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Boxes and paradoxes

By Marjorie Senechal


It was eerie, a gift from the grave. But I thank serendipity, not spooks. The gift, it turns out, was given forty years ago. When Dorothy Wrinch cleared out her office in the Smith College Science Center, she left her books for the library, her burgeoning notebooks and contentious correspondence for the archives, and three boxes of crystal models and model parts for me. But I was on sabbatical, and whoever stashed the boxes in the basement never told me. They’d be there still had a young colleague not gone rummaging for something else last fall and found them, “For Mrs. Senechal” pencilled on the top. And so they reached me at last. Forty years ago, I would have treasured these models as she had. But what can I do with them now?

Bring them to Montreal for show-and-tell? Crystallographers from all over the world are gathering there for their triennial Congress. The year 2014 is a special anniversary. On the eve World War I, an undergraduate at the University of Cambridge, William Lawrence Bragg, walking along the river behind his college, found the Rosetta Stone of the solid state. The then-recent discovery that crystals scatter x-rays had solved for the x: the mysterious rays are waves, like light. Bragg turned this around, deciphering the structures of simple crystals from the patterns in their scattered rays. Today’s textbooks trace the path from his work on table salt and diamond to the double helix, modern drug design, and the highest of high-tech materials. We forget that the path was neither easy nor straight. The boxes of chipped and scattered model parts Wrinch left me bear witness to the early years, when scientists argued over whether salt is really the 3-D atomic checkerboard Bragg said it was, whether proteins are chains or rings as Wrinch said they were, and how to interpret the diffraction patterns of mind-bogglingly complicated crystals.

But the boxes are bulky and too heavy for airlines that charge by the ounce. So what should I do with them? I’m deeply touched by the gift; I won’t throw them out. But if they were ever user-friendly, they aren’t anymore. It’s hard to fit the rods into the balls, and the paint on the balls is flaking. And who needs real models now, when we have vivid, interactive computer graphics on our iPads? (Let’s get that one out of the way: real models are still working tools for me and I’m not alone.) No, it’s not their aged parts, it’s their aged ideas that make these models obsolete.

Figure 1. A ball from the box of model parts that Dorothy Wrinch left for me.

One book Wrinch didn’t leave for the library was a massive, gilded tome called Grammar of Ornament. It’s a cornerstone of the decorative arts, a veritable catalogue of rectangular swatches of floor, wall, and ceiling patterns created by people in all times and places. She loved this book because ornaments are like 2-D crystals. This analogy was crystallography’s chief paradigm, questioned by no one: the atoms in crystals repeat periodically in space. If you know one swatch (crystallographers call it a unit cell), you know the whole thing. A Grammar of Crystals would be a catalogue of swatches of 3-D atomic patterns. But that was then. Swatches are to modern crystallography as Pythagoras’s whole-number ratios are to √2 and pi. They’re still useful, but they’re not the whole story. The world of crystals, like the world of numbers, turns out to be bigger than anyone imagined.

Look closely at Wrinch’s wooden balls (Figure 1). The holes are drilled at the corners of squares, and at the centers of those squares, and at the centers of their edges. Six squares make a cube; if you picked up a ball and turned it around, you’d see the cubic pattern. With balls like these and rods to connect them, you can build 3-D swatches that stack like bricks to fill space. And that’s all you can build. But as the last century drew to a close, this paradigm crumbled. There are crystals, we now know, whose atomic patterns don’t repeat like ornaments. They spring surprises at every turn (Figure 2).

Figure 2. Left: To create this pattern, just fit the swatches together. Right: How would you extend this swatchless pattern?
Figure 2. Left: To create this pattern, just fit the swatches together. Right: How would you extend this swatchless pattern?

Aperiodic crystals have opened a new chapter; what will its paradigms be? At this still-early stage, we conjecture, argue, explore the new terrain from every angle. It’s fitting, and telling, that the Montreal Congress will be a double celebration. If ever a scientific discovery changed the world, x-ray crystallography did. But, paradoxically, the Congress will give its plums and prizes this year to the scientists who consigned its paradigm to history’s basement and sent us back to basics.

Figure 3. Compare the flexibility of this modern Zome Tool connector with its rigid ancestor in Figure 1.
Figure 3. Compare the flexibility of this modern Zome Tool connector with its rigid ancestor in Figure 1.
Figure 4. A model of an actual non-repeating crystal structure  made with Zome Tools by my students at the Park City Mathematics Institute, July 2014. Though aperiodic, this pattern of atoms can be extended in space.
Figure 4. A model of an actual non-repeating crystal structure made with Zome Tools by my students at the Park City Mathematics Institute, July 2014. Though aperiodic, this pattern of atoms can be extended in space.

I’ll put Wrinch’s models back in storage. She wouldn’t mind. “A science which hesitates to forget its founders is lost,” Alfred North Whitehead declared in 1916. A mature science, he explained, reconfigures itself as a logical structure from which the arguments and passions that built it are erased. Dorothy, then a student of his colleague Bertrand Russell, took the logical structure of science as a challenge. Later, when she ventured into less abstract realms, their reconfiguration was her mission. She would be delighted, I think, that so much of crystallography is automated today, and that the Grammar of Crystals is a databank. She would be delighted by new vistas to be reconfigured with modern models. And she would be delighted that crystallographers are still arguing.

Marjorie Senechal is the Louise Wolff Kahn Professor Emerita in Mathematics and History of Science and Technology, Smith College, and Co-Editor of The Mathematical Intelligencer. She is author of I Died for Beauty: Dorothy Wrinch and the Cultures of Science. She will be attending the International Union Of Crystallography Congress in Montreal 5-12 August 2014.

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Image Credit: Photos by Marjorie Senechal.

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