Sanding Away the Old

During the last three weeks I’ve had the opportunity to use several techniques, materials, and processes from the “home improvement” inventory. Linda and I have been helping our friends Steve and Sandi work on Sandi’s mother’s house preparing it for sale. Her mom passed away a couple of months ago and most people my age have experienced the issue of preparing the “parent’s home” for sale.

My father was and is an excellent craftsman. He built our first home in Lewistown, and I mean “built” as it was with his own hands. He’s always been a craftsman and an expert at many building trades including installing tile and building cabinets. If I had inherited only 1% of his skills in this area, I could have my own home improvement show. I did learn a few things from him as a child while I was busy losing his tools. I’ve had to refresh several of those memories and skills these last few weeks.

I’ve never done much painting, unless you count the motorcycles we refinished back forty years ago here in this very area of Tidewater, Virginia. With a son who is a skilled professional painter, I’ve always contracted even the simplest paint jobs to him. This time he wasn’t here to help, so I spent some time behind the brush and roller, but mostly I worked on other tasks.

There was considerable electrical work, installation of new switches, outlets, and even lights to be installed. I replaced a couple of electrical boxes and even did a little rewiring. I put new locks on the front door and painted the handle to match the new hardware. There was some plastering and spackling to be done to close all the holes not covered by switch and receptacle plates or to hide where curtains once hung. Mirrors and cabinets needed hanging in bathrooms and the living room and a large bay window had a shelf with water damage that I completely refinished in a three day orgy of sanding and applying new polyurethane. I’ve done so much sanding that I think my arms are going to fall off.

The last four days were spent refinishing the kitchen cabinets, although I haven’t completed that job. We leave tomorrow, so that task will be left unfinished for Steve and Sandi to complete. I’ve prepared all the cabinet faces and most of the doors and drawers are ready, although there is a little more detail sanding to be done on some doors. I bought 36 new nickel plated knobs for the cabinets, and I’m sad I will not see the finished project.

A lot of the sanding was by hand … I also refinished seven window sills. I resorted to power tools for the bulk of the cabinet refinishing. You should see me with my protective ear covers, dust mask, and googles. I look like an alien visiting the Earth. Plus, all that stuff is hot and I sweat a lot just watching an exercise program on TV, so you should see me when I actually do physical work.

Still, there is little to compare to the satisfaction of taking something worn and old and making it look fresh and new. There’s still plenty to do, and some choices will be made by budget restrictions. For example, there are louvered doors into the pantry that are a dark and ancient finish. I’d love to replace them with brand new doors freshly sprayed a nice white color. Ditto for the dark trim on the kitchen doorway. Whether Steve and Sandi will do that remains to be seen. Still it is satisfying to see the rebirth and restoration that a little sandpaper and sweat can do to wood … nature’s purest product. No plastic, steel or glass can equal the warmth and comfort of wood. I prefer natural finish, but paint is good too and can cover a wealth of wear. Sort of like makeup for the soul.

One can make comparisons to life and how we are gouged and scratched by the “slings and arrows of outraged fortune.” We could all use a little sanding, wood putty, and refinishing as we grow older. (Plus a little hair restoration … but I digress.)

I keep thinking philosophically while growing callouss on my hands. The ancient Greeks were great thinkers, but they eschewed physical labor. That’s their loss. The mind works better when the hands are busy.

I’ve been busy thinking while the repetitive and monotonous work is being done. Hard work is good for the mind as well as for the soul and the body. I’ve even come up with some new insights into physics during the repetitive, Karate Kid, “wax on, wax off.” I think the path to deeper understanding of quantum physics, relativity, and string theory lies in the basics of classical, Newtonian understanding. A good house, and a good finish, must have a good foundation. Preparation is the key to the final coat being smooth and glossy.

If I get lost in the manifold twisting of 29 dimensions of rolled-up-space, it may be because I don’t know the simple field of F = mA. I spent one whole day sanding and listening to “the greatest hits of the ‘60s, ’70’s, and 80’s,” and thinking about these basic concepts. (I had no choice but to think about basic concepts, since my grasp of the advanced subjects is lacking … after all, that’s the issue … isn’t it?)

