omenti

perspective

Some details, sadly including her name, have faded but others are crystal clear.  My eighth grade art teacher gave us something profound.  Interesting at the time, but years later I began to appreciate how profoundly beautiful it was.

 

Other than a straight edge, pencil and a sheet of paper our desks were bare.  She asked us to stand facing the window.   "Close your eyes"  We were to imagine being on the prairie looking an our feet. I opened an eye and saw most of us were looking at the floor.  We were to imagine the line between our eyes and the point on the ground.   Then we were to mentally draw a line to a point on the ground some distance away.  "Don't worry about the numbers, but think about the angle the line makes with the ground..."  Then we looked ten feet out, twenty, fifty, a hundred feet, the length of a football field.  "What are the angles doing?  How fast are they changing?"  "a mile, ten miles, a hundred miles..  "how are they changing now?"  Now off as far as we could imagine vanishing off into space. The angle was changing so little as we went further away.   "Open your eyes"  It may have been an eighth grade art class, but I felt myself gasp. We were looking straight ahead to the horizon and had touched infinity.  

 

Next she showed us a photograph of railroad tracks receding into the distance.  She asked us to close our eyes and scan along each of the rails and notice what happens.  Of course they meet at infinity.  But parallel lines can't do that.   What if there were tracks a hundred feet to our left?  She gave us our first lesson in projective geometry and none of us realized it.

 

We sat down the the straight edge and she showed us how to construct a grid of equally sized squares without resorting to numbers  It was just drawing straight lines to the horizon and intersecting them with straight lines from other points.  Very neat.  That consumed the period.  I couldn't wait for the next class.,

 

A few days later we met again.  Now we were to imagine a semi-transparent piece of paper between us and perpendicular to the ground.  We were told to think about where the lines to the "vanishing point" - the artist's infinity would intersect the paper.  Then she took out a cheap print of a Renaissance painting, I remember it as the Giving of the Keys to Saint Peter by Perugino, and drew the perspective lines on it.  She told us the great artists of the Renaissance had been struggling to create the most realistic painting possible and had developed perspective to bring some of the three dimensionality of the world onto a flat two dimensional sheet of paper. 

 

My mind was blown.  I started seeing the world differently.  I knew that it had three dimensions and was Euclidian, but when you looked at anything a bit in the distance the two dimensional approximation of perspective was apparent.  This was a powerful tool.  It explained why movies and photography looked realistic.  It started me thinking about what a lens does.  

 

Years later I learned more about projective geometry - an area of math pioneered by the Renaissance artists.  I can't appreciate it to the depth of an artist, but it is a good metaphor for other things.

 

Some concepts in physics seem strange - even bizarre - to non-specialists.  Examples and popularizations often aren't clear enough, leading to even more confusion.  The trick is to find metaphors and analogies that preserve enough of the core ideas while being easy to understand.  Projective geometry - perspective to an artist - is rich ground for tackling some of the basics.

 

Four that are central to physics and art come to mind:  relativity, invariance, symmetry and complementarity.  Forget about any associations you might have heard about those words and physics (especially relativity).  This is more of a fundamental level of how to look at Nature in general and learn to ask the right questions.  It applies to art and probably many other fields.    So taking them one at a time:

 

Relativity You can paint an object from many perspectives.  All of them represent the same information about the object, it's just they're encoded differently.  The same subject can be faithfully represented without any loss.  (for physics think about simultaneity)

 

Invariance  Bits of the subject can look different when you change your perspective, but there are some that never change.  Consider straight lines on the subject that meet at a point.  The fact that they intersect at a point doesn't change in different perspectives,.  Those features that don't change in all possible representations are invariants. 

 

Symmetry  Now think of moving the subject rather than the artist -  rotating a subject on a turntable for example.  Arbitrary rotations may appear different, but the the sum of all views  from all possible perspectives doesn't change.  The sum of all of the perspectives is the projective description.  Here rotation is a symmetry of the projective description.

Another way of looking at this is to consider a few special cases rather than the totality.  Consider rotating a circle looking at it from one perspective - standing over it for example.  No matter what rotation you make - 2°, 180°, 42°, 234234234234234° - it looks exactly the same.  A circle is symmetric under rotation.  Now think of an equilateral triangle.  It only looks the same if you rotate it 120° - it has a three fold symmetry under rotation, a square has a four fold symmetry that pops up ever 90° of rotation  Symmetry operations can be thought of as change without change.

