The Golden Circlet

All the good things in life

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The Sum of Infinite Series (When You Don’t Do Math)

So a friend of mine got a little riled up about this video:

You know, I write and read poetry, and there’s this thing that happens when I talk about poetry, a thing that I know also happens all the time when people who write and read math talk about math. People say, “I don’t like poetry.” Or sometimes, more charitably, “I don’t understand poetry.” Sometimes — if they like me — they think my interest in poetry is adorable. But they don’t want to talk about it with me. And meanwhile, I’m thinking, what do you mean you don’t like poetry? Poetry is a big thing! It’s like saying you don’t like music! Or food! There’s so much of it, I’m sure we could find something you would like.

Well. That’s poetry. Because you know what my reaction was to my friend’s curiosity about that video up there? It was, “I can’t help you with that.” I might as well have said, “I don’t like math. I don’t understand math.” And in some ways that’s true. But it’s not my best answer.

I’d like to live in a world where my mind was alive. I want to be able to walk up to whatever I don’t understand and sit down and learn from it. I want to sit up late and learn from the people I love about the things I don’t, until I have some of their passion for their subjects reflected back to me, if not actually lighting me on fire myself. My vision of a life well lived includes dinner tables full of people talking, teaching, learning. And the responses, “I don’t like ____” or “I don’t understand ____” or even “You’re so cute when you talk about ____” do not belong at that table. Because the only thing that can come after those responses? Is silence.

Which is maybe why I spent most of the evening and dreamed off into the night reading everything in sight and thinking hard about the sum of a divergent series of natural numbers, trying to make sense of the claims here, and whether they made any sense to me. So here’s what I came up , with my mind that doesn’t understand and doesn’t like math. (And since I don’t understand and don’t like math, you’ll have to put up with any errors I make here as I learn. And notice how I felt the need to put this disclaimer here, even, which tells you something about how people including me feel about looking stupid when they play with numbers.)

It’s a little weird to talk about adding up an infinite series of numbers at all, and getting a result of any kind. After all, you can never be finished adding up to infinity, can you? You can always add one more. So how could you ever get the sum of an infinite series? But apparently people have tried. And in trying, they’ve observed that some infinite series of numbers seem to be *almost* getting to a single definite answer, and others do not.

For example, if I add 1/2 + 1/4 + 1/8 + 1/16+ 1/32 …. and so on up to infinity, I’m going to get closer and closer and closer to 1. I’ll never quite get there, of course — I can always add yet a smaller fraction — but I can certainly see my way to my unreachable destination. That’s apparently called a “convergent series,” because it converges on the answer as it heads toward infinity.

In a divergent series this doesn’t happen. And it doesn’t happen in several different ways that I found kind of interesting. In some divergent series, the number just keeps getting bigger as you add more to it. 1 +2 + 3 + 4 + 5 and so on. Hard to imagine ever finding the *answer* to that infinite series, despite the weird video, because you can always add one more. But other divergent series seems to be equivocal about their “answers,” to have answers they are “considering” but can’t quite ever commit to . Like 1 – 1 + 1 – 1 + 1 – 1 + 1 and so on. Depending on where you stop adding, the answer is always 0 or 1, and if I wanted to know what the “answer” was to the infinite sum of that series, I’d have trouble making up my mind between the two. In the video the answer to that divergent series is the assumption on which everything else rests, and they simply tell you the answer is 1/2. Which makes a certain kind of weird sense. 1/2 is just the average sum over the series. But they don’t actually prove this in the video; they just tell you it is so.

Here’s a more scholarly article that goes into a lot more detail about it, more detail than I can follow, myself. But what I gathered from that is that these divergent series are seductive precisely because you can readily do the simple algebraic manipulations they show in the video, but when you do that, weird things sometimes happen. For example, you can rearrange the terms of that 1-1+1-1+1… series in such as way as to get 0 as the answer, and there are other divergent series where you can rearrange the terms and get any number you want as the answer. No wonder one of the mathematicians whose name is linked with these series described them as an “invention of the devil”!

Still, that paper describes a wish list for how to derive an answer to these infinite series: 1) you should be able to do linear operations on the series without changing the answer, and 2) you’d better not allow re-arranging the terms, since clearly that creates problems, and 3) whatever rules you come up with to sum divergent series, if they work at all, you should get the correct answer when you apply them to convergent series, too. And I guess (at least?) two systems have been invented to meet this “wish list,” one called Cesaro summation, and one called Abel summation. And these systems do indeed yield the answers shown in the video above, which I guess lends credence to the idea that those simple algebraic solutions they do on the video are not *just* hocus pocus. But evaluating those systems other than that really is beyond my understanding without instruction. Still, my cousin said that his opinion of the claim 1 + 2 + 3 + 4 + 5 … = -1/12 was that this was “true in a certain sense,” rather than true the way 1 + 1 = 2 is true, and I guess I agree with him. These solutions may not be hocus pocus of the kind an old boyfriend of mine used to prove 0 = 1 (you had to squint to see at one point he divided by zero), but I think it’s more accurate to say -1/12 is the best answer we can assign to the series, rather than its sum in any ordinary way.

