Geometry: back to work 


It’s been a good long while since this particular project occupied my attention and focus.  However, I’m currently motivated to finish it — or at least finish the nine pages that I already have outlined and planned.  There are six more pages that are unplanned except for the margins, which means that I have a total of fifteen pages left to write, and maybe a card or panel to put in the pocket of the book, an afterword of sorts to explain the project a little better than I did at the beginning.

What project am I talking about? This one, the geometry book that I began a long time ago practically in a galaxy far, far away.  In fact, from the earlier entries from 2013, I can tell that I was already about sixteen pages into it.  Now, I’m thirty-seven pages into it, and I have fifteen left.  I’m almost the opposite point in this project as I was four years ago.  Funny how these things circle around, right?

The current pages, #36-37

Of course today is the day that I made a mistake.  I drew out the process of comparing 1:√2, and didn’t discover my error (on the right-hand page) until I had already inked the diagram and written the explanatory text.  Always check your work in geometry before you render it in pen!

The next pages laid out (and upside down for some reason)

No matter.  I had the room to be able to describe the process incorrectly, add in A WARNING IN CAPITALS AND RED, and then offer the correction. Typical medieval manuscript at this point, really — sometimes errors creep in, and the lowly scribe has to figure out how to offer the correction clearly and legibly in less space.  I managed.

As I said, I have nine pages remaining in this project that are already laid out.  A lot of this project is me working through Andrew Sutton’s book, Ruler and Compass from Wooden Books.

Why did I return to it, though? Well, first, I’m trying to clear my desk of unfinished projects. This one has been a big one, and it’s been on my mind to complete for a while.  But for another, I recently took up the opportunities and challenges of tutoring again.  And I’m tutoring a few young people in geometry.  So this project is serving to lubricate and rub the rust off of my geometry skills. Even so, I’m finding that the knowledge of actual geometric proofs isn’t quite as useful as one might imagine.

A lot of the work that students do in geometry class these days appears to be algebra. There will be one diagram (with a note beside it to say, not to scale or not rendered accurately), and then a lot of algebraic notation, and the student is expected to work without a ruler and compassed just their brain power and maybe a calculator, to solve the problem.

Say what??

I don’t understand.

Are we teaching geometry, or geometric algebra?  It looks like the latter, rather than the former.  And I understand that teaching actual geometry is challenging, and that it involves looking at a lot of diagrams and working out a lot of constructions by hand… but heck, that’s what we do as human beings. Isn’t it?

I said to someone on Twitter today that

Screen Shot 2017-05-15 at 2.43.25 PM

pardon, I can’t figure out the ’embed tweet’ system for my server.

But that’s (more or less) true — we use our hands to instruct our brains, and vice-versa.  How do we actually learn geometry if we’re not using the tools that geometry has used for thousands of years (or reasonable electronic replacements, though I’d argue that such tools are not as good as actually using hands to manipulate a compass)?

In any case, here’s a place where abstraction often gets the best of us.  I think it’s time to bring back some actual geometry to the classroom, and not simply ask students to do it algebraically.  This is a set of skills that belongs in our students’ hands, and not just in their heads.

The river and the column

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Overlooking Astoria, WA is a copy of Trajan’s Column, adorned not with the sculptural story of the conquest of Dacia, but the sgrafitto (layers of brown and white paint scratched through to produce figures of realism and shading) history of the conquest of the American West: Captain Gray’s discovery of the Columbia; ​​Lewis and Clark’s 1804-07 expedition; John Jacob Astor’s 1811 fur trade colony; the Oregon Trail of 1846; the railroads of the 1880s. The First Nations get a 3/4 turn at the base of the column. The rest is given over to a celebration of American conquest and control of the Pacific Northwest, and mastery of the mouth of the Columbia, visible beyond the bridge. 