The object of my “a-ha” was Lagrangian Mechanics. The "Lagrangian formulation" of Newtonian mechanics is based on an alternate form of Newton's laws which is applicable in cases where the forces are conservative. (Note that, in nonclassical physics, the forces may not be conservative. They may be “liberal.” — just a little political humor. ) Lagrangian mechanics adds no new "semantics" to Newton — it's just a mathematical change, not a change in the physics.

You see, Newtonian mechanics has a problem: It works very nicely in Cartesian coordinates, but it's difficult to switch to a different coordinate system. Something as simple as changing to polar coordinates is cumbersome; finding the equations of motion of a particle acting under a "central force" in polar coordinates is tedious. The Lagrangian formulation, in contrast, is independent of the coordinates, and the equations of motion for a non-Cartesian coordinate system can typically be found immediately using it. That's (most of) the point in "Lagrangian mechanics".

Before we go on I should hasten to add that the Lagrangian formulation also generalizes very nicely to handle situations which are outside the realm of basic Newtonian mechanics, including electromagnetism and relativity. I actually was thinking about the most fundamental equation of all time … at least in my opinion. That is Pythagorus’ famous equation for a right triangle. You know, the one about squares and hypotenuses. It is at the foundation of coordinate systems and measuring the curvature of non-Euclidean, Riemann space that is the basis of the General Theory of Relativity. If you have trouble understanding the complex stuff, it can often be traced to a root cause of failure to really, REALLY understanding the simple, basic, fundamental stuff it is all based on.

That was my epiphany as I repeatedly rubbed the wood, always with the grain, into its natural basic beauty found underneath all the dirt and worn finish from sixty years of wear and tear. They built houses well back in the fifties. Time and wear has added scars and wounds, but the underlying structure is still sound. It just takes a little elbow grease and 120 grit sandpaper, plus — maybe — a little help from electricity and Black and Decker.

I am going to miss the hard work and transformational effort, but I won’t miss the sore back and arms that hang helpless at my side from exhaustion. I’ll heal. The house will be beautiful. Happy memories will merge with fresh paint and polyurethane to start new memories for the next home owners. And my new insights will be applied as I get back into study mode. It is hard work to renovate a 60 or 70 year-old home. It’s much harder work to stuff knowledge and concepts into a 66 year-old brain. I need a Black and Decker brain sander.

That was the year that was!

It was a very good year.

Germany surrendered on April 29, 1945 and Japan gave up on August 15. It took a while for the boys to get home. Some were already married. Many, like my dad, married shortly after returning safe. That made 1947 the first year of the baby boom. Oh, sure, a few were born in ’46 by those that had a head start, but, with nature's delay of nine months, it is 1947 when the action started. That is the real start of the bubble that passed through the demographics like a pig being digested by a boa constrictor.

Nineteen Forty-seven. That’s the year that was. It was a very good year. And just who might be a famous person born in ’47, you ask?

Well, there’s Arnold Schwarzenegger, actor, governor, body builder, and general tough guy. Although not born on American shores, he’s a ’47 baby. Also Kevin Klien, Jonathan Banks, and James Woods. If you’re into scary movies, there’s Robert Englund and Stephen King. Also Peter Weller from Robocop and Sam Neill from Jurasaic Park. Richard Dreyfuss and Glenn Close, not exactly character actores. Then there’s Alan Thicke and Ted Danson as well as Farrah Fawcett and David Bowie. My favorite is Edward James Olmos, but also Teri Garr (remember Young Frankenstein?) and Rob Reiner — Meathead. Oh, and also Meat Loaf. There’s Anne Archer, Stephen Collins, Joe Mantegna, Larry David (Seinfeld guru) and Jane Curtin. Remember Michael Gross … Family Ties and Tremors? Well, Merideth Baxter too. Cindy Williams and Tom Clancy; Barbara Bach and Wes Studi; Kim Darby and William Atherton; Sally Struthers; Elton John and Iggy Pop. Ben Cross and David Mamet as well as John Larroquette and Takeshi Kitano … 1947.

