Two weeks ago I took my dog to the vet. He needed orthopedic work on his back legs. As I sat in the waiting room, reading up on Ilizarov, another man's dog sniffed at my dog. So the man and I fell to talking. Turns out he was a petroleum engineer from Baku, in Azerbaijan.

I said, "Do you know about a doctor named Ilizarov?" "Oh, yes," he answered, "Ilizarov is my wife's relative." This engineer had once worked in the Azerbaijan oil fields. Rough work. He'd seen many open fractures and lot of Ilizarov surgery.

"The men would walk away from the operating table," he said. Later, as the vet worked on my dog, I mentioned Ilizarov to him. He said, "We've done external fixation on animals for a long time.

Anyway, I might've thought that that Azerbaijani gentleman at the veterinary clinic was telling tall tales. But just the day before, Dr. Gugenheim had taken me in to watch one of his operations. And I'm reminded of something G.K. Chesterton once said.

A woman asked Chesterton if he believed in infant baptism. Chesterton lifted his eyebrows and said, "Believe in it! Why, Madam, I've seen it done!" Well, now I'd seen it done, and it made a deep impression on me.

The operation I watched was reconstructive surgery. As I entered the operating room I saw the small leg of a seven-year-old boy protruding from the surgical linens. His deformed foot was bent at a right angle. All his life he'd limped about on the side of that twisted foot.

With obvious pleasure, the circulating nurse told me, "He's such a sweet boy." And that's when it sank in: This seemingly hopeless foot was actually going to be put right. But for me, a greater revelation lay within the operation itself -- something even beyond its radical promise to heal a lame boy.

First consider the problem Dr. Gugenheim had to solve. I don't suppose this is surprising to you all, but try for a moment to "make the familiar strange." See what I saw through my engineer's eyes, if you can.

The foot had be rotated almost ninety degrees on an axis through the ankle and normal to the tibia. At the same time, the foot had to be straightened -- it had to be bent in the plane of the foot itself.

That compound motion -- rotation and bending -- had be done by a mechanism that would ultimately throw all the reactive forces back on to the tibia. A machine with about 150 parts would be built during the next three hours without anything recognizable to me as an engineering drawing of the assembly.

Now: this stands in stark contrast with what's happening in our whole technological, educational, and business establishments these days. Our thinking is moving in exactly theopposite direction, and maybe you Ilizarov people can help turn things around. The problem centers on the way we visualize things.

Watching TV and computer images these days has become a glorious experience. It's become like falling into space. We wheel and turn in three dimensions, seeing an object as though we were some mad dervish swirling above it, below it, around it.

We couldn't've dreamt displays like that 40 years ago. Now the mathematical logic behind geometry and perspective is built into our machines. Once we used drafting to translate the pictures in our mind into pictures on paper. Now we build the picture on a computer screen without first seeing it in our heads.

Today, we need to do less and less mental construction. Instead, we call up finished images on 2-D screens. When that happens, the gains are so great we forget to count the cost.

Let me tell you about another Russian -- a colleague of mine. As a boy, this man had lost his lower leg to a German artillery shell during the siege of Leningrad. Later, he worked at the great Russian think tank of Akadiem Gorodok -- the academic city outside Novosibirsk in Central Siberia. This man is an inventor with remarkable powers of visualization. (And, just by coincidence, he actually met Ilizarov in Siberia.)

So I told him about the operation I'd watched -- about the complex mental manipulation of forces and moments. His response jolted me. He said,

Why doesn't the doctor just put a three-dimensional picture of the bones and the mechanism into the computer? The computer could do most of the stress design for him, and specify much of the configuration of the fixation structure as well.

He was right of course. But did he realize he was giving away the store? Sure, you all really could preplan those surgeries in three-dimensional virtual reality on the computer. I see no reason why you might not even ask the machine to optimize external fixation for you. The possibilities are there, and they're enormous. In fact, I just found out before lunch that those methods are already in the works.