In typical American fashion, the gift shop and the car park for buses have both been put on the fore-lawn of the Column, meaning that both tour buses and this ugly mobile home structure will be in all your panoramic photographs, trying to capture the grandeur of empire on display.  There’s also this replica Chinook canoe of cast concrete that blocks the view of Fort Clatsop, Lewis and Clark’s winter camp. Sight lines matter, people. Would a site plan and some symbolic planning kill you people?

But it’s part of the reason the magical thinking of this place is so confused. This Astoria might have wound up the capital of the American West with a street plan more like a Paris or a Washington DC. Yet it’s always been a company town committee to making money — a gift shop and a bus park and a cannery instead of wide boulevards and cannon pointed out at the river from a fortified Esplanade. Instead, there are trolley cars, used bookstores, and crust punks from Kansas and from California, and little boutiques selling what my dad calls landfill — mementoes of a trip that don’t really hold any memories, destined for the transfer station (i.e. The Dunp) the next time someone in your house picks up a book about cleanliness and minimalism. 

Orange cones, the symbol of empire

Astoria is a town at the edge of the American empire. It knows its own history as a fur trading station and cannery town; and it prizes that history. But it also has a self-sufficient streak that’s visible in a preponderance of pink and purple punk haircuts, punk fashion statements among both natives and visitors, coffee houses more than halfway toward the Amsterdam model (but not quite there yet either — no open pot sales, more of the dispensary model), medical dispensaries of the herb, and section 8 housing in the Astoria Hotel. The optics are… interesting.  There’s a fair bit of the Midwest here, too — Kansans and Nebraskans and Okies living out of their cars while they look for work. 
The tourists ahead of me in line for ice cream chatted in fluent English with the scooper about his gulf war era tattoo in Arabic script; then they bring ice cream to their hijab’ed wives and children by the trees a short distance away, lounging in the sunshine after a meal of fish and chips, cheerfully chatting in Arabic the whole way. There was no sense of animosity here, just people going about their lives in a complicated time, under a rare burst of sunshine. 

Astoria feels like middle America, a Michigan accent crossed with New York, trying to become cosmopolitan while retaining a Pacific character. The Chinatown is largely gone, but the bookstore has the largest collection of Buddhist and Taoist books and trinkets I’ve ever seen outside of a retreat center or a Tibetan’s shop. How many translations of the I Ching does one really need? As many as one’s tarot decks, I suppose. Or dictionaries. 

So yes..  Astoria feels like it’s in America. But. It doesn’t feel like New England, and it doesn’t feel like the South. And it doesn’t feel like California or Chicago either. It feels a bit like an occupied place, in some ways, a place where the cruise ships come when they’re transferring from the summer cruises off Alaska to the winter cruises off Acapulco.  Which it is. 

 The US Coast Guard maintains its Lifeboat School here, because the Mouth of the Columbia is the Graveyard of the Pacific. The river mouth is so dangerous that there are two different pilots’ guilds to handle the two challenges: crossing of the bar, and the navigation to Portland. Our captain says that pilots make the best money in the industry, but their skills are non-transferable: the examination is a pass-fail drawing of the river mouth on a blank sheet of paper that has to be accurate to within about five feet at the chosen scale; currently only twenty people have that honor in this river mouth. 

South of here are the Cascade and Olympic ranges of mountains. Its temperate rain forest over there: massive amounts of nearly constant rainfall.  We visited the remains of Fort Clatsop today where Lewis and Clark spent the winter of 1805-6: 106 days of constant rain, twelve days excepted.  A pair of fifty by fifteen foot log cabins facing one another across a twenty by fifty “parade ground” of  mossy wood chips; a fifty yard walk to drinking water; a two hundred yard walk to the canoe embankment. Four years after the encampment here, no one could find it in the rain forest. It had been swallowed whole. 