Richard Lewis, James Hunt, David Letterman, Kareem Abdul-Jabar and Bill Smitrovich as well as Harry Reems were born in that very good year. Cheryl Tiegs and Jimmi Walker as well as O.J. Simpson, Karen Valentine, Martin Ferrero and Jameson Parker. Don’t forget Paula Dean or Sammy Hagar and Jill Elkenberry or Don Henley and Joe Walsh. Also, Gregg Allman and Michael Flynn. I’m just getting started. There’s Carlos Santana, Hillary Rodham Clinton and Salman Rushdie. Remember Jon “Bowzer” Bauman?

My wife liked Danielle Steel and I liked John Dykstra who did the special effects in Star Wars. There’s Ron Wood from the Stones and Mitt Romney from the … well … Then you can add Ian Anderson from Jethro Tull and Bill Lancaster who wrote The Thing. The lovely Emmylou Harris was born in 1947 as was Warren Zevon and Ry Cooder. Include George S. Clinton who wrote the sound track for Austin Powers and Bill Hayes who wrote Midnight Express. Arlo Guthrie, “The NY Express way is closed, man.” and Judge Joe Brown from daytime TV. Both Melanie and John Stossel as well as Johnny Bench and P.J. O’Rourke from that very good year.

Comedian and columnist Dave Berry and Jim Plunkett as well as Steve Forbes and Peter Waterman … also ’47. Who remembers Chip Moody from talk radio or Dan Quayle form … again … talk radio. Or the WCW “Giant Haystacks.” (I’m not making this stuff up, folks.) How about “Lacy Legends,” billed as a “busty” actress.

Then there’s also Ron Fleming, Jack Barney, John Barr, Gary Hornseth, Steve Miller, and Gary Murphy. Oh, and there’s also Mickey Cheatham.

Album Art

It is the nature of us folks of slightly advanced age: the Medicare Generation, the Baby Boomers, Old Farts, Former Hippies, and other general terms for anyone over fifty, to compare our times to today in such a way that it is obvious that civilization has gone to h-e-double-toothpicks in a shopping cart. We had it so good in our day … flushing toilets and the electric-light-bulbs. We rode our bikes with no helmet, drove our cars with no seat belts, and had to call up our girl-fiends (or boy-friends) from pay phones or suffer the possibility that our younger siblings would listen in on the extension.

Yup, it was a simpler time of black and white TV and AM radio. We gradually graduated from LPs to 8-tracks to cassettes, improving portability along the way, but nothing like today’s iPods and smartphones that seem to be surgically attached to every runner, walker, and stroller we see on the street and in the stores and restaurants.

Let’s talk about LPs. That’s Long Playing … as in “Long Playing Records” or, as we quaintly called them in comparison to old picture books: “albums.” Of course, we also had “45’s,” but it was album rock that really set the stage for the sixties. Soon we were consuming concept albums and enjoying the art work as we shared the music and the experience.

Now days we have CDs. Never mind that they don’t match the LP in sonic capability, what about the art work? You need a magnifying glass to really enjoy what was once a great part of the hip scene. And don’t get me started on downloads and iTunes. Sure, Apple has some of the album artwork and on a large, hi-def screen it does approach the intimacy of an album cover, but you don’t pass the laptop around the room as you listen to the music and an iPod is a terrible visual tool and isn’t about sharing at all, unless you count giving one of the ear plugs to your buddy to enjoy the left channel. And albums had more than just a front cover. I spent many an enjoyable listening hour accompanied by liner notes and intimate inner art.

Oldsters and geriatric hipsters will wax poetical about sitting around the listening room (that’s LR in architectural parlance) and passing the album artwork around while listening and possibly passing around something else.

