The Golden Circlet

All the good things in life

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Dancing Meditation 1: Body Heat

41501536_10215970590604996_5891155766588473344_oEarlier this month, I signed up for Jo Anna Dane‘s RootCoreCrown dance meditation series. You can hear Jo Anna describe the practice and how to do it here.

I did it every day, and I liked it, a lot. It was a 13 day series, though, and now it’s over, which is sad. Jo Anna offers guided meditations like this in her virtual salon, but finances are tight around here just at the moment, so that’s right out.

So I made a deal with a friend to trade meditations and playlists. If you’d like to dance along to the first meditation I made, here’s the Spotify playlist.  If you listened to Jo Anna’s video, you understand that the playlist is to be used with a list of questions for contemplation during the pauses and as you dance. So here are the questions for this playlist:

  • Song 1: What is your body hungry for? What does your body long for and desire? What does your body want?
  • Song 2:  What would wake your body up? What would make it feel alive, invigorated, excited, thrilled?
  • Song 3: What is holding your body back? What frustrates your body, makes it feel hurt, restricted, deprived, shamed? Where do these restrictions come from?
  • Song 4: What comforts your body? What would make your body feel relaxed, warm, open, nurtured?
  • Bonus question for the final pause: What do you want to take with you from this meditation today? What seed are you planting?

If you enjoy dancing, you’ll probably enjoy this practice. If you try it, I’d love to hear about it, and if you want to join in the trade, make me a playlist with meditation questions and I’ll make one for you!


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How to Make Someone Happy with $1

Cake mix. It costs a dollar for a box. The store brand’s as good as the name brand. And with it, you can make most folks happy.


Some tricks to make cake mix into something more magical:

If the cake mix is a chocolate version, when you grease and flour the pan, don’t use flour. Use cocoa. Better looking and tasting.

Chocolate ganache makes everything better. Melt chocolate with a little butter or cream until it is melted and glossy. Let it set up just a bit and then drizzle it over the cake.

These stars and lots of candles turn everything into a party. This fancy, rotating, singing candle is also cool AF. Chanukah candles are thicker and more substantial (and longer-lasting) than birthday candles, and sparklers also up the dazzle.

Cake mix is a solid template for creativity, too.

I made a pumpkin spiced chocolate gingerbread cake by taking a devil’s food cake mix, substituting pumpkin puree for the oil, and adding 1 and 1/2 teaspoons pumpkin pie spice. When it came out of the oven, I made a glaze with 1 cup of powdered sugar, 3 tablespoons sweetened condensed milk, 1 teaspoon of cinnamon, 1/2 teaspoon of nutmeg, and a 1/4 teaspoon each of ground ginger and cloves. (Whisk them all together and add a tiny dab of milk if it isn’t drizzling consistency. Then drizzle over the cooled cake. This glaze is very intense, in a good way.)

You can make a booze-soaked cake (rum cake, whiskey cake, liqueur cake)  by substituting 1/4 cup of liquor for the same amount of water when stirring up the cake using the box directions. Bake the cake in a (greased, floured) Bundt pan, and then stir up a soaking sauce: Combine 1/2 cup of butter, 1 cup of sugar, 1/4 cup of water in a saucepan and bring it to a boil to melt the sugar. Take it off the heat and add 1/2 cup of liquor. While the cake is still in its pan, cooling, prick it all over, deeply, with a fork and pour on 1/2 cup of this sauce. Let stand 15 minutes. Turn it onto a serving platter and prick it all over again and pour the rest of the sauce onto it.  Yum.

You can make a crazy multicolored marbled layer cake from white cake mix adapting the technique for coloring the cake batter from this recipe. Cake mix is a blank canvas for happiness! Make some cake mix art for your dollar (or your Nana’s birthday) today!


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Our Favorite Experiences in Venice

There are a number of neighborhoods, sestiere, in Venice. Here are the things we loved most in each of them.

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On Burano

Dorsoduro is full of art galleries, the Accademia, the Peggy Guggenheim. At Christmastime, the Christmas market is here, and the Ponte de Accademia, a wooden bridge that spans the Grand Canal.

San Polo has the Rialto bridge and markets. We wished we spent more time here.