 

Complementarity  This one is related to the story telling from a few posts ago.  There can be many views of the subject that are equally good, but when you describe the subject you usually have to pick one - hopefully one that makes sense in the context of how you describe it.   This seems almost simple at first, but consider trying to take a picture of something very small - say an electron.   The electron has a position and some momentum.  When you make your picture you have to shine some light on it.  The light its own energy and momentum and, in the process of interacting with the electron, manages to muck things up.  If you're really clever you can set up your measurement to focus on one aspect of the electron - say it's position.  Now the interaction will disturb the electron's momentum a great deal.  You can also try to focus on it's momentum, but that will disturb the electron's position - ah the uncertainty principle.  The point is you decide which description you want and the other descriptions are rendered invalid.  Picasso would not approve.

 

Math can render physics incomprehensible to non-specialists, but principals common to art are often involved at a deep level.  It's nice to frame them in terms of something much more tangible - perspective in art.

 

And if you wonder why I keep arguing for education in the arts...

 

 

 

the dawn of programming to serious hidden tech

I love the music, but usually ignore the technical part of SXSW, but this year a few people reported progress in the Google's Project Jacquard initiative with Levi's.  It's an interesting project in a potentially rich area where most have been failing - weaving computation into clothing. 

  

At the beginning of the 19^th century Joseph Marie Jacquard invented a powered loom that could manufacture textiles complex patterns.  An amalgam of inventions going back nearly a century, the cleverness was a chain of cards. Punched cards laced together forming a sequence of instructions - a program if you will - that instructed the machine how to weave.  These sequences could be arbitrarily long.  Suddenly brocades and damasks could be made at scale. Jacquard's name became something like Xerox - a company, a machine and a process.  Like Xerox the term became generic.  

 

The complexity of patterns could be astonishing.  Jacquard was an inspiration to Charles Babbage who commissioned this portrait to be woven in silk on a Jacquard machine.  Jacquard is frequently mentioned as one of the earliest bits of computation.  An interesting symbol for Google's project.

 

Mixing local computation and fashion has been a tough area. Most companies have been trying to show off their tech paying little attention to use and practicality.  Whatever is done it must be useful at some level,  machine washable several hundred times, comfortable, easy to manufacture, inexpensive and aesthetically pleasing to some non-trivial segment.  Personal style is important and no one worries about what kind of ARM processor is up their sleeve.

 

The Google-Levi's collaboration aims to weave touch and gesture into a variety of textiles using modern industrial versions of Jacquard looms.  They're currently working with a Japanese firm that does 3x1 denim for Lanvin and Prada.  The reports suggest the wires are almost invisible and the material still feels like 3x1 denim.  Taps and gestures are sent to a little module that talks to your smartphone via Bluetooth.  (it has to be removed for washing) The jacket gives some tactile feedback with a little vibration and it also lights up in different colors.  It's a simple input and display.

 

The first product due out in the Fall is a $350 cycling jacket - a variation of their Trucker Commuter Jacket which currently aims at the twenty something urban cycling market.¹ It strikes me as an interesting interface research project (something I've been involved in at times), that is narrowly focused at this time.  Software would have to be tailored to a variety of needs - of course that is possible, but to get it right isn't trivial. Reports were that Google's software was incomplete and a bit rough.   I have to wonder if augmented reality from something like Apple's AirPods makes more sense for this niche.

 

It is interesting that the killer app here is the same as what many claim for the smartwatch - fitness.  These devices are really only useful as approximations of what your body is doing.  The relationship between work and weight is poorly understood and the wrist isn't necessarily the best place to measure work.  The measurements can be very useful if it is an encouragement, but they need to be taken with a grain of salt.  It's unlikely athletes will take a serious interest at this point.  

 

A more accurate fitness future may be suggested by Leomo's Type-R with it's small network of wearable sensors.  I think the future is much richer than just a watch and a phone.   A cluster of  appropriately located sensors using a local secure network to communicate with your smartphone.  A watch may well be part of this cluster.