Whether it was the special covers of “Yes” or “Santana” or the “Beatles,” these were works of art worthy of the sonic art contained on the black disks. Some were very expensive productions. I remember a Stones cover that was one of those 3-D things that changed when you moved it or a CSN cover that was almost leather in duplication of a picture album. Later, these were changed to simple photographs of the more expensive materials to save on production costs, but still the artwork was prominent.

And it wasn’t just limited to the front cover. It extended to the back, and the sleeve, and often the covers opened like a book providing two more surfaces (sometimes even more) for graphics and photos and liner notes. These are all things long gone in this age of downloaded and digitally compressed musical product that is about as tasty as a tomato shipped from Chili to us hipsters used to tasting the music fresh from the garden.

These covers were real art: Joan Miro, Max Ernst, Mondrian, Picasso, Dali, John Martin, Rene Magritte, Jean Tinguey, Bruce Pennington, Eddie Jones, Jack or Josh Kirby, and … of course … Derek Riggs.

A band’s first contact with the public is the picture on the front of the cover. This was before MTV … and I mean back when MTV was “Music” TV and before late night concerts and Saturday Night Live. You knew the band personally from the cover. Sure, there were also posters for concerts and clubs, but that’s pretty much gone too now days. A record release was a cultural event. It was a big canvas and had a big art to go with it … and then there were the “gatefold” covers that had inner surfaces and there were included notes and other goodies and even the record sleeves could receive art to complete the concept. In those days a album included a lot of art, both sonic and visual, to excite and entertain and inform the listening public. It was part of the art and the experience. As Jimi would ask, “Are you experienced?” Well … I am.

That all ended with the CD crystal case. In marketing terms the crystal case was just a complete disaster. The covers went from being impressive objects of art to practically insignificant additions. Reduced cover pictures were all there was, and they were singularly unimpressive like miniature ceramic birds.

The rise of the MP3 download has finished off the art form. Now there’s practically no avenue for you to advertise your new album and link it with a certain image or style. Your music has no covers. There is no way to link merchandise with these products; this is an important source of revenue for all bands. The customer cannot identify the product by the image or link himself to that product with a certain style. T-shirts and other tour merchandise substitute, but not at the level of exposure of the original record cover.

Yup, in my day we had it good. Sure we had to walk to school … through the snow … up hill … both ways. But we had album covers. You young whipper snappers don’t know what you’re missing. Now, about hip-hop music and rap!! Oh, don’t get me started.

Scalars, Vectors, and Tensors -- Reloaded

In 300 B.C., Euclid created (or some would say documented) Geometry as a deductive, axiomatic system. In 1637 René Descartes introduced the coordinate system in Euclidean Geometry, combining algebra and geometry into a single mathematical structure. A logical procession of this line of thinking ultimately led to the development of the Tensor.

One can say that tensors came of age with the appearance of the remarkable paper by the famous mathematicians Ricci and Levi-Civita called Methods de Calcul diferential absolu et leurs applications, Mathematische Annalen, 1901. But it is primarily due to Einstein's use of tensors in his General Theory of Relativity that was responsible for the sudden emergence of the tensor calculus as a popular field of mathematical activity.

Tensor calculus is concerned with the study of abstract objects, called tensors, whose properties are independent of the reference frames used to describe the objects. A tensor is represented in a particular reference frame by a set of functions, termed its components, just as a vector is determined in a given reference frame by a set of components and if a new coordinate system is introduced, the same tensor is determined by a new set of components and the new components are related to the old ones, in a different way known as the covariant or covariant way.

Since tensor analysis deals with entities and properties that are independent of the choice of reference frames it forms as an ideal tool for the study of natural laws. In particular, Einstein found it an excellent tool for the presentation of his General Theory of Relativity, and he spent a couple of years in correspondence with Levi-Civita to get the math right. (We all need a little help from our friends.) As a result tensor calculus came into general prominence and in now invaluable in its applications to most branches of Theoretical Physics, it is also indispensable in the Differential Geometry of curves and surfaces in Euclidean space.

Tensor is a generalization of the term "vector" and tensor calculus is a generalization of vector calculus.