So: Will you have to pay a price for this?

You see, creative thought means building in our minds, and that takes many forms. We can build strings of logic or poetic images. We can sift and rearrange recollection. We can construct every kind of relation among objects or shapes or quantities. The computer can now do some of that. The problem is, we've been doing far more spatial construction in our minds than the computer does or is ever likely to do. For millennia we had to use that spatial ability. And that's meant much more than just drafting in our minds.

But what's to become of generations that've never formed the habit of visualizing -- to math students who've never built graphs in their minds, or to medical students who've learned gross anatomy on a two-dimensional computer screen?

My favorite image of futuristic medicine is Dr. Beverly Crusher on Star Trek, The New Generation. Dr. Crusher passes that little hand-held sensor gadget over the patient and it makes the diagnosis -- it executes the cure. You wind up wondering just what it is she actually had to learn in medical school.

So what's wrong with that? Nothing -- except that, if such a machine is ever really made, it won't be in the next millennium. We and our grandchildren will have to deal with far more imperfect machines than that.

And as our machines create the illusion of doing our work for us, they create all kinds of mischief. Already laparoscopic gall bladder removals, for all the misery they alleviate, are causing new troubles. When doctors do surgery on a two-dimensional computer screen, they make more mistakes than they once did when they worked with their hands right inside the human body.

The statistics on Ilizarov surgery show that its popularity skyrocketed during the late ‘80s and up to 1992. Then the number of surgeries actually dropped, and it's only just beginning to pick up again. Why is that?

I'll bet I know what happened: A lot of orthopedic surgeons without the ability to build in their own minds found external fixation wasn't serving them and their patients nearly as well as they'd hoped. I don't know, but I'll bet that there was a lot of disenchantment from people who couldn't gain the fluency they needed in three-dimensional space.

Now, lest you think I'm trashing computers, let me say, very clearly, that I'm not. Spatial thinking is only one part of all they do for us. Computers, with their two-dimensional screens, are here to stay, and thank God for them! The question isn't whether to take them up. We'd be crazy not to. But, make no mistake, they are changing the way we think.

And the place that comes through most clearly is in education. Our thinking is moving away from the old conceptual models our schools are based on. And that gulf will grow wider. The new electronic media are leaving a great vacuum. It's a vacuum we have not yet figured out how to fill.

Let me tell you something you may not know about Gothic cathedrals. Suppose you were asked to design a quarter-mile-long structure, as tall as a typical downtown office building. Then suppose you were told it was to be made of stone -- no steel beams, no concrete -- and that the only available power had to be at ground level: horses, oxen, and maybe a three-horsepower water-driven saw. Finally, the straw to break the camel's back: You have to do the job with no working drawings.

Well, that's what medieval cathedral builders did, over and over. Yet we seldom weigh cathedrals as engineering construction projects. In fact, they took remarkable coordination of people and materiel. The master mason was both architect and builder. He often directed a work force numbering hundreds. But he also worked among his people. He cut stone and he raised scaffolding.

The subtle grace of Gothic cathedrals, which touch us so powerfully on so many levels, was a stunning feat of engineering design. Barrel vaults, spiral stone staircases, flying buttresses, Gothic arches! They were all born of concepts that were only fully expressed in the mason'smind before they were actually built.

Masons had no symbolic mathematics. Many could not even read and write. As we comb the rich medieval record, we don't even find architectural drawings -- only the crudest sketches. Yet medieval cathedrals are filled with geometry and proportion -- from labyrinths in mosaic floor tiles to the criss-crossing ribs that hold the ceiling.

What masons did have were dividers and a straight-edge. They had mathematics, but it was visual math. The cathedral itself was the master mason's geometry. With his fingers touching stone, he used stone to express geometry. The balance of mass and space in a cathedral goes by square roots of 2 and 3 and the so-called Golden Section.