The thing that really riled up my friend, I think, was the idea that these weird “results” have applicability, apparently in string theory and physics. That doesn’t trouble me as much, and not (or at least not only) because I don’t understand the applications. Somewhere I used to have a list of technological innovations that were based on scientific theories that were later shown to be incorrect — but the technologies still worked. Perhaps applying -1/12 to string theory is similar.

Or perhaps this is just one more example of God’s sense of humor, which I gather is shot all through both higher mathematics and theoretical physics anyway…..


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Homeschooling: Herbal Marshmallows

Floppy and I have been working through this book, a sort of children’s herbal.


I have mixed feelings about herb books. Part of me loves them, loves the wise-woman magic of making medicines and cosmetics and foods and rituals from plants. One of my failings as a parent is that I don’t do well with Floppy’s little injuries and illnesses. I tend to get irritable with him for having gotten sick or hurt — a blame-the-victim mentality if ever there was one, but to be fair, he usually gets boo boos from doing things like tearing through the house at top speed after having been told 10 times to cut it out. But I think some of the irritability is really just my own frustration with not being able to fix it, not being able to help. I’d like to have a repertoire of wise-woman tricks — even if they are only placebos — at the ready to care for my loves. I love the idea of having a witchy little apothecary of things I’d grown or collected to share in difficult or painful times.

Also, Floppy loves this stuff, in much the same way I did when I was his age. The natural world has magic in it, and making aromatic herbs into mysterious things that you can tell magical stories about is appealing to him as it was to me.

On the other hand, a lot of the medical claims they make in books like these are, you know, pretty much totally unfounded. I get the feeling that most of the adult people who are “into” herbs don’t have a very high standard for scientific rigor of medical claims, you know? And, well, fair enough. Even if you want to be intellectually rigorous about it: There are a lot of legitimate reasons for herbalists to be cynical of science as it is actually practiced, such as the fact that scientific studies of herbal medicine — as with every other kind of medicine — don’t happen unless someone stands to make a lot of money, and are biased in various ways against herbs that cannot be used to make anyone any money. Also, it’s very hard to research herbal medicine and traditional folks beliefs of all kinds, because these things work — if they work — synergistically, embedded into systems that are hard to study. You can’t isolate one active compound from a traditional herbal medicine practice and do a randomized controlled trial on it without feeling like you’re searching for the needle in the haystack. For example: Imagine an herbal compound for blood pressure and cardiac health, let’s say, that works when one part (which part? that’s a whole study right there!) of the whole plant is brewed into a tisane given to you by your herbalist, but does nothing when the compound is isolated from the plant and synthesized into a swallowable pill given to you by your pharmacist. Some plants are perfectly edible, delicious, and nutritious, when consumed by a healthy, non-nutritionally stressed human as part of a balanced diet, and fatally toxic when consumed in quantity by a hungry human. Some medications — not herbs, plain old Western medications — do nothing when the patient doesn’t know s/he’s taken them. This stuff would be terribly difficult to research with enormous resources. But there are very few resources to study herbal medicine traditions, and there never will be.  

So I understand why herbalists are cavalier about science. Nevertheless, using a medication — any medication, even one made out of a plant and brewed up into a mild herbal tea or applied as a wise-woman poultice — that I has nothing more than folk wisdom to recommend it, or even ensure its safety, gives me the creeps. And teaching my son to do this strikes me as unscientific, unthoughtful. So I have mixed feelings!

I’m resolving this in the short-term by sticking to those portions of the book that feature food herbs and somewhat schlocky-sweet magical stories. With which we are having an excellent time!

This week, we made marshmallows using real marshmallow root:

I was SO excited for these marshmallows. I’ve always wanted to make marshmallows with the actual botanical, and the recipe had no gelatin in it, which I also liked, because I dislike the fetid smell gelatin gives homemade marshmallows. But, truthfully, these turned out more like meringues than marshmallows — whipped egg white is the main ingredient — and Floppy thought they were disgusting. I think they might be good floating in hot chocolate — but haven’t tried that yet,

After we stuck the marshmallows in the oven, we read a silly little story about two children exploring a garden where the spirits of herbs grow and beg humans to use them. Floppy thought this story was wonderful, magical and inspiring, and after we read it, he wrote a little plants-and-magic tale of his own. A successful evening all around!