 

But that is fitness.  Where is fashion - regular clothing - personal style?  I've been excited about developments in smart fabrics, both active and passive.  There's been little in the way of aesthetics.  Functionality will be addressed.  This is another post, but imagine if you could keep a person comfortable in a larger thermal envelope.    It doesn't make sense to heat and cool very large air spaces if you can keep people comfortable.  Comfortable, normal looking clothing that keeps a desk worker comfortable at 17° or 18° C in an office -- or at 28° or 29° C.  HVAC bills could be greatly reduced. Local humidity and UV exposure are also being tackled.  You can also make functional clothing that helps people with disabilities as well as the infirmities of age.  A long list of possibilities exist and the known unknowns and unknown unknowns lists are larger still.

 

This sort of work cries out for diverse collaborations. Google clearly has the idea by teaming with Levi's, but that's a baby step.²  I'd love to see some universities that are strong in materials get involved with clothing designers, engineers and others - I can think of some folks doing animation I'd like to see involved.

 

So cheers to the collaboration, but there is an enormous amount of blue sky and a $1.5 -$1.8T apparel market to think about.

 

And then there's the question of doing it with humanity.  The apparel industry has been involved in human right's abuises for at least two centuries.   

 

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¹ The promotion video

compare with the video for a similar Levi's commuter product that paints a different person  --  I don't know if this is by design or if accidental, but a very different message.

 

² The folklore is Levi's was not the first choice.  Experience in tech/design collaborations tells me they're at least as important as Google in this and possibly more so.

 

emergence

I love this image.  

Dama con l'ermellino is one of four works Leonardo painted of women.  I've only seen reproductions as the original hangs in a Polish museum.  The paint and canvas of the original have weathered the centuries and, with careful conservation, are supposedly close to what they must have looked like over five hundred years ago.   It's  a collection of atoms arranged in a very specific way.  In theory you could make a list of each atom and re-create a nearly perfect replica, but that's not the only way - the only story - we have to talk about the painting.

 

When I see it, I see a young lady from the Italian Renaissance with a ferret.  I've been around many ferrets so I know it's male and a bit overweight.  They look very comfortable with each other.  A bit of research tells me she's Cecilia Gallerani, the mistress of Leonardo's patron Luidovico Sforza - the Duke of Milan.  Her dress is simple - she is not of nobility.  A white ferret means there's a lot of symbolism of the time.

 

I didn't need to see the original arrangement of atoms. A very different arrangement causes the same story to form in my mind.  The arrangement can be very different - the pixels on your monitor for example.  Their position, color and intensity are sufficient and the arrangement doesn't need to be perfect.  Even a hand sketch might be enough.   

 

All of these stories are emergent properties of the original arrangement of atoms. 

 

Here's another example.  A few years ago I was in a large room with Juliette.  We were talking about complexity.  A quick mental calculation gave the number of air molecules in the room - about 3 or 4 x 10²⁸.   All you need to know is the type and position of each molecule and it's momentum vector, how the molecules react when they hit each other and, of course, the position of the walls.  Once this is known you simply plug these values into equations that describe how molecules behave when they bang into each other and run the simulation ahead in time.   Of course no one does that.  No computer is fast enough to handle the situation for the number of air molecules in a cup.

 

Fortunately there's another way to look at the air in the room.  It's a gaseous fluid.  We can take it's temperature with this amazing device called a thermometer that measures the average kinetic energy of the molecules. We can measure it's pressure and viscosity.  If it is compressed or expanded, say by opening a door, we can get tricky and use equations of fluid dynamics to note how it flows and say something about it's turbulence.  

 

The room was big enough to play volleyball in so we could use those same equations to calculate the lift and drag on a spinning volleyball coming off Sarah's serve and calculate the trajectory she sent it on.  Or we can sort out how a Dreamliner will fly before it is made.

 

Some of these calculations can be performed with no more than a slide rule or pocket calculator while others might require serious iron.  The point is we have a description of Nature at the macroscopic level that let's us calculate.  We can design airplanes or even understand why volleyballs often don't go where you might naively think they should.

 

Both descriptions are, depending on when they're used, accurate stories of how air behaves.  Each has a specific domain. The fluid description is used when we have a lot of molecules and we're worried about macroscopic objects - say anything larger than a millimeter.  The molecular motion description - physicists call it kinetic theory - works when the molecules are diffuse or you're at a small scale, but still larger than where quantum mechanics becomes important.

 

These stories use different words: density, viscosity, temperature and pressure in one case and position and momentum in the other.  The models and equations to think about them are very different, but describe the same reality.  They're both true in their own domain.

