Elizabeth Chen assembles a tetrahedron from connectors and straws. (Photos by the writer.)

In that way, at least, Ann Arbor is densely packed.

This is a story about that town-gown connection. It’s a story that connects a recent UM mathematics PhD thesis defense to the Ann Arbor planning commission – and takes a continuous path though topics like Klingons, grocery bags, affordable housing, yard waste collection and Valentine’s Day.

We begin with Elizabeth Chen, who successfully defended her PhD dissertation last Friday in East Hall on the UM campus. Her presentation included several hands-on assignments for those in the audience of around 30 people – several of whom assured The Chronicle that hers was an “unconventional” thesis defense.

Chen exhorted the assembled mathematicians to paste together plastic pyramid shapes with gummi putty to help them get an intuitive feel for the shapes: “It’s not so scary!” she admonished them. After half an hour, one member of her thesis committee prodded her to get to the mathematics part – he really had “better things to do.” The Chronicle, however, did not.

Packing Pyramids: Background

Never mind the answers – many of the questions themselves that mathematicians work at solving are completely inaccessible to (even very clever) non-mathematicians. That’s not the case with Chen’s work. Her dissertation title sounds almost like it could belong in the children’s section of a bookstore: “A Picturebook of Tetrahedral Packings.”

Certainly even small children can grasp the basic concept of the question Chen works on: How tightly can you pack pyramids together?

Example of a model of a regular tetrahedron in the form of a die. The way you tell which number is "up" on such a die is to look at the one that can be read in its usual orientation. For this one, someone rolled a "4."

The specific kind of pyramid Chen works with is a regular tetrahedron (plural: tetrahedra). Each of the four faces of a regular tetrahedron is an equilateral triangle – one with three congruent sides.

For longer than a little while, it was believed that tetrahedra could be packed together perfectly to fill all of space, leaving no gaps at all. It was Aristotle (384 BC-322 BC), writing in “On the Heavens,” who suggested that regular tetrahedra were space-filling.

But by the 1400s, German mathematician Johannes Müller had countered Aristotle’s claim. And by the end of the 1800s, another German mathematician, Hermann Minkowski, had begun looking at the general problem of packing convex shapes. [A tetrahedron is convex – if you take any two points in a tetrahedron, the straight line connecting those points stays completely inside the tetrahedron.]

In 1900, David Hilbert, also German, included the problem of tetrahedron packing as a special case of Problem 18 in a list of 23 problems he had identified as interesting.  Hilbert’s list has guided much of mathematical inquiry for the last century. From Hilbert’s paper [emphasis added]:

How can one arrange most densely in space an infinite number of equal solids of given form, e. g., spheres with given radii or regular tetrahedra with given edges (or in prescribed position), that is, how can one so fit them together that the ratio of the filled to the unfilled space may be as great as possible?

Already in the early 1600s Johannes Kepler had conjectured that the most efficient way to pack spheres was in a way that Chronicle readers would recognize as the same approach that any produce clerk would take to stacking oranges. What Hilbert was asking for, though, was an actual proof that this was the optimal configuration. That (computer-aided) proof came in 1998 from Thomas Hales, who began his work at the University of Michigan.

A cluster of 9 tetrahedra gummied together. Chen's approach to maximizing density involves taking copies of these locally dense clusters and fitting them into lattices.

The density of an optimal sphere-packing is approximately 0.74048. That is, given an infinite number of identically-sized spheres, about 74% of space can be filled up with them – and we know, per Hales’ proof, with 100% certainty that there’s no configuration of spheres that would be any denser than that.

The 0.74048 number is thus a kind of a benchmark against which tetrahedron packing can be measured.

In 1972 Stanislav Ulam, a Polish-American mathematician who worked on the Manhattan Project, conjectured that spheres were the worst-packing of all convex bodies. So from Ulam’s conjecture, it should follow that tetrahedra should pack denser than 0.74048.  But in the mid-2000s, investigations of tetrahedron packing that used computer simulations, as well as experiments using  physical tetrahedral dice, could not establish any configuration of tetrahedron packing that clearly surpassed the 0.74048 for spheres. Maybe tetrahedra were worse-packing than spheres?

Was Ulam wrong? No. We’ll get to that in a moment. Now’s a good chance to think about how very wrong Aristotle had been – wrong about tetrahedra and their ability to completely fill space. How did he manage to massively miss that one?

Part of the reason could have been that Aristotle had no ready source of tetrahedral dice and gummi putty to try pasting models of tetrahedra together – the way that Elizabeth Chen asked the audience of her thesis defense to do. Once you have them in your hands, it’s easy to paste together models and convince yourself that they will fill less than all of space – a pastes-great-less-filling experience.