Reference frames are important in Newtonian physics as well as Special Relativity, but it is the distortion and curving of space that Einstein's General Theory predicts that make tensors such important tools.

In mathematics, a manifold is a topological space that near each point resembles Euclidean space. More precisely, each point of an n-dimensional manifold has a neighborhood that is homeomorphic to the Euclidean space of dimension n. Lines and circles, but not figure eights, are one-dimensional manifolds. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, which can all be realized in three dimensions, but also the Klein bottle and real projective plane which cannot.

Although near each point, a manifold resembles Euclidean space, globally a manifold might not. For example, the surface of the sphere is not a Euclidean space, but in a region it can be charted by means of geographic maps: map projections of the region into the Euclidean plane. When a region appears in two neighboring maps (in the context of manifolds they are called charts), the two representations do not coincide exactly and a transformation is needed to pass from one to the other, called a transition map.

The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows more complicated structures to be described and understood in terms of the relatively well-understood properties of Euclidean space. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. Manifolds may have additional features. One important class of manifolds is the class of differentiable manifolds. This differentiable structure allows calculus to be done on manifolds. A Riemannian metric on a manifold allows distances and angles to be measured. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model space-time in general relativity.

As a kid interested in cars and souped up cars called "hot rods," I find a good example of a manifold in the metal parts of an engine that share the term. Both the intake manifold and the exhaust manifold are good examples of the twists and turns that are part of the geometric representation of mathematical manifolds.

The notion of a fiber bundle first arose out of questions posed in the 1930s on the topology and geometry of manifolds. By the year 1950 the definition of fiber bundle had been clearly formulated, the homotopy classification of fiber bundles achieved, and the theory of characteristic classes of fiber bundles developed by several mathematicians, Chern, Pontrjagin, Stiefel, and Whitney. Steenrod's book, which appeared in 1950, gave a coherent treatment of the subject up to that time.

Around 1955 Milnor gave a construction of a universal fiber bundle for any topological group. At that same time, Hirzebruch clarified the notion of characteristic class and used it to prove a general Riemann-Roch theorem for algebraic varieties. This work led to methods for mappings between various manifolds and ultimately led to a mathematics course that combined Fiber Bundles and Cobordism with topics of principal bundles, vector bundles, classifying space, connections on bundles, curvature, topology of gauge groups and gauge equivalence classes of connections, characteristic classes and K-theory, including Bott periodicity, algebraic K-theory, and indices of elliptic operators, spectral sequences of Atiyah-Hirzebruch, Serre, and Adams, Cobordism theory, Pontryagin-Thom theorem, and calculation of unorientated and complex Cobordism. I am now the freshest graduate of that class, and ready to map the reference frames until my slide-rule melts from the friction.

Now I'm ready to tackle that General Theory. I don't have Levi-Cevita to help me, but I do have my trusty Dale Husemoller "Fibre Bundles" text, even if the spelling is British. Good old Springer-Verlag, New York, Heidelberg, and Berlin. Das ist gut.

Scalars, Vectors, and Tensors

A student’s first physics class, often encountered in High School, will usually begin with a discussion of “vectors.” We assume these students are already schooled in simple mathematics such as arithmetic and geometry and even some trigonometry. They are quite familiar with the Pythagorean Theorem which gives values for the sides of a right triangle with the famous square of the hypotenuse is equal to the sum of the squares of the other two sides. I suspect all students, down through the ages, good student or bad student, it doesn’t matter, all students know this basic formula which was discovered by all the ancient cultures that existed before the modern Christian era.

For some reason, I don’t recall what it was. It may have been due to a broken bone or possibly a trip I took with my parents. In any case, I missed the first two weeks of Physics at Fergus County High School in my Junior Year. So when I showed up for class, all the other students knew about vectors and how to manipulate them, often using the ancient Greek theorem I described in the previous paragraph. But I didn’t!

While the rest of the class was learning how to add forces and manipulate the press of gravity and pushes and weights and measures, I was scrambling to add vectors to my sphere of understanding. It was a traumatic few weeks until I got caught up.

