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.



I have a much better appreciation for the volvelles, or circular computers, that survived from the Renaissance and the Enlightenment to the present day.  Fragile, finicky and prone to moving right went you don’t want it to, the volvelle is the brainchild of Raymond Llull, a Catalan Catholic theologian of the mid-1300s AD. 

I want to make a volvelle to go on the inside front cover of a hand bound book I’m designing. As you can probably tell, this volvelle is astronomical in nature, but Llull’s was intended to be logical and grammatical, designed to explore theological concepts and train missionaries to work in Islamic regions (he failed to win many converts).  

The volvelle remains. This one has pointers for the seven visible planets of medieval astronomy (less the Moon, because I lost the paper cutout between cutting it out and assembling the volvelle). It also has a horizon line, and a “sphere of fixed stars” that includes both the Decans of the Zodiac and the Mansions of the Moon; as well as the fixed ground of the twelve houses of astrology. 

And it doesn’t work as smoothly as I’d like. I need to replace the brass brad with a paper system, as is used in medieval and renaissance volvelles. The brass brad is too thick, and doesn’t allow for smooth or independent rotation of the parts. Back to the drawing board. 

Podcast: Delphi and the Pythia

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A number of years ago, I did a group of podcasts for my students on various themes in ancient Greek history.  This one, about the Oracle of Delphi and the Pythia, was one of the longer ones, about 7 minutes.

Something came up elsewhere today that made me realize that someone out there might have use of it. And so I have posted it, so that I have a URL to point to.

Latin & Illustration

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Zentangle Decorll of my students in Latin lately have been doing a good deal of Zentangle work.  They’re working on a project on Greek and Roman deities. I’ve done this project before, with good results, as part of their effort to produce a miscellany together of their Latin texts.

Their project has five parts:

  1. An illustration of the god or goddess — a hand-drawn picture
  2. A series of borders or other decorative elements
  3. Their own handwriting — for good or bad
  4. The written Latin texts, in four sections:
    1. A descriptive bit, that explains who the god or goddess was to the Romans (using a lot of declarative sentences, with transitive and intransitive verbs, and complementary infinitives.Zentangle Decor
    2. An explanation of the god or goddess’s attributes — that is, lots of use of the linking verbs est and sunt, and proper use of adjectives.  This is also a chance to show off the three styles of poetic form that we’ve practiced this year — parallelism, chiasm, and epithet (through the use of the genitive case of nouns — you see how I’m using this project to demonstrate learning?).
    3. A ‘Prayer to the god or goddess’ — that is, demonstrating the use of the imperative/command form of Latin verbs: “O Juppiter, Domine Fulveorum, adiuve et fert auxilium! Noli me Necare!” That sort of thing.
    4. The telling of a myth or story about the god or goddess — again, an effort to use all of the styles of sentences we’ve learned this year, and maybe throw in some ablative of means or instrument or time, since my sixth graders basically won’t have time to learn any tense but the present tense.

But the truth is, that if you want your students to do this sort of project, you have to give them the tools to do it.  And I’ve been remiss.  I should have been including some lessons in hand-writing from the very beginning — not because kids need to know how to write in cursive these days, or even in Italic (although Italic is very beautiful, and easy to learn), but because it’s a thing worth caring about, making writing beautiful.

Zentangle DecorAnd that means teaching them some of the patterns and borders that you might like to see on their written pages.  It doesn’t matter if you want checkerboards, or egg-and-dart, or vines, or acanthus leaves, or waves, or sea-shells.  If you want this sort of thing in their work, you have to make it beautiful, and you have to hold the students to a standard of beauty, too.  The aesthetics matter.

For each of the visual samples here photographed, there was a lesson: in how to draw a ribbon; in how to draw a checkerboard; in how to make egg-and-dart designs, and vines, and abstract wave patterns.  Someone is doing Hades/Pluton, so there was even a bit of discussion of how to draw skulls as part of an ongoing border in an illuminated context.

No really: there was. I have pictures, there’s proof. Zentangle DecorAnd that’s cute, as far as it goes.  But what’s important about this, is that it doesn’t really matter if you’re teaching kids to draw pictures of ancient Greek gods or if you’re teaching them to write about Catholic saints in Latin, or if you’re teaching them to write about heroes and emperors.  You’re teaching them to write — and therefore to think — in another language.  And learning a foreign language requires four practices, which I learned from a colleague to whom I’m deeply indebted:

  1. Read it
  2. Write it
  3. Speak it
  4. Hear it

When we’re talking about a ‘dead’ language like Latin, the speak and hear components of that training are hard to come by.  How do you teach kids to care about that?  But more than that, how do you keep it alive in their memory six months from now, ten years from now?  The answer seems to be, teach them to make it beautiful.

