omenti

measuring coastlines to helping someone see with sound

 Late in May of the year I was in forth grade we were given a map of the United States and Canada with outlines of the states and provinces.  Our task was was to correctly label it with capital names and rough locations along with the names of states and provinces.  We had a few for practice and the shapes caused my mind to wander.

 

States like Kansas, Colorado and Wyoming, were boring rectangles.  Or were they?  The supposedly parallel lines on the East and West would converge on the North Pole.   My Dad had had this neat little mechanical map measuring tool.  You'd zero it and then mechanically trace out a path and write down how many millimeters it had gone (his was German and labeled in millimeters).  Then you'd look up the scale on your map and convert to miles or kilometers.  I quickly learned the school maps weren't very good and started playing with road maps from the gas station.  In the day you'd pull up and have someone pump your gas, clean your windows, check your oil and water and tire pressure and then pay about thirty cents a gallon for the gasoline.  They had a pile of maps for the asking. I had a nice collection.

 

These were better, but didn't answer my basic question.  I was starting to wonder what represented ground truth.  If you're admiring the  Slartibartfast's crinkly edges of Norway you have to wonder about scale.  From space there may not be as much detail as from an airplane and you will get somewhat different results.  Down to ground level, where you can easily measure in centimeters rather than meters, it's much more complicated and is now dynamic with forces like the tides, continental drift, erosion and so on.  Where you do draw the line, so to speak?

 

At about the same time I was getting boggled, Benoit Mandelbrot was investigating fractal geometries.  He was something of a polymath and the only person I've met whose personal style included a piece of fresh broccoli in his suit pocket.   He loved self similar shapes..  A lot of math is just drawing things to see what's going on - a neat little example is the Koch curve.   Draw an equilateral triangle and the add three a smaller triangle with a sides a third the length of the original.   Now place each midway on the original triangle with so they're pointing out and just keep the shape along the outside boundary (you may want to think of cutting them out of paper and just using the outer edge.)  Now it looks like a Star a David, but you've just started.  Add new even smaller triangle along each of the edges of the Star of David and you start to get a snow flake.  Keep going. (a computer may be useful:-)

 

The edges are much more complex that anything smooth like a line or a circle.  Mandelbrot and others came up with a way to assign a dimension to any shape.  A line has one dimension, a plane has two, the space we're used to has three and math is just getting warmed up.  The Koch curve has a dimension of 1.2619.  The definition give 1.0 for lines and 2.0 for planes, but is also a way to talk about complexity and repeating complexity.  

 

Just in case someone asks,  the dimensionality of the Southern Norwegian coastline is about 1.3.  The Southwestern coast of Britain is very close to 1.25. 

 

Drawing these curves by hand or computer could and should be taught in middle and high school because it's playful.  Understanding how to calculate dimensionality of these shapes would take too long for me to describe, but I think it could be done as part of a high school calculus class.  Playing with these concepts builds your tool kit.  A tool kit that allows you to stitch seemingly unrelated concepts together.  

 

Our brains are remarkably "plastic."   If you push them with new mental activities they rewire so as to become more efficient.  Profoundly deaf people use some of the brain normally associated with sound to increase their visual acuity.  Musicians enhance regions of their brains and the mental component of many sports leads to very measurable changes. If you work at several different things your wiring may become uniquely creative -  I suspect some of the most creative athletes spent a lot of time studying an instrument at some point. These enhancements are readily seen in brain scans even in people in their sixties. It's easy to measure the changes even in people who are in their sixties.  You can teach old dogs new tricks.  (this is such a rich area, but back to the equally rich math)

 

What if you wanted to help a blind person by encoding the information from a camera in sound?  How would you approach the prolbem?

 

To get an idea of what's going on I take a very simple picture..  in this case 256 pixels on a side.  I start with my avatar. (pretend it's 256 x 256...  this one is twice as large)  I make it black and white..  256 grey levels in this case.  I don't know what to do with more detailed information...  I'm just dinking around a bit with the problem.  

 

How to present this to the brain? Don't worry if this would actually work - just consider it a little thought experiment.  We could associated each pixel had a frequency?  The loudness could be associated with the brightness.  In this case white could be very quite and black the loudest.  You might play them all at once, but it might be too confusing..   Try adding a bit of order.  The question is what kind of order?  (again. . there are smarter ways to do this - you may be thinking of some now, but let's work on this one)

 

It could be some randomly chosen ordering.  After all, the brain is good at finding order in a mess.. as long as we use the same random ordering.  But that presents a problem..  what if we use a better camera..  like 3000 x 4000 pixels?  Then we need a new random ordering and the brain has to start all over again.,

 

There is raster scanning .. like an old CRT ... you move back and forth t across the image starting at the bottom and shifting up a pixel each time you reach an edge reading the brightness of each pixel you encounter until you've scanned the whole image.   Simple but it throws away some good intuition.  An image tends to have a lot of neighboring similarity.  It would be nice to make use of that. Hang onto that thought for awhile.  

