Two days ago a friend told me about a wonderful auroral display over Greenland and Canada as he flew from Paris to San Francisco. We're used to the notion of great circle paths. The shortest distance between two points on a sphere is a geodesic - the fancy pants generalization of the concept of straight line to curved spaces.^{1}

I didn't think about this until a professor who I corresponded with as a kid sent a letter that told me to draw an equilateral triangle on a sphere, the seat of a saddle (I did have a Montana address and knew what a horse was), and a sheet of paper and measure the angles. Next draw two parallel lines. Finally I was to draw a circle on each and tell him what π meant. I won't offer a drawing. If you haven't thought about this, give it a try. Euclidian geometry falls apart.

People started thinking about this seriously a few hundred years ago. A specific geometry was developed for one class of curved spaces - a certain type of space where the equivalent of a plane would fit hug the surface of a saddle seat. Interesting, but limited and mathematicians lust for generality.^{2}

Bernard Riemann went off to college to study theology - one of the few fields where a college degree offered reasoanble employment prospects. He went to a lecture by Gauss and gave it all up to study math. A few years later, as the Civil War neared in America, Riemann was going for Habilitation - a sort of Ph.D.++ in Germany. He suggested a few dissertation topics and Gauss told him to work on the one Riemann thought boring - namely push the foundation of geometry. After a year and a considerable amount of time the heavens opened. A short paper that allowed you to understand the geometry of a space of any number of dimensions with any curvature at any point in space. My mentor said it was like Euclid had a few lines in the sand for counting, the first non-Euclidian geometry mathematicians moved to a hand crank adding machine and Riemann developed a supercomputer that we'll have in a hundred years.

People still wonder what kind of coffee he drank

He thought about the intrinsic geometry of space .. not how it might look from the outside. For him each point in the space would have information relevant to the space. I won't go into the math, but tensors are perfect containers for the book keeping. But the real breakthrough (actually the first item - looking at space from the inside was a huge breakthrough) was realizing that everything you needed to know about the geometry of a space was encoded in the distance along any curve you might draw in the space and that distance could be written in the form of a tensor he called the metric. So if you tell me any segment of a curve of a space you're fond of, I can use the metric and tell you how long it is. From this you can build up and calculate areas, volumes, geodesics (like flying from Paris to San Francisco), angles and everything you'd like about geometry. You could even talk about the path of a bug walking on an apple. Geometry from the ground up.

It was beautiful math and for about 50 years mathematicians had it to themselves. Then this guy named Einstein comes along with this crazy idea that gravity is really the bending of spacetime by a mass. (This is where all of those lame demos with balls and rubber sheets come from.) He had no idea how to approach the problem mathematically. After all, this space-time would have constantly changing curvatures. Riemann's work was an almost tailor made for the task. He worked with a people more conversant in the math for awhile until the training wheels came off.

If you've played with gravity this is all old hat. I've avoided the math attempting to describe the beauty of Riemann's work - the notion that any kind of space can be understood by looking at it from the inside and using this metric thingamabobby. Some may be too difficult to calculate, but in principal it's always possible. It turns out to be just the tool for a revolution in physics. Physics that is necessary if the GPS in your phone is to work properly and physics that still stands after over 100 years as our deepest understanding of gravity.

There are no lone geniuses in math or science. It's all connected. And the future of technology depends on it. That part scares me given the current anti-science sentiment in government.

Oh - the book. That's the undergrad textbook I had for General Relativity. Beautifully written with a sense of humor. One of the authors was very young at the time. In a few days Kip Thorne gets his Nobel Prize for his work detecting the sound of some of the most violent events in the universe... events that generate ripples in spacetime as predicted by Einstein with a theory made possible by Riemann.

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And one more thing. I've made this point several times, but it bears repeating.

People tend to talk about some theories as replacing others. While that's often true, sometimes the "superior" theory is just more general and reduces to the old one under certain conditions. If you're large - say over a millimeter in size - and moving slowly - say less than a tenth of a percent of the speed of light, Newtonian physics is just fine thank you. It may not be a deep way of thinking about the world, but for some practical matters like engineering, driving a car and playing volleyball it's just dandy and is much easier to use calculate than relativity or quantum mechanics.

The stories we tell are are equally valid to use if they're relevant to the domain we're using them for and if they don't contract physical evidence at the domain level.