 

Fluid dynamics emerges from kinetic theory.  We usually think of going from microscopic to macroscopic rather than the other way around.  You can derive fluid dynamics from kinetic theory, but not the other way around.  

 

Different stories are not equally useful!  In kinetic theory we've got something that is broadly applicable, but difficult to compute.  In fluid dynamics we have something that isn't quite as applicable (although it is enormously applicable at the human scale!), but is computationally easy.   We pick the story that allows us to easily describe Nature richly. 

 

Other emergences aren't quite as smooth.  We can have atoms that form into groups of organic molecules moving from physics (several layers if we want) to chemistry.  Turning up the complexity dial we get into biochemistry as the molecules are substances associated with living organisms.  At some point complexity is sufficient that something is alive.  That's not exactly a clear-cut distinction.

 

We can look at the changes of stories as phase transitions.  Heat up ice and it becomes liquid water - there is a very well defined phase transition where adding heat energy to the ice no longer increases it's temperature but changes it's molecular structure.  We use different language and models to describe these two forms of H₂O molecules. A stream of water and a block of ice.

 

Sometimes the sciences are seen as different story types for looking at Nature .. physics → physical chemistry → chemistry → biochemistry → biology → psychology → sociology or anthropology   (these aren't crisp and can be reordered at times).   A string of emergences.

 

There are difficult fascinating questions - when does life emerge?  intelligence?  how about consciousness?   Very tricky and even the basic questions are somewhere between difficult and impossible to nail precisely.  What is real and what is imagined? No time to touch on them now... or on specific types of emergence, or sparse vs rich ontologies ... the area is fascinating.  Enough, as Jheri says., to make your head spin.  

 

People often say Einstein (it's his birthday today!) proved Newton wrong.  That isn't exactly right.  He discovered more general and  accurate versions of gravity and motion, but at the human scale we rarely need or even can measure, the added accuracy.  Computationally Newton's Laws are much more tractable and we usually use that story.  

 

Just pick the story that is appropriate for your needs - it's probably a good enough representation..  at least until you need something different.

the invention of the future machine

mauve

 

I found myself walking by the Fashion Institute of Technology on a very warm February day last week.   Even for me fashion and design students are easy to spot, but three sported solid mauve outfits which triggered a few neurons.  

 

If you've had organic chemistry, you've probably heard about mauveine.  William Perkins had been challenged by his professor to do something useful - namely synthesize quinine.  There were failures. In one the teenager noticed some purple bits of something when he washed the gunky blackish failure with alcohol.   He played around with it and found it to be an excellent dye.  Soon the first synthetic chemical dye was patented and entered volume production.   Mauveine was everywhere.   England's emerging middle class could afford color and fashion changed to reflect the new choices. 

 

Before the 1800s most of what we'd call science was done was done by solitary or small groups of usually wealthy enthusiasts.   In the early 1800s Germany chemistry in the support of consumer goods emerged becoming a big thing by mid century and spreading to England and France.  The product was any number of chemical "improvements" on hard to get or low quality natural products -  food products like beef extract, dyes, paints, and so on.  It became clear that a functioning industrial laboratory using the scientific method could be profitable - very profitable.

 

Around the time of the American Civil War England began to worry about it's place in innovation.  The 1851 Exposition was something of a triumph, but progress since then had slowed with Germany and France catching up and even surpassing them in many areas.  Something had to be done and the trigger was telegraphy.

 

First used as a signaling mechanism for railroads, telegraphy rapidly spread as business recognized its value as a general communication tool.  A transatlantic cable was a major goal in the 1850s, but early attempts failed with great financial losses.  Physicists identified the problem -  there was enough of a  basic understanding of signal transmission, but they didn't know how to scale it properly.  Also the people designing and building the cables had only the sketchiest notion of signal transmission and were making design decisions that were known to be bad.  The cure was to study the problem in a laboratory setting.  There it was possible to strip the problem down to its bare essentials and experiment at a practical scale away from the messiness of real world conditions.  It was possible to watch how it behaved varying one variable at a time.  A deeper understanding emerged with results that could be carried back into the real world where, mirabile dictu, transcontinental telegraphy finally worked.  More than a few businessmen profited handsomely not so much from the cable, but from the more agile business it enabled.