Alex Mueller (foreground) and Zach Scherr (to Mueller's right), both graduate students in mathematics at the University of Michigan, stayed pretty well focused on the hands-on tasks provided by Chen during her thesis defense.

In 2008, Chen showed how to arrange tetrahedra to achieve a packing of around 0.7786 – clearly beating the maximal packing for spheres, and in some sense vindicating Ulam.

Since Chen’s 2008 paper, other researchers have ratcheted the number upward, to 0.855506. But in early January of 2010, in a paper published with Michael Engel and Sharon Glotzer – both faculty in the UM department of chemical engineering – Chen nudged that number a bit higher, to 0.856347. [The more recent activity in the field of tetrahedron packing is succinctly covered in a New York Times article by Kenneth Chang: "Packing Tetrahedrons, and Closing In on a Perfect Fit"]

The January paper’s result, which is not included in Chen’s PhD thesis, was all that some in the audience wanted hear about: “What about the ‘champion’? I want to know how you did it, and then I’m going to leave.”

Chen eventually produced what they were there to see, which was the culmination of her systematic investigation: how individual copies of clusters of tetrahedra can fit densely into lattices. And her committee gave her a passing grade on the thesis defense.

Ann Arbor’s Kind of Density

When the topic of dense packing shows up in the pages of The Chronicle, it’s typically not in the sense of how densely you can pack space with tetrahedra. It’s usually something less esoteric, like a caution from the city’s public services area administrator, Sue McCormick, about packing the city’s yard waste containers too densely with leaves. From a recent Chronicle report on a city council budget meeting:

McCormick cautioned against compacting too many leaves into the containers, as it sometimes made emptying them difficult. [The automated arms tilt the carts upside down – whereupon the contents are liberated from the confines of the cart through a physical attractive force, a so-called "gravity."] McCormick pointed to the benefit of bagging as (i) providing more control, and (ii) limiting the amount of disruption in the community.

Or, if not densely packing leaves, then it’s densely packing people that’s the topic of discussion. We reported resident Lou Glorie’s remarks made during public commentary at a June 2009 city council meeting this way:

She suggested that urban sprawl had been replaced by the desire to pack 1,000 more souls into the downtown of some city. “Concrete is the new green,” she concluded.

Discussion on the merits of planning for greater population density in the city of Ann Arbor has dominated the local political conversation at least over the last decade. So it’s worth noting that a former Ann Arbor planning commissioner, Eric Lipson, attended Elizabeth Chen’s dissertation defense on dense packings of tetrahedra.

Eric Lipson, former planning commissioner with the city of Ann Arbor, hold the hands-on materials provided to audience members at Elizabeth Chen's Feb. 12 thesis defense.

Lipson did not attend by random accident. He’s the general manager of the Inter-Cooperative Council, a housing cooperative started in 1932 by UM students. Chen lived in ICC housing, at the Georgia O’Keeffe House, from 2005-2008. She was the O’Keeffe work manager for most of her time there.

That’s how Lipson knew Chen, and knew that her dissertation defense was coming up.

But Chen and Lipson aren’t just linked by the ICC connection.

Lipson himself has a practical interest in geometric shapes. He holds a patent on a connector for construction panels, which can be used to create 10-sided dome-shaped buildings.

And those 10-sided buildings can be shipped flat-packed wherever they might be needed. The company formed to manufacture and sell the product is called DecaDome. Lipson has prototypes set up in his backyard. While the audience was waiting for Chen’s dissertation committee to confer on her presentation, he showed us images of those prototypes from his Blackberry.

Eric Lipson showed The Chronicle images of DecaDome protoypes while we waited for Elizabeth Chen's dissertation committee to confer.

Part of what makes the connector special, said Lipson, is that the opening doesn’t require absolutely perfect alignment in order to accept a panel, which makes the task easier. As far as tools, all that’s needed is a screwdriver – though he allowed that a cordless power screwdriver would be recommended.

Panel material for DecaDomes ranges from foam core, to fluted polycarbonate, to pressure-treated plywood, to foam core panels covered with resin cement and fiberglass mesh.

Klingons

Different kinds of material is also the basis of the Klingon connection to Chen’s thesis. After Chen’s presentation, Sharon Glotzer, a UM professor of chemical engineering, helped clarify for The Chronicle why she and chemical engineering colleague Michael Engel were co-authors with Chen on the world-record tetrahedron-packing paper.

Glotzer and Engel are interested in designing new materials with interesting properties – properties that could, say, affect how we visually perceive objects made from them. That is, they’re interested in materials that have some kind of cloaking property. Glotzer told us that the various tech blogs take their speculations on this kind of scientific work in the direction of the Klingon cloaking device from the Star Trek series. [A cursory look into the Star Trek archives suggests it's the Romulans who pioneered cloaking technology, not the Klingons, who may have simply stolen it, but that's an issue that lies beyond the scope of this article – in any case, the proof is left to the reader.]