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On Printing Poetry (and Being Supportive)

I’m taking a letterpress chapbook printing class. I’ve wanted to learn printmaking for a long time, for a project I have in mind, and the opportunity came up to learn in this way, so I grabbed it. Everyone in the class is printing a chapbook of poetry, for a practical reason. Setting type into paragraphs isn’t a good beginner letterpress exercise, because one error and you have to redo the entire paragraph. Which, at the rate I set type, might take me 8-12 hours. For one paragraph. So the instructor doesn’t let you set prose. Poetry it is: Make a mistake in typesetting, and you only have to fix one line.

There’s another restriction set by the instructor, too, which is that — given that we are publishing an edition of 20 copies of each chapbook — the poetry set has to be free of problems with copyright. Either you own the copywright, or you have permission, or it’s in the public domain. I’m setting my own poetry, but I’m the only one. The other women in the class — the class is all women, except the instructor, this time — are setting various things.  Some Gerard Manley Hopkins, some Victorian doggerel, some other things I haven’t kept track of. But this was the interesting thing to me: There are eight women in the class. Two are setting their husbands’ poetry, one woman her boyfriend’s poetry, and one woman the poetry of a friend.

Here’s the thing: I’m a feminist. I’m a skillful apprentice writer and poet. I’m a talented behavioral scientist. (Does that sound arrogant to you? Maybe so. I’m also a little bit arrogant.) Not only that, I’m used to being extremely well-supported by the men who care about me. My husband is a stay-at-home parent for our son. He makes nearly all of my meals. He does most of the child care and always has. He does most of the housework. In the most fundamental sense, he supports my career.  In fact, almost all of the men I’ve loved have been feminists, all of the men I’ve ever loved have cared about me and wanted my success in whatever endeavor I put my mind to.  I’m privileged to be able to say I’ve always been very well supported by men.

I’ve been well supported by men (and women) specifically as an artist and scientist. I’ve received lots of encouragement and praise for my writing and scientific work from male teachers and colleagues. I’ve received excellent criticism, extended, time-consuming, focused criticism over multiple drafts  of my writing, from male poets at my online poetry workshop, and detailed reviews of my scientific work from male scholars and editors. All of my mentors, of both genders, have always done their jobs for me and given me tremendous opportunities (although with my sample n = 1 I can say that the female mentors have always been more generous, and occasionally the male mentors have hit on me, which the female mentors never have done). Male editors have published my poetry and my scientific work. Male judges have awarded me prizes. The men who love me have — when I’ve asked — read manuscripts, made copies of grant applications, run things to the post office at midnight to meet a deadline. I’ve had plenty of privileges given to me by my betters and loved ones (of both genders) as an artist and a scientist, and I couldn’t possibly ask for more.

But I really, truly, cannot imagine any man in my life taking it into his head to learn to typeset so that he could set a chapbook of my poetry. If they had that kind of time or energy for art, it would be their own art they would be investing in. (And rightly so!) That kind of support — the kind of support where someone believes in your art or your ideas to such a degree that they take your success on as a personal project of their own — is not support I’ve ever received from anyone, with one exception. When I was a little girl, 9 or 10 years old, my favorite aunt, also a fine poet, submitted my poetry to literary magazines and contests. The accolades she worked for on my behalf were my only publication and prize credits for poetry until I was well into my 30s. It took that long for me to develop that kind of drive for myself.

I’ve  provided this kind of support to a few artists I’ve known and cared about, mostly men, sometimes lovers and sometimes “just” friends, gifted people who captured my imagination and whom I wanted to prod onto a wider stage. Not because anyone asked me or expected me to, but because I loved and admired the artists and/or loved and admired the art. Because I wanted the best for them.  And looking around my letterpress class, half full of women setting their beloveds’ poetry, I know I’m not alone.

Recently I approached a male poet friend about a possible collaboration, and he turned me down in a way that was mysterious to me for a long time. He suggested that I had the talent to do this work for myself, ought to develop the confidence to do it for myself, and therefore didn’t need him as a collaborator. I was mystified because my sense was that confidence had nothing to do with it. (I have my insecurities, but not so much about writing.) I wanted to collaborate because I like to collaborate — I like the energy of two or more people writing and making art together, encouraging each other, challenging each other. I get more done in groups and pairs than I do on my own, because it’s so artistically satisfying to work in collaboration. What did confidence have to do with it?