 

As scientists and engineers were learning about telegraphy in these university labs, a sizable group of young men (sadly women were rare) were being training.  Kings College and University College were major centers, but Cambridge resisted.  The maths department believed the function of a university was teaching subservience to your tutor.  An experimentalist is all about rebellion.  You were to take your tutor's word.   

 

Remember England worrying about it's place in science and innovation?  A Royal Commission on Scientific Instruction and the Advancement of Science was established.  It was chaired by William Cavendish Duke of Devonshire, a Cambridge trained mathematician who had become a successful businessman. Cavendish pushed a radical solution. Science education was to be made compulsory in secondary schools, new laboratories would be built using public rather than private money and laboratory science would take a seat advising public policy.  It was implemented in the mid 1870s with a scientific curriculum established in secondary schools along with professorships in science at the universities.  Students going to Cambridge began to demand laboratory science courses and degrees.  Resistance was futile.

 

Cavendish gave Cambridge a new laboratory named in honor of his physicist brother Henry Cavendish.   Perhaps his most brilliant move was attracting James Clerk Maxwell as the laboratory head.  

 

About a decade earlier Maxwell had created the first unification of physical forces and established field theory as way to look at Nature.  Enthusiastic about Nature in general, he worked on color vision among dozens of other things.¹  He had a belief that someone  intensely curious was better suited for physics than someone with pedigree.  Anyone was welcome, sadly as long as they were male - he had trouble being around women, and the natural sciences took off at the school.  To say it was a success would be understatement in the extreme.

 

In under a hundred years the way science was done had changed dramatically.  Originally it was accessible to untrained people who had an interest in the Natural world.  It was small and poorly organized.  It had become a sport where rigorous training - six to eight years of post secondary education at a minimum -  was necessary.  Laboratories had groups  of scientists and centralized apparatus.  A class of technician who knew the apparatus deeply emerged.  Serendipitous communication became common and it was relatively easy to perform experiments with apparatus that was closely matched to what you wanted to do.  The flexibility was astonishing.  

 

Having this specialized space was of central importance.  A great example is the X-ray.  J.J. Thomson of Cambridge heard about the German discovery and within two weeks he was able to create his own X-rays and, building on existing work he had done on charged gases, managed to discover the electron.   

 

The labs had directors with a vision for where the future might lurk, a community of local experts with a mix of skills, the flexibility to change direction quickly and a community of support staff who build, run and extend the experimental apparatus. 

 

These groups created the underpinnings for most of the technology we know today.  I would claim the emergence of these great university and later government and a few pure research laboratories was one of the defining developments of the 19^th century.

 

Sometimes discoveries takes decades to gel or unite with other discoveries before they make the leap into technology, sometimes it is very rapid and often it is the stuff of knowledge and beauty.  There is little way of knowing this upfront.  Unfortunately the move to make research more productive is shifting its focus from pure to applied research.   Both are important, but both are essential. 

 

There has been a cost.  Science developed it's own language.  It became so specialized that it took expertise to communicate results to the public.  Something was lost.  People are working on re-establishing communication.  Unfortunately that hasn't been very effective - at least in the US.  

 

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¹ It was tempting to title this post something like a color purple, but purple isn't a color in the sense of colors having a certain wavelength.  Rather it is an artifact of your mind created by two or more wavelengths striking your eyes,

Recipe corner

 

I'm still working on non-dairy "ice cream".  This is almost there and I can recommend it as a starting point.

 

Creamless Banana Ice Cream

 

Ingredients

 

° 5 ripe bananas -- like almost oozing ripe

° 2 tsp cinnamon

° 1/2 cup coconut milk (full fat)

° 1-1/2 cup almond milk

° 1 tsp vanilla extract

° 1/2 cup crushed chocolate..  anywhere from bitter to semisweet depending on your taste

° some maple syrup or brown sugar .. optional

 

Technique

 

° oven to about 350° F

° cut bananas into 1 inch pieces and put them on a baking sheet (you may want to line it)

° sprinkle cinnamon over the bananas, turn at about 15 minutes and go until they are brown and soft - about a half hour

° throw the bananas and all remaining ingredients except the chocolate into a blender or use an immersion blender.

° add cinnamon or a bit of maple syrup if not flavored or sweet enough

° stir in chocolate bits

° chill in the fridge

° churn freeze with an ice cream machine

 

 

 

 

it's stories all the way down

The phone rang on Sunday afternoon...