 

A mathematician might suggest you consider a Hilbert curve -  more specifically a Pseudo-Hilbert curve. Rather than get impressed by the fancy name we'll drop the pseudo and just play around.   Take out a sheet of paper and draw a square.  (line 1 in the figure) Now divide the square into equal quadrants.  Starting in the lower left quadrant, draw a line up to the upper left quadrant, continue to the upper right quadrant and then down to the lower right quadrant.   This upside down U is an Order One Hilbert curve.  Now it's time  to make it work.  (line 2)   Draw another square with four quadrants.  Now draw an Order One curve in each of the quadrants.  Start traveling from the lower left and you find yourself making some unfortunate traverses.  Is there any way to rearrange these Order One curves to get something that doesn't intersect itself?

 

 

 

 

It turns out there is.  (line 3).  Rotate the curve in the lower left quadrant 90° counter clockwise and it's neighbor in the lower right quadrant 90* counter clockwise.  Now you can connect all four Order One curves into something that doesn't cross itself - an Order Two Hilbert curve. You can probably guess the algorithm from here.  To draw the next order draw your current order Hilbert curve in each of four quadrants of a square.  Rotate the bottom left curve 90° clockwise and the bottom right 90° counter clockwise.  Connect them and you're done.

 

Now that you've got the recipe you can draw Hilbert Curves of any order your patience allows.. Past four and it's best to use a computers.  Here are Orders One through Five

 

 

There are two things to notice:

 

First the curve is covering more points of the square as the order increases.  We can talk about it's dimension.  As the order goes to infinity, the curve's dimension is 2.0 ... in other words the curve hits every point inside the square.   How sweet - an infinite one dimensional object that covers two dimensions.  Our job is much easier - we're using pixels and only require the line to go through a pixel.

 

Second, the curve preserves localized information as the order of the curve goes up.  Look at the nose of my ferret. If I stretch out the curve the points associated with the nose are roughly in the same region of the line.  This is useful if our blind person is listening to the string of notes coming out.  Most of them around the nose have a similar value and are played at nearly the same time.  If we use a camera with more pixels that local nature is still preserved.  The raster scan that snakes up the image don't preserve locality.

 

There are many uses for these locality preserving mappings (as they're called) and space filling curves like the Hilbert Curve are just what the doctor ordered.  And you don't have to stop with one dimension. They may get hard to visualize, but the technique works with as many dimensions as your heart desires.

 

Math is useful for calculating, but more important is the fact it's a playground that offers insight and new ways of thinking.  It is one of the glues of creativity.

 

 

the barking dog catches the car ... what next?

The 19^th of October is a big day for me.  It marks the beginning of the only time I've been close to being depressed.  A depression triggered by something that should have been a celebration.   That afternoon I had just defended my Ph.D. thesis.

 

The audience,  while nothing like Harald Bohr's, was full of friends possibly lured by the traditional cake.¹ The committee - my advisor, a theorist and experimentalist from my department and a professor of surgery who was also a classically trained violinist.  The surgeon who took his duty seriously and I had spent many hours teaching him some particle physics.  He rewarded me with some of the more penetrating questions of the afternoon.  Years later I returned the favor serving as committee member for one of his students.

 

The cake vanished and I was supposed to be preparing for the big celebration with friends.  Instead I just sat in a quiet room.  I felt directionless and a sense of despair  built.  It made no sense.  This had been a goal for years and the defense went well.  I had a post lined up.  The feeling darkened and I didn't feel like celebrating, eating or even moving.  I just sat there with a piece of cake next to me.

 

People found me and tried to pull me out.  I walked back to my place and didn't come out. The big celebration never happened.  People kept checking on me.  It went on for nearly three days.

 

The path began the month I turned twelve.

 

The structure was small but, despite the amazing view, there was only one small window.  A curtain kept out sunlight. A cluster of electric cables connected the building to the cable car building a thousand feet distant.  Twelve and thirteen is an age when most of us are still curious and society is still encourages it.  I knocked on the door.

 

My family spent a few weeks each Summer anywhere from the Bob Marshall Wilderness Area in Montana up to Jasper in Alberta.  That year it was Banff.  We had taken the Sulphur Mountain cable car to the viewing platform.  On a clear day the view was nearly a hundred miles.  The observation deck was crowded to capacity with tourists from all over the world.  My parents and sister were talking to a couple from England when I left for the unusual building along the mountain's North ridge.