__________

^{1} In physics you'd say it's a locally length minimizing curve ... the path a moving particle follows if it isn't accelerating.

^{2} only surfaces of constant negative curvature.

## x marks the spot

That's a really stupid idea Steve.It was somewhere around 1992 and wasn't the first or last time I heard something it. In fact it's part of the process of doing science - I just was applying it somewhere else. The PR people and business units of AT&T were for looking futuristic communicatons ideas in the labs that could change it's user's lives as the company moved beyond telephony.

I didn't start out as a technologist and still wonder why some people think of me that way. After some post-doc work in physics I joined the Bell Telephone Laboratories when it was still the best place this planet for certain types. A wide range of technologies were not just represented, but were invented there. My work was a bit to the side of the hardcore technologists and developers. My background going in was narrow - I'd get lost in many of the talks. But if you're curious, read and ask questions, anyone can learn. They gave me first-hand tours explaining their ideas in languages I could understand and, if they couldn't, or if they couldn't, they'd find an interpreter. That was considered part of my job. You could see the trajectory of some of the technologies -- threads moving moving forward in time and in some space. Threads that had a history that allowed you to compare your projections with what went on before. They had characteristics like cost, rate of development and dependencies on other technologies. It became a game to look for threads that could weave together in a history of the future. And every now and again a fabric began to emerge.

The rejection I began with was the most beautiful and poetic of the ideas I wrote down.

Maybe in fifteen or twenty years you'll hold a piece of glass in your hand and watch a friend's face on the other side of the world as you wish them happy birthday. The glass will be wireless and buttonless and the service will be too cheap to meter.It was a weaving of wireless, battery technology, displays, silicon, image compression, and some things that were happen with user interfaces. The need was there. I had learned it paid to create an experience people craved. AT&T had been after the videophone since 1928 with numerous failed attempts, but they kept hammering away at it. It was clear that video over the Internet was possible - heck, I was doing it then - and the other curves looked attractive. The weaving was there, but there were these pesky questions.. How could you get that much bandwidth over a cellular connection. Who would built out a wireless network?

When, How, Why, Who ...?

These were all temporal questions that deserved attention if you were looking into the future. You just need an excuse and the company arguably gave me too much freedom. There were a few ideas they liked. I started thinking about that distant point in technology space.

In math and physics when you're solving for something you often call it X.

Five years later and I was involved in a group specializing in computer mediated human interactions. About a dozen really bright people who knew things I had never thought about. People with patience and their curiosity beyond their own backgrounds. I brought my own background to the group. We did a few neat things.

My favorite was a project a few of you were involved with - Air Graffiti. I won't go into it deeply other than saying it was a realization that phones would become location aware computers with cameras. We hacked together a few computational bricks that gave us a rather clear view into a few things that happened ten to fifteen years later. We thought the basic technology of a location-aware computerphone with a camera would start to mature around 2005 - about the time cellular networks could begin to support links a few tenths of a megabit per second. That'd be a start. There were all kinds of objections, but we learned the trick was to not say much in demos and just give someone the test brick and let them imagine and invent on their own. These were the most amazing demos I had seen and the folks in the group were just amazing... Wayzen, Jessica, Gregg, Dave, Nancy and Steve (yes - we had two Steves). We all developed a healthy respect for user interface and user experience and we had dreams of special hardware. I began to think Apple was going to be the future, but had no idea it would be a phone ..

The fundamental technologies are silicon and radio. The 1992 piece of glass looked very do-able computationally. The requirements were lower than a first generation iPhone and it was just a simple Moore's Law extension. Batteries were more difficult. I was sure GPS would get there with some military funding. The camera and display were certainly possible - it was just a question of picture quality. The big IF was the buildout of the wireless network. I had been playing with folks building community networks in isolated areas in the West. The technology was there .. and the need was there. It was a question of buildout and price. Several of us thought it would happen, but we had no idea what the path and timetable would be.

Today Apple showed me the spot ... that point I called X. It took twenty five years, but the implementation is so much richer than I had imagined. Radio turned out to be an enabler, but in Apple's case amazing specialized hardware has been a driver. It is coupled with software, but that ability to do both is what makes them special.

But the fabric continues to move with new threads weaving their way in. That's what makes it so fascinating and keeps it so rich.

Oh yeah .. and hindsight is at least 20:20 .. I get a lot of things wrong too. But that's how you learn.

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