 

"Hi, Steve?  Syd has a question..."

(Syd is about ten)

"Hi Steve.  Why is the sky blue?"

 

She probably has heard a few explanations - most of us have over the years.  In fact the common explanation you are shown in high school and college isn't very good. We set up an iChat link for the sketches and facial expressions.  Words and sketches appeared that were a bit richer than the standard explanation.  No equations, but there was enough to branch off into color perception, colors that exist only in your mind, why the Sun looks yellowish and some experiments to try and so on until my battery was down to about ten percent. Her parents are probably annoyed.

 

We tend to describe reality as layers of stories.  Take your smartphone.  You push regions on it's screen and things happen.   A video chat with a dear friend thousands of miles away, a map, photos from Iceland, music from a huge library or directly from a concert, someone's heartbeat.  All of us know how to make these things happen - the incantations we make with our fingers and voices to interact with a real and virtual world.  But that's just one layer.

 

Far below is the fundamental physics of the atoms the phone is made of.  Quantum field theory provides a rich description of the subatomic particles and fields and we have general relativity to describe how it's mass interacts with spacetime and visa versa..  A story I know well - perhaps better than how to use my iPhone, but it's pretty useless when I'm trying to find directions on a map.

 

There are many layers in between.  The story of materials - metals, semiconductors, plastics, battery chemistries .. the things that make the physical device.  A bit deeper is how semiconductors work .. a bit above is how electrical circuits work.  And then there's the rich layer where artificial language meets circuitry - the programming of the device.  This story is told in code at a variety of levels.  I know some of these stories well and can use them creatively and author in their language.  And above it all are human interactions - an area we all know a bit about. There are many layers where I'm clueless.  That's probably true of most of us, but I've noticed telling stories at one level while thinking about another can lead to new ideas.    

 

It is quite possible to deeply understand how to use an iPhone or Android phone and think the Earth is flat and riding on elephants on the back of huge turtles.  In a rich real world each of these layers must be locally valid and the layers must be constant with each other.  Fake science -  global warming denial, anti-vaxing, homeopathy and so on are modern versions of the flat earth.

 

Most of these layers have incomplete stories - works in progress.  Our understanding needs improvement and that is wonderful.   As an aside it is remarkable that the lowest layer .. basic physics .. turns out to be completely understood in the part of the Universe humans inhabit.¹

 

The poet said the world is made of stories..  At least the real world can be described by them.  It makes me think of the upcoming scientist's march on Washington.  I'm torn about attending.  It probably won't have much impact and will be ridiculed an misreported by the those who need to hear it's message.  At the same time science is incredibly import to society - more now than ever in the past - and it's under direct attack.  Unfortunately scientists are mostly lousy story tellers ..  I know I am.  Scientists live in a world of coded jargon.  We don't take the time to talk about the astonishing surprise and beauty of Nature often enough.  

 

I have serious shortcomings as a storyteller. Writers, musicians, poets, artists, actors and athletes are much better.  I'm trying to get better and think it is important for more people to develop skills.  Some of them are showing real promise.   Alan Alda set up an excellent program at Stony Brook that is slowly spreading.  A bit of hope.

 

a line I offered to Syd  

 

The sky is mostly transparent and colorless, but it happens to be the stage on which the colors dance... 

 

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¹ We know this to a very high confidence level .. much deeper than in most of science.  You can even write down an expression for it ...  here's a very compact dense form.  Some call it Core Theory .. basically it's just the Standard Model with General Relativity nailed on.  You can imagine places and situations where it breaks down .. in some collisions in particle accelerators and in some of the most violent events in the Universe.  Then you need to unify gravity and quantum mechanics and more,  but we don't directly interact with them as humans.  This is probably one of the deepest advances in science and it really has only come together with a high degree of certainty in the past decade.

Just because we know what the interactions are and how they work, the details of real world problems can make them too difficult to calculate. We just know, to an astonishing level of confidence, there isn't anything other type of interaction that impacts our world.

(this may look like gibberish, but it describes the physics that we experience on Earth and most of the Universe.    Not often seen on tshirts outside the boundaries of Caltech, but one of the great intellectual achievements.  It is a capstone on a small corner of understanding Nature, but it just marks a beginning.)