But then I remembered this brilliant article, that I’ve always assigned students to read when I teach about the psychology of gender. Maccoby’s point is that gender is not mostly learned from parents and families. It’s learned on the playground, from other same-gender kids. What do little boys teach other little boys?

That life is about dominance, about victory, about success, about who wins at King of the Hill. So the men I’ve known have mostly been focused on their own careers, their own ambitions. When they think about others’ careers, it’s mostly to compare and figure out how to get ahead. They might like me, love me, care about me, be in love with me, but they certainly don’t have time to typeset my poetry. No wonder my friend assumed if I wanted to collaborate it must be because I didn’t think my work could stand on its own. Why else make the effort?

But what do little girls teach other little girls on the playground? That life is about love, relationships, braiding hair and telling stories, who likes you, who you can be kind to in return. So I know lots of women that work in groups, or that get personal gratification and enjoyment out of developing and promoting other people’s art. Think of Julia Cameron and her former husband Martin Scorsese. They’re both famous. But he’s “one of the greatest directors of all time.” And she’s “most famous for her book The Artist’s Way,” which was “written to help people with creative artistic recovery.” He makes masterpieces, and she makes masterpiece-makers. 

Scorsese and Cameron weren’t married long, but they were married at a pivotal moment in his career. Would he be “one of the greatest” without her midwifery? Maybe he would. But how many men would be the artists or scientists they are today, without the personal investment of the kind of women who would learn to typeset so they could publish a book of his poetry?

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Hotel De Glace

We’d been meaning to go for a long while, and in honor of the boy’s 10th birthday, we finally went: The Ice Hotel, in Quebec City:

Sleeping in an ice castle is every bit the magical once-in-a-lifetime experience you might hope on the one hand, and a total miracle of marketing, on the other. If they called it “ice camping,” after all, which is what it is, they probably couldn’t charge north of $500 a night for it.  That bit of bitterness aside, however, you’re still an ice princess in your ice castle, right? Right.

And what did your humble family of ice royalty do in their nordic castle? We carved ice sculptures, we drank fancy cocktails from glasses made of ice, we sat on fur pelts in front of roaring fires, we danced under an ice disco ball.

We spent an absurdly long time sliding down the ice slide, and far less time than we should have soaking in the hot tubs and steaming in the sauna before returning to what must have been one of the coldest nights of the Quebec winter thus far. (We indeed tested the limits of the -30 degree sleeping bags. We slept much less than we should have, too.) We wandered down mysterious ice queen hallways ….

 and into rooms carved into the shapes of forests, polar bear playgrounds, and avalanche caves. We took an absurdly large number of photos for our families and our blogs. When we finally got cold, we went out to dinner, where I had what must be the very best and most fun sushi I think I have ever had. We slept well, when we finally did sleep:

And in the morning we practiced our French wandering around old Quebec, buying junk and noshing. An adventure, for sure!



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Tu Bishvat Seder

Last weekend was the Jewish holiday of Tu Bishvat, the “New Year of the Trees” — a Jewish Earth Day, or Arbor Day. It’s not a major holiday, but has some lovely ritual associated with it, especially the seder, which was created by Isaac Luria in 17th century Israel as a mystical, kabbalistic way to celebrate the Tree of Life.

Just for fun — and because we were missing the Tu Bishvat seder his Hebrew school hosts — we had a seder at home, loosely following this outline. Most Tu Bishvat seders eat no fruit with the fourth cup of wine, as this is meant to evoke the realm of God, but I like the idea alluded to in the outline I linked to of choosing as a fourth fruit a “fruit of the Gods.” Chocolate, for example. We also tried to include the seven species, as well as a fruit none of us had eaten recently (papaya), so we could sing the shehechyianu.

Here was our menu:

  • First Cup: Pina Coladas in Coconut Shells
  • First Fruit: Mixed Nuts
  • Second Cup: Pina Colada mixed with a bit of Sparkling Grape Juice
  • Second Fruit: Olives
  • Third Cup: Sparkling Grape Juice mixed with a bit of Pina Colada
  • Third Fruits: Dried Figs, Red Grapes, Strawberries

  • Dinner: My Grandmother’s Special “Orange Mud” Soup (which contains barley, one of the seven species), Bread Drizzled with Honey

  • Fourth Cup: Sparkling Grape Juice
  • Fourth Fruits: Guacamole with Chips
  • Papaya, Kiwi, Pomegranate, Mango, Chocolate

It was really a lovely seder. We’ll definitely do it again next year.