Cannaregio is where the original Jewish ghetto was — the German Jews who lived and worked in Venice were required by law to be locked into the city’s foundry (geto, in Italian) at night. They couldn’t pronounce “geto” correctly, so their mispronunciation became the word we all know. There are three different synagogues in Cannaregio, for their German, Italian, and Levantine Jewish communities who lived (and some still live) there, a Jewish museum, some Judaica shops and kosher restaurants. But Cannaregio is not only a Jewish neighborhood; it’s where most Venetians still live. There are quiet streets, ordinary shops that don’t cater to tourists (marine repair shop, anyone?), and lots of bacaros, quiet trattorias, and magical storefronts with artists at work. My favorite street in Venice was the Fondamenta della Misericordia in Cannaregio. In Cannaregio, have the 10 euro lunch special at any little trattoria that advertises one on its chalkboard outside, then spend the afternoon at the Jewish museum. See the gorgeous synagogues, try the traditional Jewish pastries in the gift shop, and stop by David’s shop in the square outside the museum for the best prices on Murano glass jewelry anywhere in Venice, including on Murano itself. Then, when it gets dark, admire the giant menorah if you’re there during Chanukah like we were, and wander along the Fondamenta della Misericordia for cicchetti and spritz.


Cannaregio is also where the Fondamente Nove is, on the banks of the Venetian lagoon. Walk through the hospital, which looks like no other hospital you’ve ever seen, and look at the ambulance boats. Wander by the several furniture restoration shops until you get to the water bus stop.


The Hospital

From here, you can take a ferry to Cimitero (the Venetian cemetery, on an island), Murano and Burano, both stunningly lovely and also obviously touristy. Murano gets sleepy in the afternoon — go in the morning if you want to see glassblowers at work. Burano is like walking through a storybook: Shopkeepers spill out of the colorful storefronts speaking four languages, hands full of lace and glass and cookies. Wander away from the commercial streets to find quiet parks, medieval churches, December roses, and that incomparable red-gold-blue Mediterranean sunlight.



26023981_10213904293308855_5361053052358937817_oSan Marco is the most touristy part of Venice. The streets are full of designer chains (Ferragamo, Chanel) and souvenir shops. Scam artists in San Marco have Venetian romance — if a man falls before you on bended knee and insists you take a spray of gorgeous roses, free, just because you are that beautiful, he is looking to extort a handful of euros from your boyfriend. (Just say no thank you and keep walking.)


Every bridge has its gondolier, and you have to do that, of course, preferably at night, and paying extra for singing. We asked our gondolier what was his favorite part of Venice, a question he found too personal and ducked. But he did say that he always gives tourists the same advice: Go to San Marco and then get lost.


Getting lost is indeed the best part of Venice, in San Marco or elsewhere. Some of the streets are so narrow that only one person can just walk between them, and the stone walls of the buildings are seamless and high. Walking in Venice is like walking in a maze, that is pockmarked with loveliness. An ornately carved ancient door here, a brilliantly colored stained glass window there, shining shop windows full of magical objects, warmly lit bays where you can stop to buy a drink or a snack to carry as you walk. It’s like a scavenger hunt, a maze full of endless magical little corners to discover, and you’re wandering through it nearly alone in December, only sometimes running into other people doing that same thing: Venetians, smoking and carrying keys, tourists, looking  down at their phones for a clue.


All you have to do is walk. Walk, look, take pictures. Stop to buy a cappuccino, a Prosecco, a snack. Listen to the guy playing Sinatra standards on saxophone in the square. Step inside a church and buy a candle. Consider a mask, or a journal, or a handbag, or a feathered pen. Cross another bridge. Wander into a church full of priceless instruments. Listen to the quiet, to the miracle of footsteps on stone.


Venice is not a food city, in Italy or not. Especially not in San Marco, but even in general: It’s a small town that has spent centuries catering to the anxious bellies of the world’s tourists: the food is overpriced (especially if you have the temerity to want to sit at a table) and bland. Actual Venetians, as far as I can tell, live on a diet of coffee, cigarettes, ombre and conversation, fortified only by the kind of snacks you eat standing up — cicchetti (bar snacks). gelato, pastries, kebabs. These things are the most delicious things in Venice. So walk out in the morning and get coffee and pastry, and whenever you’re hungry, stop at an open window or a back alley bacaro and have a snack. Stay open to the secrets and conversation. We wandered down a quiet back alley late one night until we were embraced in a circle of light and conversation outside Bacaro Da Fiore  Too late for their amazing-looking cicchetti, we ordered tiramisu instead, eaten standing around a little barrel while the owner clapped his guests on the back with apparently genuine joy at having company. Love letters dangled on strings from the ceiling, and it was hard to tell whether dessert or decor or company were sweeter. Venice is a convivial place.