 

Physicists have a primal need to explain things as deeply as time and their audience's patience and curiosity permit.  Almost immediately I was learning a bit about cosmic rays, muon showers, scintillators, photomultiplier tubes and so on until my father showed up.  He had been looking for me for a few hours and happy wouldn't be the right word to describe his mood.  

 

The ride down the mountain and back to our camp was quiet - I knew better than to say anything.  Nothing else seemed to matter at the time.  I was completely engaged in what I had seem.  Most of it was over my head, but seeing the existence of the remnants of a cosmic ray shower on a simple detector someone put together solely to explain what they were doing hit me deeply.   I saw the beginnings of how people ask questions of Nature and it had nothing to do with looking up things in a book.  By the time Lyra was overhead I feel asleep knowing I would be either a physicist or an astronomer.  It was my calling. 

 

A month later I tagged along with my Dad on a trip to Los Angeles.  He was taking a ten day workshop for a certification and I was seeing California for the first time.  Friends and family lined up trips to Disneyland, Knott's Berry Farm, Hollywood, and the beaches - the things kids are supposed to see in Southern California.  To Dad's dismay I wasn't interested and somehow managed to get my way.  Instead I visited JPL, had a professor explain something about stellar explosions on his blackboard at Caltech, and visited the Griffith Park planetarium..  Gateway drugs all..

 

By the time I entered college I was focused on the narrow path of math and physics and didn’t let much else get in the way. I wasn't aimed at a career as much as just wanting to learn more, but somewhere along the way the title had become a goal. And all of a sudden I had it. All that work for something and it felt so hollow.

 

What finally worked came from a wise Scottish friend.  He said something like ~ "So you've managed to catch the hubcap you were barking at for so long.  Rest assured there are an infinite number of hubcaps of types you haven't imagined.  May you find a few."²  I didn't realize I was just at the threshold of the always new.  I was at a point where it was recognized I could ask and work on interesting questions. 

 

Some of you are working towards greatness.  Some are arguably there, not that the distinction matters.   It's just part of your path ..  a visible point where we take stock of things.  Once important and (hopefully) a just a happy memory as new hubcaps begin to glow.

 

 

One of these days perhaps I'll celebrate mine and I'm thrilled to celebrate those visible points along the paths of friends.

 

note:  Some of you have struggled with serious depression.  I don't discount those experiences.  My modest skirmish was a profound experience for me and continues to make me think.  I can't imagine the impact of deeper experiences.

__________

¹ Harald was the brother of the great physicist Niels Bohr.   He was an elite footballer representing Denmark in the 1908 London Olympics as they took they the silver medal.  He became a national hero and his Ph.D. thesis defense in math was so packed with football fans that it had to be held in a large auditorium.

² I translate  roughly - he suggested a proper Hilbert space of hubcap types.  Anything Jack suggested other than food was worth following.  His advice was spot-on. 

 

 

 

the first italian-american talkie of the universe

One hundred and thirty million years ago dinosaurs were a little more than halfway through their stay on Earth.  Underneath their feet the first plants that relied on animals to spread their seeds - the first flowers - bloomed.  Flowers had an enormous impact on life on Earth - greater than the dinosaurs - but during this long ago time in a galaxy far away a pair of neutron stars danced.

 

As with all neutron stars they had had seen remarkable times.  The result of supernova explosion, all of their matter had been compresed into a city-sized sphere with a density three times higher than the nucleus of an atom.  No electrons or protons, most of their interior was nothing but neutrons.  While violent, the process releases many of the elements that make up you and me.  In fact you are probably the result of at least a half dozen supernova explosions.  Put in pin in that thought and we'll come back to it.  

 

The pair, one of them with a mass one point one times that of our Sun, the other one point six, orbited a common center of mass for a long time radiating weak gravity waves over time progressively getting closer and closer orbiting faster and faster.   The process sped up dramatically towards the end.  Their last minute saw them moving close to the speed of light completing about fifteen hundred revolutions and in the process shaking the fabric of spacetime. 

 

Then they fused.  

 

These new fangled flowers worked out how to attract animals and the animals went about their business  completely unaware that something interesting happened out there.   

 

The signal was first detected on August 17^th during a normal look at interesting event triggers from the Fermi Gamma-ray Space Telescope.  Basically an orbiting cluster of  large chunks of sodium iodide crystals and bismuth germanate crystals, it detects and measures gamma-rays from ranging in energy from about ten million to a hundred and fifty billion times the energy of a photon of red light.¹  The Fermi looks at gamma-rays from exotic electrical storms on Earth, open air nuclear explosions, and the most violent events in the Universe.  Once every few days something thought to be a gamma-ray burst from the collision of two neutron stars takes place.²  This one wasn't particularly strong, but was noted and the team went back to work.