The Astor Column was raised in 1925 as part of the building of the railroad empire of Union Pacific.  By 1985 it had deteriorated to the point that it needed a two year repair and restoration. By 2015, it needed another complete repair and restoration. The damp gets under the stucco and plaster and paint, and the wind chips it off.  All along the river, dams chip and blister under the weight of water and the threats of environmentalists and the needs of the salmon and the obligations of the treaties that Jefferson and others made with the First Nations of the river. Nothing that empire values will last here forever —  only the mountains, the cedars, the salmon, the cedars, and the river. 

Empires rise and empires decline. Earth abides. 


The River and the Moon

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There is no camera gear in the world that was up to the task of capturing the moon shimmering on the western water this morning, as we cruise eastward under cover of night toward Portland, OR. A little after 4:30 there was a knock at my door: dad, in his underwear, beckoning me from our cabin to the stern deck, there to see the setting Moon framed between mountains. A bend in the river took it behind those self-same mountains, a few minutes later.  But it was enough — the Moon is capable of shattering our unhappiness, our fear, our terror, especially if we encounter it in the right state, half-asleep yet startled from our beds.  We wake thoroughly to encounter the world in silence.

It was the same at Multnomah Falls. Despite the crowds, the rain, the place was tremendously green and lush. Despite the fact that we spent an hour round-trip on a bus that smelled of diesel to get there, and had maybe 30 minutes at the Falls, there was a serenity there, a joy. A bus load of kids from some school trudged up past us on their way to the upper bridge, looking lonely and wet in plastic ponchos. They came down the hill again cheerful, connected, peaceful. They were collecting high-fives from complete strangers on the way down. I myself got twenty-seven high-fives; it felt like a reunion with humanity.


Water and the Moon both reconnect us to ourselves and to each other. They remind us of our humanity, our connection to each other.  And it’s often enough to wash away loneliness and fear. The Moon has a tendency to remind us that everything will be all right, eventually. Give it time. Give it another go-round. This too shall pass.

The river and life 

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The world and the river


There’s something remarkable about going to sleep on a boat on a river in the middle of the wilderness, and waking on a boat in the middle of a city.  An old city. 

Kennewick is just downstream from here. In 1996, some bones were found sticking out of the wall of silt along the Columbia River, barely a mile from where I am now. Dated to about 9000 years before the present, the local natives, the Umatilla nation, refer to him as “The Ancient One”.  People have lived here in dense settlements for a long time. When Lewis and Clark passed through in 1806, they found thriving and wealthy communities. 

Those communities are still wealthy. The dams of the Snake and Columbia rivers produce some of the cheapest electrical power in the world: glacial ice of the Rocky Mountains returning to the Pacific Ocean and the high desert between the Cascades and the Rockies as irrigation, wild fish farm, and gravity-powered energy. This water lit the fire of ten thousand suns: the electrical power of the Pacific Northwest fueled the Manhattan Project, refined the uranium and plutonium for the atomic and hydrogen bombs, and smelted the aluminum that built the ships and planes of World War II. 

Today, the electricity produced here for a penny or two per kilowatt-hour gets sold in Los Angeles, CA for 10-17 cents apiece: Hollywood’s blockbuster movies and CGiI special effects are born of this river; so are Seattle’s Amazonian online commercial wonders and Redmond’s Microsoft magic. The bathtub locks and salmon ladders make commerce and fishing possible all the way to Idaho, where Clarkston and Lewiston sit as the easternmost ports on the American Pacific coast, sending timber and grain and paper to the world. Even Silicon Valley’s technological achievements are fished out of this river system like so many salmon of wisdom — rerouted electrons shimmering like so many stars in the heavens, refashioned into to stuff dreams are made of. 
Hail, Columbia!

It occurs to me that Columbia is herself a goddess, but one who shares a certain cachet or category with Cerridwen. The poet-technologists watch her flowing, bubbling cauldron, and withdraw the gifts of prophecy and foresight and poetry and art from these waters. The goddess chases them all the way to the sea, shedding her gifts behind her as she goes, steadily and unhurriedly demanding what is hers— the right to reunite with Great Ocean. I don’t know anything about her identity in Native myth… I’m going to have to find out. 