Now those who have taken the class know that things were divided into two categories or groups. One was called “scalars.” That is, measurements or values or numbers that had no direction. Scalars are physical measurements without any direction to them. For example, temperature, mass, and bank account balance. They can have a positive or a negative value, but, otherwise, they are simple integer or real numbers (decimal points).

In contrast, vectors have magnitude like a scalar, but also direction. They are like little pointy arrows or the hands on a clock face. They have a magnitude, often represented by the length of the vector in a geometrical drawing. But they also have direction like North or South, or like the previously stated hands on a clock, pointed at 12 noon or at 6.

In the physics class they could be represented by an angle from a baseline, which was usually horizontal like the floor or table top. So a vector pointing to the right, by convention, would be zero degrees and, pointing straight up, ninety degrees. If the vector points to the left it is 180 degrees, and so forth.

Then the fun begins. You could use a coordinate system such the common Cartesian coordinates, named after René Descartes, that have an “x” axis and a “y” axis … at least in two dimensions.

Now you could describe a vector as a set of numbers or coordinates. It was convenient to put the start of the arrow at the coordinate system origin, a location with x and y values of 0,0. Then you would describe the pointy end of the arrow with some value such as 2,2. That’s an arrow from 0,0 to 2,2. Some quick calculation would show the arrow has a length of the square root of (2)² + (2)², which is the square root of 8 or 2√ 2 , approximately 2.8.

Remember?

Of course, vectors can be in three dimensions too. A similar vector from 0,0,0 to 2,2,2 has a length of the square root of the three coordinates. (The Pythagorian theorem can be expanded to multiple dimensions.) So now the length would be √ (2)² + (2)² + (2)²  , almost 3.5.

Now suppose you expand it to more than three dimensions. Now it gets hard to visualize. Assuming that the vector starts at the origin, you can state a vector as a matrix with one row (or, more common, one column). For example, our vector on the flat plane would be [2,2] and if it was in five dimensions, it would be [2,2,2,2,2], where the square brackets are the symbol for a matrix or collection of numbers. And you would take the sum of the squares of the five numbers to calculate the length of the vector in 5-space.

As I pursued advanced mathematics, I learned to use matrices as a mathematical tool and coordinate systems could be represented by a matrix. Vectors and matrices are the common mathematical topics in all branches of modern physics.

Now I’m learning “Tensors.” What is a tensor, you ask? Well that’s an interesting question. They are related to scalars and vectors. They can describe the relationships between scalars and vectors and even between other tensors. Tensors are often represented by matrices with several rows and columns.

Because they express a relationship between vectors, tensors themselves must be independent of a particular choice of coordinate system. That is one of their powerful capabilities and it is why Einstein chose tensors to represent space, which he supposed was not flat or Euclidean, but rather curved by the distortion of gravity and energy. That is the basis of his general relativity theory.

Both the concept of matrices, which were used to describe basic quantum theories, and tensors being used in the early 1920’s were new mathematics to physicists. At that time they were esoteric structures only studied by mathematicians. Now days, they are part of the basic math curriculum studied by every physics student.

Einstein actually had to seek help from mathematicians as he perfected his general theory field equations. General relativity is formulated completely in the language of tensors. Einstein had learned about them, with great difficulty, from the geometer Marcel Grossmann. Tullio Levi-Civita then initiated a correspondence with Einstein to correct mistakes Einstein had made in his use of tensor analysis. The correspondence lasted 1915–17, and was characterized by mutual respect.

(Levi-Civita was an Italian mathematician, most famous for his work on absolute differential calculus (tensor calculus) and its applications to the theory of relativity, but who also made significant contributions in other areas. He was a pupil of Gregorio Ricci-Curbastro, the inventor of tensor calculus.)

So next time you are struggling with your school homework and asking for help, isn’t it comforting to know that even the great Einstein had to find a tutor to help him? As we always said at work, “It’s not what you know, but who you know.”