Zentangle designs mixed with Latin, and maybe next year with some handwriting education, is producing an ‘heirloom of their house’, something that apparently their parents and they will treasure for at least a few years.  One of my eighth graders just found her sixth grade Latin ‘cardboard codex‘, and had a moment where she just couldn’t even. Will it work for every kid? No, probably not.  But it will be beautiful to some children, and a reminder of long-ago lessons, and maybe it will be a reminder to adults, that not all languages are spoken, and that the language of picture, of rhythmic use of line and shading, tells another story without using a single word.

Tai Chi Y3D359: The Other Side of Horowitz


Wizard of Oz puppet headYesterday, I commented on Horowitz’s remark, “When I don’t practice for a day, I notice. When I don’t practice for two days, the orchestra notices. When I don’t practice for three days, the audience notices.”

It’s true, of course.  When I give myself the right to slack off in tai chi — as I am this morning — I get a morning where I have a lackadaisical practice where just completing the task of doing tai chi is enough.  But lately I’ve found a way to move my practice forward and improve upon it; and the fact that I’m NOT PUSHING MYSELF today may be seen as somewhat disappointing.

But it occurs to me that Vladimir Horowitz’s insight about three days of no practice has an important value built into it.  Which is that sometimes, you have to go three days without practicing to your fullest extent. Sometimes there are things which are more important.

Now… I have practiced tai chi today. But I wouldn’t call it a stellar practice.  It was about as short as it is possible for a tai chi practice to be: Five Golden Coins, and then the form.  Short, as I said.

But it’s been a crazy week.  Next week, tech week for the play begins.  As a result, the production effort on the head of the Wizard of Oz has shifted into high gear. As you can see in the photograph, the head is taller than I am.   It’s heavy.  But now it has eyes and eyebrows, and lips, and a moving jaw.  It will hold together for ten days (I hope!), and it will accept a layer of green paint, and then some other colors for highlights and shadows.

It’s also been all-consuming for about five days. And basically, it has to be mostly-complete today.  So if I haven’t given my tai chi practice the intensity it deserves, that’s OK.  The days for returning to that more intense practice will return.  In the meantime, there’s a different kind of work to be done.

Design: Production


Design: production It turns out that Production is a very different mindset than prototyping.  It’s also not particularly interesting.  Mostly what it is, is doing a similar group of tasks over and over. This morning, it was gluing blocks of wood together to make “boots” for marionette-style puppets.  These will be painted black, and then glued onto the lower legs of the marionettes.  I needed sixteen boots.  So it was just a matter of taking the cut pieces that a kid produced the other day (Wednesday), and gluing them together with wood glue brought from home.  Fifteen minutes, and the task was done.

But getting to that task took four fifteen minute blocks of time — one to discuss the “boots” with the theater director and make a design for them; one to work with the kid to cut and drill the individual parts, who then took another fifteen minutes to cut the blocks; and one to try out (and confirm that it didn’t work) a type of glue that we had a lot of in the Design Lab, but that doesn’t particularly work on wood (as we discovered). And then, finally, the fifteen minutes that *I* spent, gluing the “boots” together.

Design: production Then it was time to go through the same process, going from the prototype of the lanterns that will represent the Emerald City in our production of the Wizard of Oz, to a set of six such lanterns. Designing the lantern was the easy part. Cutting out the six parts for each lantern risked repetitive stress injury, and the potential for deep cuts from a utility knife in my hands and fingers.  No cuts, thankfully — but the risk was real.

And now, there are the parts for six such lanterns, and one assembled.  A parent, who gazed upon it, breathed this sigh of astonishment and said, “But they’re beautiful.”

And they are.

They’re these elegant traceries of cardboard.  Our theater director and tech director know a way to paint them so they look like slightly tarnished, reflective copper; we’ll add on a few details suggestive of rivets or strapping, and hang them (probably from chain, maybe from cording) from poles, so the citizens of the Emerald City can carry them (and turn on the LED ‘candles’ inside of them. The cut-out windows will be filled with pale green tissue paper.  They’re going to be beautiful when finished.

And people are going to want them when we’re done. Even though they’re only cardboard and tissue paper, people are going to find them amazing and beautiful. Maybe I should try producing one in wood, something like thin luan or other simple plywood.  I bet I could sell one for a lot of money, especially if I designed a genuine lamp system for hanging a bulb inside, and provided said system.

Design: production But I don’t think people realize how much this design owes to a misunderstanding of a Easy blog post. There was a thing on Etsy’s blog about lamps for parties made out of cardboard.  I thought it only required two of these elongated diamond patterns, and nearly went off the rails during the prototyping when I discovered that I needed to make the lamp much bigger to get it to stand up flat.  Argh!