Medieval iconography regularly showed a pair of dividers in the hands of the mason. Medieval artists often showed God Himself as The Master Craftsman, holding a great pair of dividers.

So, if the mason had one overriding talent, it was an ability to see in three dimensions. Compare that with our work today. Our computers fairly sing with the ability to show us objects in 3-dimensional space. They turn objects around and show them to us from every angle. They place you close up or far away. What the medieval engineer did in his head, the computer does for us.

But having done so, it then displays that object on a two-dimensional screen. What once went on inside our heads is now presented to our eyes. And that, I contend, is a terrible loss.

Even fifty years ago, engineering students were still people who'd cut their teeth on sewing machines, model airplanes and Ford transmissions. Now they feast on the World Wide Web.

I teach thermodynamics. Ask students to visualize pressure-volume-temperature surfaces, and they look for them on their computers. They no longer seek those surfaces in their minds.

They can do what I could not have dreamt of doing. But the cathedral builders did what is unimaginable to us. And their skills were fading fast by the late 19th century.

When James Clerk Maxwell set the theoretical foundations of electric field theory in 1873, he said at the outset of his treatise,

Before I began the study of electricity I resolved to read no mathematics on the subject until I had first read [Faraday].

You see, Faraday's pioneering work made little sense to mathematicians. Now Maxwell, a greatmathematician, systematically went back and climbed inside Faraday's head. There he found a great garden of delights. Here's what he said about the experience:

I found that ... Faraday's methods ... begin with the whole and arrive at the parts by analysis, while the ordinary mathematical methods were founded on the principle of beginning with the parts and building up the whole by synthesis.

Faraday was virtually uneducated, but he had an ace up his sleeve. He was dyslexic! And one typical dyslexic trait is a powerful visual sense. Faraday forged a finished image in his mind's eye, then broke that image down into parts that people could understand. Maxwell tells us that Faraday built a mental picture of lines of force, filling space, shaping themselves into lovely arrays.

Faraday drives his biographers crazy with the seeming irrationality of his thought processes. How can you start with the finished skyscraper, then build the foundation below it?

But Maxwell converted Faraday's vision of force fields into mathematical language and then plotted the equations. They form wild, graceful spider webs. And we see at last what Faraday had seen first.

That leaves a nagging problem for us. Faraday's work came first. Maxwell cast that vision into mathematical terms, then laid it down on two-dimensional paper. The problem is, that's not how the vision came into being.

Nine months ago, just before I spoke to the Houston radiologists' organization, my host called me and asked what slides I would bring. "None." I told him. The man was very nonplussed. "Oh, he said, we radiologists expect to see slides."

A few days later, a crazy hit-and-run driver tried to kill me while I was walking my dogs. He broke both my legs. Doctor Chris Coufal assisted with the surgery that put me back together, so I told him my problem.

He very kindly made me a set of slides of my mashed tibia before and after they'd pinned it -- and slides of the pulmonary embolism I suffered afterward. So, by a most unfortunate act of God, I was able to show slides to the radiologists after all. They needed explicit pictures to look at.

I'm terribly afraid that the computer is so relaxing the eye or our mind that that eye will no longer serve us in the 21st century. And so I'm talking to you without using any visual aids. I do that because I so badly want you to follow me with the eye of your mind, not with the eyes in your forehead.

Now what about the matter of visualization in our teaching? I'll break my discipline and show you one visual aid -- just one. Here: This full page ad for the Educational Excellence Partnership appeared in the New York Times this Thursday. It shows three children at the operating table, and it says, "Joey, Katie, and Todd will be performing your bypass." It goes on to talk about the complex tools they'll need to understand.

The clear implication in the text is that 20 years from now, when Joey, Katie, and Todd do your bypass or mend my broken leg, they'll do it with the help of hi-tech computers. They'll function far more like Dr. Beverly Crusher on Star Trek than you do today.

And we're back to the matter of visualization. Many people simply look at me as though I were crazy when I say that computer simulation is robing us of the ability to visualize. The stuff on the screen is too lush. It's too compelling.