Sonnet for Shakespeare on his 453rd Birthday

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Really, he doesn’t look a day over 380…

O Bard immortal by the Avon born,
in humble cottage to ambitious dad:
I give you greetings on your birthday morn
with tidings: The world so wide still is glad
that the work of your life and pen yet lives.
The curtain never comes down in this globe
but there is applause; each hearer forgives
some tin-tongued actor in a worn-out robe,
when your Hero emerges from the grave
or Hamlet drinks down the pearl of great price,
or Hotspur leaps to war, foolish and brave,
or Antipholus’ friends see him twice.
The faeries in their revels bless us still,
and your fame? Endures forever, sweet Will.
Composed 23 April, 2017.

Magic: chops

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Owl chop

I recently read Benebell Wen’s brilliant book The Tao of Craft after hearing her on Gordon’s show.

One of the things that really caught my attention in the book was the idea that a Fu sigil or talisman should really be signed and sealed — that is, in the same way that a decree from the emperor would be signed and sealed, a Fu sigil carries the authority and mark of the creator. I explained this idea to my partner, who thought it was equally interesting. And so I decided to make a chop — a seal stick — for her, in part as practice for making my own. Which is part of the reason I made that captive ball a few days ago: I want to get better at wood carving, because it’s a useful skill to pair with woodworking generally, and because it will be a nice fit with the automata work I intend to do when the woodworking shop is up and functional again.

My partner frequently uses an owl as her emblem, so I went online and searched for how to carve an owl. This isn’t the one that I used, but it’s close enough — a series of photos of an owl carving.  I used basswood, because I don’t carve jade or soapstone (slightly different and sharper tools, more patience and care required); and I had the tools for this already. It’s a fairly simple procedure to carve an owl. It was also fairly simple to reverse and cut the runic-style emblem my partner uses for her magical sign, into the base of the chop. Except, I still screwed it up — and I’m going to have plane the bottom of her chop flat, and then cut it again.  I don’t have the tools out to do that yet, butI can certainly do something else while I wait to make that set of tools available.

img_3232From there, it was fairly simple to find a procedure for carving a bear. His advice about frequent sharpening is good — I sharpened my cutting tools about six times in the course of carving the bear, and I still wound up using too much force and chipping his right arm off.   I chose to do a statue against a pillar for my bear, because I want to have a place to practice chip carving, on the back of his pedestal; I already completed the small chip-carvings around the base, and I cut my own version of an emblem into the bottom of this seal.  I still think this won’t be my final seal, for me — the missing arm is a bummer.

So here we have an object that shares kinship with a magical-scribal-calligraphic tool from Chinese Taoist magic: a seal. But it’s carved in a Western style, with a Western character sign that indicates a person. And it could be used in western magic to do the same thing it does in Taoist magic: sign and authenticate and command the results in the name of the practitioner.

Is it cultural appropriation — Or cultural inspiration— to draw on the techniques and tools of other traditions to add to and build to the existing Western tradition? Ironically, I think this is part of the reason why I broke my bear at the last possible moment… because the character on the base of my chop is one I chose for myself from Chinese characters a decade or more ago.  But it’s not my emblem to use, nor my tradition. And so, this one is broken, and I’ll have to make it again.

But I still think it showcases what’s possible, what’s within the realm of workability.  The folk tradition of carving already exists in the West; the tradition of ‘enlivening’ statues and statuettes has been part of Western magic since Egypt; the idea of a personal seal to accompany a signature has fallen out of favor in the last two hundred years, but there are still signet rings in the Middle Ages, the Renaissance and early Enlightenment periods… and we have evidence of signets or seals all the way back into Babylon and Sumer, at least.

Maybe it’s time to re-awaken the idea of a personal seal.

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