The tetrahedron connection to Glotzer’s work is this: Starting with tiny tetrahedra composed only of a few thousand atoms and suspended in a liquid medium, they can self-assemble into ribbon-like lattices. Exposure to light causes these ribbons to twist. And it’s the twist that holds the potential for cloaking. The twist – or chiral property – makes a compound optically active. That is, it will rotate the plane of polarization of light that’s passed through it. Glotzer stressed that the key to these compounds is the starting shape of the nano-particles – it only works with tetrahedra.

Glotzer told The Chronicle that she’s focused on the purely scientific aspect of this work – she’s not hoping someday to run a private company manufacturing cloaking devices.

Groceries and Valentines

Glotzer’s perspective on tetrahedra is not that the densest packing of tetrahedra is the most interesting packing. Rather, it’s that an interesting packing of tiny tetrahedra is the one that results in a larger object with desirable properties.

It’s a similar principle that applies, for example, to packing grocery bags. The goal is not to fit as much as possible into each bag. The goal is to pack each bag so that the resulting larger object – the packed bag – has desirable properties. A commonly desired property of a packed grocery bag is that it will stand up on its own – a property that’s a function more of the way its contents are packed than of the bag itself, something that’s especially true with plastic grocery bags.

And in Ann Arbor, at least, properly packing “square bags” can lead to love. From a 2006 interview with former mayor of Ann Arbor Ingrid Sheldon, in which she describes how she met her husband, Cliff:

HD: So you were a checker at the Kroger in Lower Town and he was a produce clerk?

IS: He was. He was doing his management training. He had just gotten his MBA from Michigan and as a part of his training, he was anticipating going into finance, they had him work in the stores.

HD: So did this unfold … was it the break room, where you first met, or?

IS: It was five o’clock rush. And these were the old columns of numbers, you know, we didn’t have a nine-key or a ten-key. We had columns for one’s and ten’s and hundred’s. I was noted for being very fast! And for packing square bags! I could ring up blind, and do the division 3-for-79 in my head, and you had to just do it. So anyway, I turned around one day, during the five o’clock rush, and there was this scrawny kid, packing round bags slowly. Ugh! So, of course, I had to assist him. But I realized he was youngish and I thought maybe I ought to pursue this guy, and find out more about him, before I totally blow him off! … it was love in the produce aisle! … and we started dating.

Happy Valentine’s Day from The Chronicle.

More Photos

Additional photos from the thesis defense that could not be densely packed into the layout of the above text:

Prof. Jeffrey Lagarias, who chaired Elizabeth Chen's dissertation committee.

Julian Rosen, a graduate student in mathematics, holds the 17-er tetrahedral cluster he pasted together during the dissertation defense.

Igor Kriz, UM professor of mathematics and a member of Chen's thesis committee.

Elizabeth Chen distributes materials for the hands-on portion of her thesis defense presentation.

Tetrahedral die held by professor of mathematics David Winter.

Elizabeth Chen (far right) receives the verdict on the oral defense of her dissertation: she passed. Standing with documents (far left) is her committee chair, Jeffrey Lagarias. Standing to Chen's right around the corner from Chen is professor of chemical engineering Sharon Glotzer.

Jo Mathis

Ghazi Abuhouleh is one smart man.

I spotted the Ypsilanti resident Friday morning carefully choosing a romantic valentine for his wife, Dalal. And this was after he’d bought her a diamond ring, as well. Though he’s been married less than two years, he knows the No. 1 rule of Valentine’s Day: This is not the time to be practical.

“I don’t buy that boring stuff,” said Abuhouleh, who opts instead for perfume, a pretty blouse, or jewelry.

Some guys, however, have a practical streak. And some guys shop for practical women who’ve told them not to spend money this year on chocolates (the calories!), flowers or jewelry.

Some of these men will be tempted to make a mistake on Sunday, and I feel it’s my duty to warn them. Men: If you plan to give your woman anything that is associated with a chore, save it for another day.

Even if it’s a $300 set of cookware. Even if she seems to love it at the time – and there’s a 10% chance she may – it will come back to haunt you.

“You know what Matt got me for Valentine’s Day in 2010? Pots and pans!”

“You think your guy’s unromantic? Derek once got me a rice steamer.”

Especially now that we all post pictures on Facebook to prove we’re enjoying a Wonderful Life, it’s important to give photo-worthy gifts.

Ghazi Abuhouleh of Ypsilanti choosing a valentine for his wife, Dalal, at Norton's Flowers & Gifts in Ypsilanti Township. (Photos by the writer.)

