled actors can be astonishing.  

 

It struck me the "hard" sciences like physics are different.  Advances in physics come with  "no and..." constructions.   You see something that doesn't seem right -  how a Slinky falls for example.  It breaks your model of how things work.  "No!"  something's wrong with your model or your observation.  You have to sort things out -- the "and" part.  This is where discovery comes.  "No and" has been a fundamental driver of scientific knowledge.

 

So what's up (or down) with the Slinky?

 

It's a simple pre-tensioned spring.  Its natural resting state is where all of the coils are touching.  Set it on its side and it's fully compressed.  When you hold it up it stretches to where the pull of gravity exactly matches the upward tension of the coils.  Release the top and the bottom just sits there as the top accelerates toward it.  The upward pull from the tension in the spring keeps balancing out gravity.  The bottom of the spring is the only part that wasn't deformed so it has no pull.

 

Working out the physics is a bit beyond first year mechanics, but if you have that under your belt it's fun.  What you learn is the levitation time is depends only on the stiffness and mass of the spring.  A standard Slinky has a 0.3 second hang time on Earth, the Moon, Jupiter or whatever..  Of course the center of mass of the Slinky falls exactly as if it were a point object.

 

For the record I've always liked the coyote more than the roadrunner.  Inventiveness and perseverance no matter what.

 

 

local mobility

Our family physician sent a note to his patients over 40 recommending cardiovascular exercise and losing excess weight as additional defenses against COVID-19.  They won't prevent you from getting it, but you'll be in better shape to fight it.  Among other things he recommended dusting off your old bike for trips to the store.  

 

I've been an advocate of human and human-electric hybrid transport for approximately forever.  Of course transportation is a system rather than a vehicle and countries like the US and Canada much needs to be done. That said, local errands and commuting are within the range of many.  There's an enormous amount of new activity in and around this area these days. Unfortunately there's a lot of bad information and products that won't meet your needs.  You need a compass to find your way.  I'll recommend two that are gold standards.

 

Horace Dediu has been doing great work promoting micromobility - vehicles from two passenger 500kg intracity electrics down to scooters as well as the physical and information infrastructures that support them.  He has a blog,  podcast, and newsletter. If you're looking for where all of this is going and it's impact on society and business, Horace is the best person around. (I'm biased - he reads this blog:)

 

But what if you're looking for something like a cargo bike?  There's been an explosion of new models, both pedal only and pedal-electric hybrid.  Unfortunately there's a lot of garbage and sorting the wheat from the chaff is difficult task.  Most bike shops cater to people interested in recreation.  They might be great at repair and understanding carbon-fibre frames and aerodynamics, but practical cargo bike experience is hard to come by.  For this flavor of micromobility my strong recommendation is Arleigh Greenwald a.k.a. BikeShopGirl.    Her page has links to her blog, useful how-to podcast, honest cargo bike reviews and more. Arleigh has huge amounts of practical experience and as a mechanic knows how well various models will hold up. She's also into getting kids started on bikes. Just consult with and buy from her if you're close to Denver.  She offers consultations for local people .. if you're out of the area you might try and write and offer to pay something for her specialized wisdom.  

 

Of course you can just dust off that old bike for some exercise to keep from going stir-crazy.  A good bike shop can tune up anything for not that much.  

 

 

a 33 year old olympian writes a letter to her younger self

Sarah's letter is excellent - even if you're not a sports fan. 

It seems like a good exercise in self-reflection and I briefly considered writing one to myself.  Of course if I received a letter I'd think: "OMG! time travel exists!" which, of course, it doesn't, but..   I'd probably tell myself not to focus on a goal, but rather pay attention to the path.  Next would be to recognize serendipity in the sense it was originally intended - to have the sagacity to recognize and opportunity and act on it. Listen more and finally become more well-rounded - to have more range.  In the end my life has played out very differently than I expected and has been much more interesting.

Quarantine or not, many of us will have a bit more time to reflect.  Writing a letter to your teenage self might be a good way to spend some of it.

 

a dive into pizza pt. 2

Now that we know why we hold pizza the way we do, it's time to move on making a great pie.  There are dozens of varieties. I'll stick with one I have some (perhaps too much) experience with - Neapolitan.  A few  other thin crust styles like New York (which has a higher fat content in the dough) will be somewhat similar, but deep dish and other types have dramatically different baking requirements.

 

Folklore and my experience suggests a brick oven does the best job.  These are usually a set around 330°C (about 625°F).  The cooking surface is brick with a large brick dome covering the space.  Many of these use burn gas, but wood is sometimes used to impart a bit of a smokey flavor.  Much more common are commercial steel electric or gas ovens.  Chain pizza shops? I'm not a fan.

 

There are three heat transfer mechanisms: conduction, radiation, and convection.  Conduction is heat transfer by direct contact between the oven and pizza bottom. Radiation is infrared energy that largely comes from the rest of the oven - the broiler element in your oven uses this trick. Finally convection is the transfer of heat from the oven to the pie by the air.

 

Convection turns out to play only a minor role so we'll skip it.  Conduction is another matter.  In our 330°C brick oven the interface between the pizza and brick is about 210°C (410°F).¹  In about two minutes the bottom cooks nicely and the temperature at the top of a half centimeter thick pizza approaches the boiling temperature of water.

 

Most pizzerias use steel pizza ovens.  Steel is a much better conductor of  heat than brick.  With the oven set at the same 330°C,  the temperature at the bottom of the pizza is now much higher - about 300°C (570°F) and the bottom burns quickly.  Dropping the steel oven's temperature to 230°C (445°F) will give you the good 210°C crust cooking temperature, but there's another problem - what about the top of the pie?  For both brick and steel ovens we need to look at radiative heat transfer..

 

Infrared radiation from the 330°C brick walls quickly raises the temperature of the top of the pie to the boiling point of water allowing water in the cheese and toppings to evaporate so they can cook properly.  For a simple Margherita pie two minutes gives a nearly perfect "balance" with everything cooked just right.  The cooler 230°C steel oven has a much lower radiative heat transfer.  So low the top of the pie doesn't reach boiling by the time the bottom cooks  Cooking the top requires another 90 seconds or so leaving the crust overcooked.

 

You can partly get around these problems by adjusting the dough (New York style pies have a higher fat content), using a broiler element (some steel ovens have them) or physically lifting the pie from the oven floor with a wooden pizza spade for a minute or so.  A good pizza cook with a brick oven will also use the spade on a Neapolitan pie if there are a extra toppings.

 

If you're lucky enough to visit a shop with a brick oven and good cook, try to go when they aren't busy.  It is common to raise the temperature to at least 390°C during busy hours to increase throughput by turning around pies in about eighty seconds.  These fast and hot pies are difficult to judge and frequently have slightly burned crusts.  It's even harder if ask for anything more than the simplest pie.

 

You can get a ceramic or brick pizza stone if you want to back a pie a home.  Set your oven as high as it will go and experiment - it will take more cooking time.  I've tried with lackluster results .. I tend to wait and visit real pizzerias.  Alternatively you can work on types of pizza that have very different cooking techniques. 

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¹ You can calculate this with a differential equation using the thermal properties of the brick and pizza. It gets a little messy and the 210° figure could be off by a couple of degrees, but it's in the right ballpark and that's a good temperature for baking the crust.

 

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. 

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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.