Also, you have to see the cathedral for which San Marco is named, from the outside at least, and at Christmas there’s an enormous tree in San Marco square. Dress up and go have a drink at Caffe Florian, the oldest cafe in the world. Go to the bar in the back and order a drink; much cheaper than a table, and the bartender’s flair is continental and unmatched. The glasses are elegant, and there are free bites with your cocktail. Try the perfume at the gift shop counter: for tourists, sure, but the scent is evocative and memorable.




In San Marco we also went to our palace ball, at Musica a Palazzo, a private, members-only opera appreciation society (you can join, and buy opera tickets, at the door, but must make reservations in advance). We saw La Traviata, done here with relentless romance as a chamber opera that takes place in a 15th century palace, and that places the audience right into the text. You’re a guest at Violetta’s party; she’ll kiss your cheeks and pour you Prosecco. You’ll hold her hand in the drawing room while she cries and writes her fateful letter to Alfredo. You’ll be sitting bedside as she and Alfredo say their last goodbyes. In between acts, you’ll drink bellinis under the chandeliers and look over the grand canal. Walking home after that adventure, we looked up at all the chandeliers winking out through the windows of half the palaces in Venice and felt as if the whole city was a fairy tale.


Castello is the largest sestieri in Venice, centrally located, and has both touristy parts (where it abuts San Marco and along the Grand Canal) and more residential areas. We stayed here, first in a lovely, simple, inexpensive airbnb, which had everything we wanted including charm. Everything in Venice is full of art and history, and our apartment was no exception. It was on a peaceful, quiet street that immediately felt like home, with the canal and gondoliers right out our bedroom window.

The last night in Venice we splurged on much more luxurious Castello accommodations: the Damascus suite in the Metropole hotel. I’d been turned on to this place years ago by (sadly now offline) Luxirare; it was this room that made me want to go to Venice in the first place, so, although staying there was absolutely not in any sane budget I could draw up for us, we had to find a way. The rack rate on this room was upwards of 4,000 euro; luckily, in the Christmas offseason it was available for 660, which is still absolutely insane for my little family. Still, for a once-in-a-lifetime-surprise I booked it.

26116435_10213904742680089_670158453529197583_oThe degree of luxury we enjoyed in this little hotel was almost too rarefied to be tolerable to ordinary mortals like us: My teenager took one look at his private room and bath in the suite and changed into a suit just to feel comfortable sitting in it. The bellman monsieur and madamed us charmingly up to our rooms, gave us elegant, heavy keys that weighed down our hands with their antique charms (what hotel still uses keys?), and left us to enjoy the stunning view of the lagoon and the Church of San Giorgio Maggiore and our ridiculous rooms.

25734418_10213904759320505_1798280131946620867_oWe were welcomed with a tray of treats and a note from the hotel’s owner — elegantly-wrapped chocolate, biscotti, Christmas cake, and six clementines so absolutely perfect in scent, appearance, and flavor that eating them required a meditation.

26170338_10213904761200552_6410096488069475740_oThe room was tiled in mirrors and iridescent Murano glass tiles; there was an enormous Moroccan-styled bathtub in the center of the space, with an honest-to-god gargoyle-spitting fountain in it. There are three more bathrooms besides in the two bedroom suite, including a steam room and needle shower with a ceiling lit up with pinprick lights to look like the stars of the night sky. Breakfast in the lush, romantic garden of the hotel includes everything you could ever dream of wanting on a giant buffet, and the common spaces are full of antiques and collections of mystery objects, crucifixes, handbags, cigarette cases in vitrines, like a museum. Moroccan furniture and giant Murano glass chandeliers. The scent of incense and camellias. It’s the kind of place that allows you to briefly convert money into fantasy. Worth every penny.





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Venice is Something Else

American friends who had been to Venice said to me, before we went, “It’s like a theme park.” This was said in kind of a disparaging way — Venice is a tourist trap, a fake place.

This isn’t fair. Venice is more a memory of a place than a place itself, an anachronism, a place that quietly stepped out of everything else happening everywhere on earth and decided to be first above all beautiful, and second silent, as mythical as a mermaid.


There are no bikes or cars or mopeds in Venice. Venice is dead quiet. There are motorboats, but they have to share the narrow waterways with gondolas and crumbling foundations and they mostly go no faster than your granddad out trolling for bass. You can walk the narrow silent mazelike streets at night and hear your own footsteps.