Eventually you can switch to something a bit more creative. And cheaper, which is important if you’re married and she considers that it’s coming out of her pocketbook as well.

I asked my Facebook friends to name the best non-traditional Valentine’s Day gifts. Guys, here’s what they said: A foot rub. Wii sports. Books. Seeing a play. A new kitten. (Make sure she wants a furry companion for the next 20 years.) An evening or even an hour just to talk, with no distractions. A donation in her name to a cause she supports. A clean house. For the kids to do what she asks, when she asks.

One woman met her partner in an Ann Arbor Observer personal ad and thought it would be cute to rewrite each other’s ads, adding new traits they’ve discovered.

When I had four kids at home, the best thing my husband could get me was some free time in the house by myself. I didn’t need to go anywhere. I needed him to go somewhere – and take everyone with him.

Now that I’m working from home, I’ll take any sentence beginning with the words, “Why don’t we go out to …”

♥ ♥ ♥ ♥ ♥ ♥

You know what I love? The fact that you can Google the name of any song and its artist, and within seconds you can be listening to that song. Whether it’s a ’60s hit like “Norman” by Sue Thompson, the theme song to “The Rifleman,” or the catchy tune you heard yesterday on the radio. You want it. You got it. Thank you, Google and lala.com.

What’s your favorite love song? There are so many great ones: “Layla.” “If I Fell.” “And I Love Her.” “The First Time Ever I Saw Your Face.” “Unchained Melody.” “Happy Together.” “There’s a Kind of Hush.”

My favorite is probably “Lost Together ” by Blue Rodeo. I love everything about it: The melody, the harmonies, that voice. And those lyrics: ”If we’re lost, then we are lost together.”

It’s not a promise to feel a certain way or a declaration of giddiness. It’s the conviction that we’re in this thing together no matter what.

That, my friends, is love in action. And that’s the kind that counts.

♥ ♥ ♥ ♥ ♥ ♥

As long as I can remember, my mother insisted that if she died before my dad, he would find a new wife within a month. I would think: Nah. Give him six months, anyhow.

Like many men of his generation, my father depended on his wife for his very survival.

As his health has deteriorated, my saintly mother has been by his side. About a year ago, she’d spent the day driving him to one appointment after another and was stopped at a light near a Sprint store. Not exactly a technology buff, my mother said: “Sprint. Hmm. That must be a running store.”

My father, who had been catered to all day by my mother as he has been for the last 62 years, sighed, shook his head, and said: “See what I have to put up with?”

He might have been serious.

So now my father is 84 and frail. He uses a wheelchair, barely talks, needs help doing virtually everything. And guess what my mother said the other day?

“If I die before Don, he’ll be married within a month.”

She still thinks he’s a catch. That’s amore.

♥ ♥ ♥ ♥ ♥ ♥

Of course you’re one of the 42,267,993 people who watched Jill and Kevin and their wedding party dance down the aisle last year.

Check it out at here, and you’ll see people are still commenting about it every few minutes.

When I watch it, I think what a fun couple they are. And that it’s too bad the song is sung by domestic abuser Chris Brown. And how there’s no way that even one person in my wedding party of 15 would have donned sunglasses and danced down the aisle. Duds, every one of ‘em!

♥ ♥ ♥ ♥ ♥ ♥

Every year during the week before Valentine’s Day, I drive over to the Ann Arbor PTO Thrift Shop and stock up on half-price Valentine goodies.

It’s all seasonal stuff nobody needs – pink teddy bears and heart-shaped candles and other knickknacky nonsense – but it’s perfect for Valentine’s Day junkies like me who insist the day should be all-inclusive. Romance shmomance. Of course the dog deserves her kibbles on a heart-shaped plate.

A mug of love.

My favorite find on Monday was a pale pink mug with the word Love.

Not “Luv ya lots!” Or “Love is in the Air” written on a balloon. Or even “I Love You.”

Just … Love.

Simple is good. And who doesn’t love love?

I was going to send that mug to my coffee-loving daughter. But that evening as I removed the mug when the steamy dishwasher, I had an Oprah Aha moment. Reader, that cup was telling me to love.

I no longer saw the word as a noun. It was an action verb. A command. Something like “Surrender Dorothy.” But more chipper.

And here’s the great part about that: I can love more just by willing myself to do it. I can’t suddenly become fluent in three languages, or a great bowler, or 25% prettier.

But I can love.

What matters more than what you feel is what you do. So you don’t feel you love your mate anymore? Love her or him anyway. Wash her car or bite your tongue or pick up your socks. Think about what you can do to make your mate happy, and do it. Do the loving thing, and quite often the feelings follow.

So I’m keeping that mug.

And if you’re the one who donated it to the thrift shop, thanks. I needed that.

About the author: Jo Mathis is an Ann Arbor-based writer.