But then I realized, we wanted them carried on the stage by our puppeteering crew.  They actually needed to be narrower than originally planned, and it was better if they didn’t have flat bottoms; then they would reflect light downward, as well as outward and upward.

So these lanterns look different than Etsy’s design, though they were inspired by that design.

But I wouldn’t have done it at all without the impetus from the Stained Glass exercise I did during the Thirty Days of Making last year.  Consider my St. Theresa of Avila stained glass panel, shown here.

She’s awful, right?  You wouldn’t want to leave her up in the window for too long.  It’s kind of a sad experience for me even looking at her again.  I could do so much better now, although I’d have to go out and buy up some new tissue paper for the ‘glass’ to fill the windows.

Thirty days of making: "stained glass" panel But the lanterns, admired today as “beautiful”, wouldn’t have been possible without the knife skills acquired from making that. Making this representation of her. As terrible as she is, she’s a starting point for the growth of a particular skill set which does not happen in a vacuum. You can’t plop one of these lantern templates down in front of a kid and say, “cut this out for me.” You can’t expect that the resulting cardboard form will be beautiful; if beauty matters, you have to do it yourself.

Or, you have to start training your students to have the knife skills to do this work in sixth, seventh, eighth, ninth grades… back in third and fourth grades.  Otherwise, you get sloppy work.  You have to cut out a lot of shapes, and learn to have an appreciation of positive and negative space, before your knife skills will be up to producing the parts for five lanterns in a day.

Which is why I say that Production appears to be a different mindset from prototyping.  When you’re prototyping a pair of tables, for example, the prototyping requires a lot of thought; but when you jump to production, some corners get cut at times; the results are sometimes a little wobbly.

Production requires care and attention, yes. But it also requires speed, and a “get it done” mentality which is different from the meandering nature of the prototype or even the brainstorm.  But the production mindset is every bit as important as the prototype.  Prototyping just means, “I know how it fits together.”  Production means, “I’ve made it before, and I can make it again exactly the same way.”

Tai Chi Y3D261: Grab the Needle

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Today’s movement was pretty easy and yummy.  I had an easy time of it, although as always I’m concerned about whether I’m spending enough time on each movement.  Today I begin a new panel in the tai chi ‘booklet’, which runs from Grab the Needle at the Sea Bottom up through False Close. Commenter Quin would find this sequence, from Opening through False Close, to be a pretty good sequence to try to learn in his thirty-day project. Assuming I can finish all the drawings and the revision of the sonnets to his satisfaction and usefulness.

Grab the needle

Grab the needle

Of course, that’s proving difficult. I thought I was a pretty good artist, but these drawings are difficult. Their very simplicity is deceptive — what’s the right amount of information to convey in these drawings? How does one know when they convey the right amount of information?

I don’t know the answers to these questions.  Grab the Needle at the Sea Bottom is a squat, of sorts, with the left hand resting on the right wrist, and using that pressure to force the hand down.  Most of the chi is in the left hand and arm, and most of the squat or sink-down is in the right leg:

Circle left hand clockwise until it rests
lightly on the wrist of the right hand; then
extend that force and diligently press
your weight through that channel. Be surprised when
your right knee buckles: but let it convey
your thumb and forefinger down to the floor.
Don’t make mistakes: first listen, then obey
how far your mass sinks; go that far, no more
nor drive your body farther than it goes.
Pick up the needle and then smoothly rise:
Allow each muscle to do what it knows,
and a little more.  As dozens of tries
you collect, your muscles will learn this move —
and in time its perfection you will prove.

I said at the time that I wrote this poem that I wasn’t happy with it.  I’m still not.  But I also think one of the key elements to remember about this move is that there’s this push to have the thumb-and-forefinger of the right hand touch the floor and yet have the upper body remain straight.  It’s taken me three years to get to that point without pushing the issue, though.  Don’t force your body to achieve this; let it find its way naturally there.  On the druidry list, we were talking about one of our rites, and a very senior member said something to the effect of, “Some would-be magicians and occultists, nearly always men, always try to force the energy to flow by tension or force, and it’s a bad idea. Not only doesn’t it work, but the actual skill takes much longer to develop.  So it is with Grab the Needle, I think. Your weight sinking down in the squat will gradually work out the tension and resistance in your legs; and then you’ll find this move happens naturally. Go as low as you can without bending your waist or back; and expect that each day that you perform the action correctly, you’ll be able to go a little lower — or not.  The goal is to achieve flow and flexibility, not forced descent.

Quin, I feel like there’s some piece of commentary or a sonnet I need to fix or revise, but with WordPress’s new interface, I can’t seem to find it. What do you need help on currently?

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