But visualization is where concept begins. The computer can't do it for us. Medieval masons literally built castles in the air, in their minds, before they built them on the ground. If we don't reclaim that ability, the brilliance of our computer displays will grow increasingly empty.

The drama I'm describing played itself out once before in human history. It went like this. The invention of printing, like the invention of the computer, began as an extension of old thinking.

The first printed books, beginning in 1455, were only facsimiles of the old hand-written manuscripts. It took about thirty years for printers to create a new vocabulary of illustration. By the mid-1480s, the first scientific illustrations turn up in printed books. And they were only crude sketches cut into woodblocks.

But, by 1525, the artist Albrecht Dürer had published the first textbook on descriptive geometry. He'd also honed the technology of copperplate engraving. Almost immediately, a new and accurate technology of printed illustration replaced the old mental constructs of the Medieval world.

Printing had, by then, reached the point the computer has reached today. The result was the creation of an array of new sciences: anatomy, geography, botany, zoology -- all in forms that hadn't existed before.

But for 265 years those new sciences limped along -- becoming more and more blatantly descriptive. Every now and then a Galileo or a Newton would plumb his own capacity for visualization and kick us forward.

Go back and read the old books. So many pictures and so little understanding. Finally, around 1790, the Romantic poets began saying that (and I quote William Blake here) "Without man, nature is barren." They all took 18th-century science to task for all that undifferentiated observation They told us that our minds must be part of the equation.

Some 19th-century scientists and engineers understood. And the scope of their work took a dazzling leap forward. Now we face the same problem. The computer creates such a powerful illusion of visualization and creativity that we could well be in for another long dry patch.

So I call on you today to tell the rest of the medical establishment about the place that you have found for the eye of the mind -- the human mind -- in medicine. See to it that Joey, Katie, and Todd do not become technicians who've mastered one machine. See to it that your children in the public schools aren't allowed to avoid thought by pushing buttons. For, believe me, the eye of the mind is under attack. And it's the most precious thing that you doctors, we engineers -- or any educated citizen -- has.

As Dr. Gugenheim tightened the last bolt on that one-of-a-kind machine he'd built, I looked at the little boy on the table and thought about the opera Amahl and the Night Visitors. Amahl, poor and crippled in one leg, was visited by the Magi traveling to the manger. And as the Magi told about their mission, Amahl was miraculously healed. The word Magi comes down to us through the iatrochemical tradition of science and medicine -- the medieval tradition before printed books. And the word Magi turns into our word magic.

Like Chesterton, I saw the magic that will make one Amahl walk again. It is, quite simply, the fleeting and invisible magic of analytical three-dimensional thought. 
 

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SOME NOTES AND SOURCE MATERIAL

The surgery I mention was done on October 25, 1996, by Dr. J. John Gugenheim at the Texas Orthopedic Hospital in Houston, TX. I did not see the young patient's face, nor did I feel it proper to ask his name. I am grateful to Dr. Gugenheim for instructional commentary during the operation. That commentary revealed the enormous geometric and kinematic complexity of the operation. For more on the Ilizarov method, see articles on its use in traumatic injuries in Techniques in Orthopedics, Vol. 11, No. 2, 1996. I am also grateful to Mr. Jeffrey A Russell, President of the Joe W. King Orthopedic Institute, for providing a document titled, "The Ilizarov External Fixator: General Surgical Technique Brochure" (I believe this is a manufacturer's document. I have no bibliographic information for it.)

Coldstream, N., Masons and Sculptors. Toronto: University of Toronto Press, 1991.

West, T.G., In the Mind's Eye: Visual Thinkers, Gifted People With Learning Difficulties, Computer Images, and the Ironies of Creativity, Buffalo, N.Y.: Prometheus Books, 1991.

Maxwell, J.C., A Treatise on Electricity and Magnetism, Oxford: Clarendon Press, 1873.