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I love to dance. Dancing is more fun with a crowd, so I’m always trying to convince people to go dancing with me. It’s always kind of a task. Dancing, like singing — this basic, primal, human thing that people have done in all societies since the beginning of time — somehow got turned into this thing that you had to be a certain kind of person to be allowed to do: Young, probably female, definitely beautiful-looking, and with that particular nonchalant, sullen superiority that passes for hip in this broken country. I’m only one out of the four, and most of my friends are also acutely aware of where they fall short on the checklist. It’s a hard sell.

But I’m a great believer in what Morrie Schwartz called “create your own culture,” and I *do* love to dance, so I keep on selling. It’s gotten extra difficult since I moved to the middle of nowhere. Even if I can get willing partners, sometimes there’s nowhere to go. I’ll dance to anything with a beat, honestly, but part of getting people comfortable with dancing is going somewhere they feel comfortable, and the choices  are limited.

Yet, really, what’s not to love? A crowd of people all moving together, arms waving, feet stomping, colored lights flashing, music shaking your veins — if you don’t think about it too hard, you can rise right up off the floor and float until morning.

So I was excited when a friend invited me to a dance party at a quirky local museum. It was in honor of her friend’s birthday, and dancing was the main attraction in a night that also featured meditation, a powerpoint history of social justice, and a potluck. That’s how we do our revolutions in Northern New England in January. We brought the kids and danced to amazing West African music courtesy of Landaya. It was fun to watch the kids — at 10, the boy is on the cusp between heartless adolescent embarrassability and a child’s readiness to get down and boogie. Initially he was all frowns, but he told me later he simply wanted to see “what I was supposed to do.” When he concluded that “dancing is just like airbending” (and somehow telepathically communicated this to his friend), the two of them airbended their way up to the front and center and took it on down to the floor. Talk about rising right off of the ground.

I didn’t airbend — that I know of, anyway — but I did enjoy the music, lights, and being out with friends and family on a Friday night. And I can recommend it: So would you. Create your own culture. Go dancing. And invite me, next time you go.

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The Brown Paper Bag Project

A while back Jo Anna Rothman sent me a mysterious little brown paper bag that turned out to have 10 bucks in it — 10 bucks that I was supposed to use for service, in whatever way I saw fit. I puzzled over what to do with that 10 dollars, how to make it bigger, make it expand into something that could really serve the world. Because lately, it seems to me like the entire world is a bucket with thousands of holes in it, and I felt like anywhere I could pour in my 10 dollars it would simply drain right out the bottom of the bucket without doing anything.

I needed a superhero to help me. So I turned to my daughter, whose present employment involves riding a pegasus, armed with a light saber, defending truth, justice, and liberty everywhere. At least, that’s what my father tells me.

Perhaps I should back up a bit. In July or early August, a pregnancy test told me what I already knew: I was expecting. We were so excited — 10 years of trying had finally paid off. I tried to ignore the *knowing* I had almost from the minute she was conceived: that I was pregnant, that she was a girl, that there was something wrong with her. I told my mom and my husband what I knew, and they told me what you have to say in those situations: You worry too much. She’s fine.

Except she wasn’t fine, as a bitter array of miraculous medical testing was able to show us in painful detail. The pregnancy ended at 16 weeks. I birthed a daughter in late October, but I didn’t get to take her home.

And then my milk came in — a strange, sad gift from my body to my little Amanda who didn’t need it and couldn’t use it. The doctors said this probably wouldn’t happen — my pregnancy ended too early for my body to make milk, they said. If any came in, they advised hot compresses and tight bras and it’ll go away quickly. But I didn’t want it to go away, I wanted it to be used. I thought Amanda would want to share what she didn’t need with other babies like her.

There are premature babies so young that their own mothers can’t make milk for them — and so fragile that without human milk they will die. And — maybe there’s hope for this awful world yet — there are also milk banks that collect human milk, pasteurize it, and give it to those babies. My daughter’s spirit whispered in my ear that her milk belonged to those fragile little babies. So I began to pump my milk.


I didn’t have much — a few ounces a day at most. But the Mothers’ Milk Bank of New England generously told me that they would accept whatever I had to give. So I saved and froze my milk from the end of October until now — and finally fed-exed them a box of 120-odd ounces of human breast milk, a check for $100, and JoAnna’s 10 dollar bill.


Hopefully somewhere out there in the stratosphere, a baby ghost in a blue nightgown on a Pegasus with a light saber is smiling, and hopefully some little ones as precious to their mamas as mine was to me, will get the chance to smile because of it.


(You may not have milk to give, but the Mothers’ Milk Bank is a non-profit, and testing, processing, pasteurizing, and delivering milk safely to babies is expensive. I’m sure they’d be happy to have your donation, 10 bucks or otherwise.)