Venice is a city of artists. Tourists are everywhere, sure, on holiday, spending money, looking to be delighted, and they have been coming with their open eyes and hearts and wallets for generations. Every other shop window belongs to some small-time artisan, often the great-great-great-granddaughter of some other artisan, who has left the legacy of speaking at least four languages and making any number of things. Masks, leathergoods, lace, marbled papers, blown glass jewelry, ornaments, and once, a whole storefront of miniature storefronts, tiny facades carved by the wild gray-haired man in the back.


Unnecessary things are beautiful. We stumbled on a paint store, with a giant rack of wallpaper parked out front, because it wouldn’t fit in the narrow shop inside and leave shoppers anywhere to browse. Stacks of 10 foot rolls of gilded brocade wallpaper. Silver, gold, bronze flowers blooming all over the street. There are costume shops everywhere, as if there’s a deep abiding necessity for everyone in town to have a tweed Santa suit, a dress that makes you look like a glass angel atop a Christmas tree, a ballgown fit for Cinderella just before midnight.


People with instruments are just walking around, opera singers. The gondoliers all sing. Artists everywhere, quietly working. Morning until night, when the artists take out fancy keys and lock up their shops and walk quietly down the dark streets, heels clicking. Medieval churches, emptied by modernity of their worshippers, hold vitrines full of priceless Italian violins.


And the artist ghosts walking through Venice: Vivaldi. Proust. Thomas Mann. Shakespeare. If Shakespeare woke up in Venice tomorrow night and walked out looking for a drink, everything he saw would be entirely familiar.


And that’s why Americans say it looks like a theme park. Because the only things they’ve ever seen that looked like that were fakes, copies of what Venice actually is and always has been. No wonder there are so many decadent costume shops. No wonder it’s famous for masks. All of Venice is a stage set, and you whirl and sneak through it making scenes.

My handsome teenage son bought himself a handmade Venetian mask for a pretty penny in one of these generations-old shops. Elegant and gilded, this mask was hanging from its ribbons, hungry for its own story. The artist who sold it to him wrapped it up in marbled paper, and started to tell it: “You could wear zis to a costume party, or a birthday party…..” and she paused, and gave him a long look, a wink, a small smile. “Or….. Sometheeng else?”

That is Venice. It’s something else.



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Venice for People Who Usually Stay Home in Saint Paul Where They Belong

25438875_10213835984761184_2987213332634451532_oFor many years Venice has been on my life list. I changed jobs about a year ago, to a teaching job, with summers off and winter and spring vacations, which is not a luxury I’ve enjoyed before. So I reorganized my life list to slot in things I really wanted to do in that vacation time. This summer I wrote two books (forthcoming this year from Routledge Press). Top of the list for Winter was Venice. I’m not sure I really planned to pull the trigger, but then TAP Portugal had cheap tickets, and one day last Spring I called up my husband and he couldn’t talk me out of it.

25488487_10213847716294465_4641911123334297781_oSo we went to Venice, as a winter break surprise for our fourteen-year-old son, who had never been abroad. We hadn’t been overseas in more than 20 years ourselves.

But airline tickets are now so frequently so cheap, and the world now is so globalized, I can’t imagine why we let it go so long. Was it just fear? I guess so.

I feel like there are travelers, who think nothing of going all over the world, and there are non-travelers, who stick close to home, or at least travel mainly in the U.S. And I’m sure there are lots of reasons for that — no money for travel, no vacation time — but I wonder how much of it is just fear. Worry about managing in a foreign country. Worry about all the things that could go wrong.


I had lots of worries. We didn’t speak any Italian, how would we manage? What if our credit cards didn’t work, and we couldn’t get money, what would we do then? Would our cell phone work? What about the electricity for a charger? How would we get around without a car? Would we get lost? Everyone says you’re always lost in Venice…..

When you’ve never been to a place, and can’t picture it, all the conflicting stories about it on the internet are just distressing noise. It’s hard to know what to plan for. And because this trip was meant to be a surprise, I planned it all myself. My husband knew we were going, but there was little opportunity to discuss it with him when our son wasn’t around, and it seemed nicer to let him enjoy the surprise as well. That meant that any mistakes would be my fault!

I’m obviously not an experienced traveler, but the world has become so globalized, and technology has made everything so easy (at least in Europe) that my worries seem ridiculous to me in retrospect.


Language: Venice has been a tourist trap since the Middle Ages. You can get around in Venice with no Italian as easily as you can get around in Montreal with no French. (And if you’ve avoided going to Montreal because you don’t speak French, let me assure you, if you speak English in Montreal because you have no other choice, people will find a way to help you.) In Venice, some shopkeepers came to the door of their shops asking us “English, French, German, or Italian?” (This query was always delivered in English, oddly enough.) Most people spoke English. When they didn’t, they seemed entirely comfortable with pointing, gestures, sign language, and the general practice of being patient and kind around a language barrier. Venice is very used to tourists.