 

One point seven seconds before the gamma-ray burst detected by the orbiting crystals a gravity wave rolled through Pisa, Italy changing the dimensions of everything generating a signal at the Virgo gravity wave detector at Cascina. Twenty-three thousands of a second later the wave rolled through Livingston, Louisiana and continued along shaking Hanford, Washington two thousandths of a second later.  The quake in spacetime continued for over a minute. Soon the most carefully conceived data handling the world has seen indicated a gravity wave had rolled through and the gravity wave folks placed a phone call to the gamma-ray community to check if they had seen anything 

 

The Italian site had only been active for a bit more than two weeks.  With the timing of two sites you can roughly determine where in the sky something happened.  Add a third and the uncertainty drops dramatically.   Numbers were crunched and an alert went out to the astronomy community to point anything they had at a patch in the southern constellation Hydra.  Over the next few weeks about seventy ground based and ten orbiting observatories were involved.  

 

We've talked about LIGO and gravity waves before  and soon a friend will be splashed with Swedish holy water for his part.  To date there have been four solid detections.  The nutshell is this is a new way of looking at the Universe. Gravity waves are ripples in spacetime, but it is so rigid that the distance between Earth and our closest star shakes about the thickness of a human hair.  Designing and building working detectors is my candidate for the ultimate human technical achievement to date.

 

What makes this one special is there was enough information to know where to point optical telescopes and the gravity wave signal is undoubtedly associated with the electromagnetic signal.  As the stars finished their dance they shook space time enough for us to hear it (I use that analogy as it is a compression and expansion of a medium just like sound is).  Then the massive gamma ray burst with a light show that went on for a few weeks.  

 

 a talkie - audio and visual!

 

An enormous amount of information was taken as the event progress. This image samples the range from acoustical gravity wave signals and various optical signals ranging in wavelength form radio to gamma-ray. 

 

So much has been learned.  It turns out (skipping the details) it's a good way to measure the Hubble Constant and with it hone in on the expansion of the Universe.  It also provides an accurate distance measure to a distant event - we have the red shift from the optical signal and luminosity from the gravity waves (again, ping me  if you want details, but it gets into too much physics for this blog - for the physics types the proposal is to call these standard sirens).  And Einstein was right...  gravity waves travel at the speed of light.³  We now know that to considerable accuracy.  

 

 ... one last thing...  You may be wearing gold.  A ring, a watch or some other jewelry. Gold has an atomic number of 79 making it a heavy element.  The Big Bang didn't create any and the burning that goes on in normal stars doesn't either.  You can produce very heavy elements in supernova explosions and this was the standard story until about five years ago.  Now it's understood the violent collision of orbiting neutron stars is a rich generator of the heavies.  They're blasted out in a jet that runs along the axis of rotation of the pair.  An event like this generates a mass of gold a bit larger than our Moon.

 

Most of the gold in your jewelry is from the collision of neutron stars.  

 

Just like what we heard and saw on the 17^th of August in that first talkie of the Universe.

 __________

⁰ ok  - the title has problems.  The acoustic part is gravity waves from the US and Italy, but the video part is from all over including Antarctica.  And researchers from all over have their thumbprints on all phases.  When you spend an hour writing, titles are issues.

¹ you don't want to expose your body to much of this stuff.  Biochemistry happens at energies close to that of visible light ..  much higher and molecules are damaged and atoms ionized.

² there are about one hundred million neutron stars in our Galaxy... way more in the Universe:)  About a tenth of a percent of the stellar population 

³ all massless fields travel at the speed of light.

 

 

 

 

 

 

the songs of quartz crystal that echo through your body

A good deal of fantasy depends on magical properties of crystals.  It's easy to see where that comes from.  They're often physically beautiful and can do wonderful things to light but there's no magic.  On the other hand some allow us to do what was considered fantastic. .  Quartz for example..  Squeeze a crystal and it deforms a tiny amount generating electricity in the process.   Tun electricity through the crystal and it mechanically deforms.   The piezoelectric effect (squeezing electricity).