Credit cards: I made sure I knew the pins on my credit cards before we left, and I had absolutely no trouble getting any Venetian cash machine to cough up money. Credit card transactions, on the other hand, were sometimes declined, for reasons that were entirely unclear to me. (Trouble talking to satellites in space?) Once, my credit card let me pay a very large hotel bill, and then immediately declined to  pay a second smaller bill to the same hotel. My card refused to pay for Vaporetto tickets from a ticket machine, but was happy to pay a ticket seller five minutes later. A few times during the trip my credit card queried me about fraud, but never for the same transactions it declined. I was glad I was carrying more than one card, and glad I had cash as a backup on the one occasion none of my cards would work (not the hotel bill, thankfully!). Venice seems to prefer cash to credit for small transactions. If you’ve got more than one adult traveling together, carry several cards, is my advice, and get some cash.


Cell phone: We have a cheap newish Samsung and are on Verizon. I discovered before I left that Verizon lets you pay $10 a day for a travel pass that will allow your phone to function in most other countries (including Italy) as if it were at home — data, text messaging, phone calls — with no additional fees. We set it up while still at home, and it worked seamlessly out of the country and stopped charging us when it detected we’d gotten back to New York. Absolutely worth it to not have to worry about WiFi and still have access to Google Maps. Newish cell phone chargers generally will adapt on their own to the differences between U.S. and European electricity,  but you will need to buy an adapter plug to stick the damn thing into a European socket. A local to us travel store sold us the adapter before we left.

Navigating, transportation: Google Maps works in Venice, and is a comfort to have in a city that is more like Brigadoon than any normal place, but it works approximately. We encountered lots of other tourists lost in back alleys and dead ends looking at their phones as if we were all on some sort of demented scavenger hunt. You share a laugh and back out, cursing the phone. It is helpful to have a map in your hand, though.

I couldn’t picture how the transportation worked in Venice from my advance reading, so let me try to explain for anyone in the same boat, hah. To get from Marco Polo airport to wherever you’re staying in Venice, my understanding is that the three options are bus-to-public-waterbus/vaporetto, water taxi, or Alilaguna airport water bus/vaporetto. The water taxis are super glamorous, gorgeous mahogany vintage speedboats with leather interiors. I suspect you will feel like James Bond if you take one; I wish we had. They are more expensive than the Alilaguna, but not as much as you might think — it was 27 euro for a round-trip Alilaguna ticket, per person, and we bought three, whereas I understand it’s about 110 euro to have a water taxi take you from airport directly to hotel. A little more than twice the price, in other words, for all three of us to take a water taxi. A water taxi will seat 10, so if you’re going in a big group it might even be cheaper to go the glamour route. I’m glad we didn’t bother with figuring out how to catch a bus to the public waterbus; I don’t think it’s much, if any, cheaper than the Alilaguna, which was just one step. You can purchase tickets for the Alilaguna from a machine in the airport in theory, but we couldn’t get it not to decline our credit card. Luckily, a kiosk also sells them as you leave the airport. Your hotel or airbnb should be able to tell you which line to take and where to get off; it’s straightforward, I promise.

I had figured we would need waterbus tickets in Venice to get around, but this was not the case. All of Venice is about the size of four or five Central Parks, and it is all connected by bridges. You walk in Venice, everywhere. We bought one 24-hour waterbus ticket to go out to the islands — Murano, Burano, Cimitero — but otherwise all you need to get around any part of Venice is feet. (Venice seems like an impossible place to work a wheelchair, incidentally — there are bridges everywhere, and only the biggest have ramps instead of stairs. I did see a few people using wheelchairs, so there must be a way, but if you’re a wheelchair user, you’ll need to do more due diligence than I can provide to figure out how to work the place.)

I’d booked the major things in advance: this airbnb for most of the trip, a spectacular hotel I’ll talk more about for a one-night finale, and a few activities. All of that worked beautifully.

I’m planning a couple more posts to talk about the trip itself, and the things we did and loved, but I wanted this post to encourage you to go in the first place — or to go someplace else — to be a traveler if you want to be. The world has been made global and small. There are downsides to that — we are losing our sense of place and uniqueness — but there are upsides, too. It’s easy. You can do it. I promise.




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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!