 

Quartz happens to be one of the most common minerals on the Earth's crust.   It's a regular arrangement of silicon and oxygen atoms (SiO₂) that forms a repeating shape in three dimensions  forming the larger crystal.  In most crystals (metals for example)  the repeating structure is arranged symmetrically, but piezoelectrics have asymmetric structures.   A quartz crystal just sitting on your desk is electrically neutral.  Although the arrangement of atoms isn't symmetrical, the distances and angles are just such that all the charges balance out: a non-symmetric neutrality.  Squeeze it and you ruin this perfect arrangement creating a net electric charge.  It takes mechanical energy and turns it directly into electrical energy.  It works in reverse too.   Apply current to a crystal and the resulting change imbalance causes the whole crystal to deform just a bit.  

 

This is extremely useful and a good deal of technology is based on it.

 

Attach a quartz crystal to a needle and place the needle on a revolving phonograph record,  and you get an electric current that dances to the tiny wiggles the needle traces.  You can speak at or or sing to it and produce a current that can be transmitted or recorded.   The same crystal sings and listens. Put a piezoelectric on something that vibrates and you can harvest a bit of power -- potentially important for tiny wireless Internet of Things devices where batteries are impractical.  There are other technologies that can be applied, but quartz crystals can be very effective sound transducers when you move up to million of deformations per second - the land of ultrasound.

 

Light or sound waves at the boundary of dissimilar materials can reflect as well as transmit. They can change angle and even scatter into a number of directions.  Materials can be identified by observing what continues through or what is returned.  Animals are good at this.   Our brains process information from our eyes and ears to sort out a tiny bit of the reality that surrounds us.   (silly us - we consider that reality when, in fact, our brain is construction a good enough virtual reality).   If it's dark we can use a flashlight (or torch for those of you in the UK), but some things are frustratingly out of sight

 

Imagine feeling yourself and finding a subsurface lump.  What is it?  Is it just your body messing with your mind or is it something serious?    It may only be an inch or so below the surface but X-rays may not good enough to diagnose and physically accessing it isn't exactly fun or safe.  You need a safe "flashlight" that lets you see what's below.  

 

Medical ultrasound is a flashlight that can be effective in soft tissue and even bone.  A burst of ultrasound travels through you and does something when it encounters anything different.  The busts are repeated regularly bounding comparatively long periods of silence when the crystal that generated the sound is listening.  A clock starts when the sound is emitted and the round trip time for each bit of sound reported by the crystal, along with it's strength, is recorded.    A distance for each echo is calculated - just half of the time of flight times the speed of sound in the tissue.¹  A digital map is made and  usually displayed in real time on a video scree in a form more useful for diagnosis. Processing is usually done on a PC.

 

A critical bit hasn't been described.  How do you make an image with more than one dimension?  In the old days the crystal was moved mechanically.  Getting this right and reliable was expensive and the scanning heads weren't easy to handle.  The innovation was to use a number of crystals in a phased array.  

 

Dip two fingers in still water at the same time you'll see expanding waves moving out from contact with interesting interference patterns forming where the waves intersect.  If you have a lot of equally spaced fingers you'll find that the interference conspire to form linear waves.

 

An ultrasound array many have fifty or a hundred crystals in a line.  Here the transmitter (Tx) generates the power to make the crystals sound off and the little antenna thingies are the quartz crystals. If the crystals all sing at the same time the flat wave from moves off to the right.  But if you apply a bit of delay to each crystal increasing it as you move to it's neighbor, you can control the angle of the wavefront.  Smoothly change the angle and you sweep out a beam.  

 

Make it a two dimensional array and you can scan out an area with the time of flight depth information adding the third.  You can even get clever with the phase delay at each crystal and focus the beam.  Some ultrasound machines only use a linear array.  It's application and cost dependent. 

 

Resolution improves with increasing frequency, but range decreases.²  A range of frequencies are used depending on the task at hand.  I was given a tour as reward for telling the operator what one of the display algorithms was doing.  We looked at several parts of me with a $150,000 brand new machine and used frequencies ranging from 2.5 to 18 megahertz.   A six and a half octave range ...  most pianos are seven.

 

A singing quartz crystal flashlight that augments the reality you perceive.  

 

Oh - and the special lubrication they use.  It has a speed of sound similar to human tissue to minimize reflections.  Trapped air would produce nasty reflections coming and going - it's all about minimizing noise and maximizing signal,.   You can buy expensive ultrasound gel, but I'm told most people find K-Y Jelly is an inexpensive substitute.

 __________

¹ This varies with tissue type and increases with rigidity.  Bone has a much higher speed of sound than soft tissue and breasts tend to be a bit slower than muscle.  The average speed for tissue is about 1540 meters per second.  

² Specifically resolved size goes as the inverse of frequency.  At the commonly used 3 megahertz resolution in tissue is about a half millimeter.  That tells it it is overkill hoping to reliably see smaller detail.