Geometry book: end of prep 

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I’ve been working on this hand-written book of geometry since at least 2013… maybe since 2011. There’s a total of fifty pages or leaves in it, although it’s an accordion-style Japanese album from Moleskine.  I recently started working on it again due to some recent geometry work in my life, and I’ve put in a few longish days.  The work itself is a manuscript to teach myself the material from Andrew Sutton’s book, Ruler and Compass, available from Wooden Books Press (a division of Bloomsbury).

Several years ago, it might have been early 2014, I laid out most of the remaining pages — the margins of each panel, the lines for the text, and the two or three geometry figures for each page.  For reasons passing understanding at this late juncture, I failed to lay out the last six pages of the book, or plan for the inside front cover.  The result was that I created a milestone, of sorts, in this project — the end of already-laid-out pages, six pages before the end, when I’d have to plan the remaining six pages and finish the inside front cover.

I’m now at that point.  My goal was to get here by Memorial Day weekend, and I’ve achieved that goal a bit earlier than expected.  I probably won’t be able to get back into this work until after the weekend, but I’ve made good progress.

Geometry Book


Some of the geometry book I'm working on...

Eight pages of geometry

I forget which post Gordon said it in, but at one point he noted that nearly all books prior to the invention of printing were books of magic.  Sure, on the surface they might be called medical textbooks or scientific textbooks or books of geography or mythology or history. But at some level, all these books were books of magic — they were intended to change consciousness at some level.

Rufus Opus said something similar about making lamens. A lamen is usually a disk or a square that you wear on your chest during the conjuration of a spirit.  The act of writing one, of punching a hole in the parchment, and putting it on a string or a chain or a lanyard, is a creative act.  If the emblem you write on the lamen is the signature or symbol of a spirit, your hand is going through a kinesthetic meditation on the nature of the relationship between the conjurer and the spirit.

Something similar is happening as I create this book.  It’s a Moleskine Japanese Album, the larger size, so the pages fold out into this lengthy ‘wall’ or ‘screen’ of emblems — about 5 1/4″ x 8 1/4″ inches per panel, but about 115 1/2″ long — call it about 9′ 7 1/2″.

I think about this project from time to time — more lately, since I’ve been working on it the last few days — and every time I do, I’m somewhat more dismayed at the current state of geometry teaching in the United States.  By all the accounts I’ve found, and by the anecdotal evidence I’ve collected on my own, we’ve stopped teaching students to use rulers and compasses in the study of geometry.  It’s too hard to remember procedures, or students don’t know how to use those flimsy plastic compasses well and the good ones are too expensive, or Euclid isn’t widely available, or … or… or…

The excuses multiply like dandelions after a rainstorm.

I don’t know that this book “will become an heirloom of my house forever,” as one of the somewhat-more-fictional sagas would have it. But I do know that I learned more geometry from the construction of the book than I ever learned in a class.  And I wonder if there’s not a better way to teach geometry embedded in that discovery?

  • Each student gets a good compass, a good ruler, colored pens or pencils, and a blank notebook.
  • Each student learns the construction for a harmonious page layout
  • Each student learns a set of procedures for:
    • Perpendicular bisectors
    • duplication of angles
    • construction of parallel lines
    • construction of similar triangles
    • construction of polygons from given sides
    • construction of polygons within circles
    • transference of a given length or distance to another angle
    • construction of nets for 3-dimensional solids
    • construction of the root-2, root-3, root-4, and root-5 (phi/Φ) proportions
    • division of lines into thirds, fourths, fifths, eighths, ninths, and sixteenths
    • construction of grid and tile patterns
    • construction of simple polygonal combinations to find the sides of super-polygons.

This benefits future craftspeople, because they’re receiving an education in proportions and common mathematical relationships, and it’s not all algebraic notation.  It brings back the beauty of geometry to the mathematics classroom.  It gives all of society a common language for seeing mathematics in the natural world.  It trains future architects and engineers in precision diagramming, and gives future laypeople practice in reading such diagrams.

And it creates hundreds of unique copies of books of practical geometry that are themselves handbooks to a forgotten magic — a magic of beauty, of proportion, of color, of relationship, of graphic design. Students would get to learn ALL of that in the process of producing their own books over the course of a semester or a year. The quality of their book would gradually improve, as their understanding of the geometry improved, and as their love and care of the book improved. Think of all the other studies that could be folded into the creation of the book, too: handwriting, color theory, graphic design, book design, clear writing about mathematics, methodology.  The book is a grade — and students who kept their book up to date would find it useful while taking tests to remember what they had created in their own handwriting. The book itself would be a palace of memory for all the geometry they had learned, just as mine is.

All of the actual constructions are covered in Andrew Sutton’s book Ruler and Compass.  But actually implementing it is on the individual teacher.  And it’s likely the case that the teacher will need some substantial support from an administration that sees and cares about quality instruction.

But it can be done.

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.

Estimation and Geometry

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This afternoon, I got into a discussion about why I spend more for milk and eggs certified as produced in Connecticut. In order to do so, I had to rely on geometry.

“Look,” I said, “Connecticut is just about 100 miles wide from east to west, and about fifty miles wide from north to south.  I know it has that weird little tail in the southwest corner, but let’s call it a box, with more or less right-angle corners, and leave it at that.”

“Ok,” said my conversation partner.

“So that means 100 x 100 equals 10,000.  And 50 x 50 is 2500.  So 12,500 square miles should be c-squared.”

“You mean the Pythagorean theorem.”

“Right. And … please don’t make me find the square root of 12,500 in my head…” fumble with calculator… “that’s 111.8.  SO none of these eggs and none of this milk is produced more than 112 miles away from us.”

“As the crow flies.”

“As the crow flies, right. Though some of these roads are pretty twisty,” I said.

“You realize we’re going to pay a lot more for eggs and milk, now, right?”

“Yes. And it will be especially more delicious because it won’t have sat in a storage facility for weeks.”

(And the closest road-to-road comparison I can find on Google Maps says that it’s about 137 miles northwest to southeast, and 144 miles southwest to northeast.)

Make Summer Camp: Geometry Paintings Progress


This is part of the Make Summer Camp project. The rules are pretty simple: Make ten things between June 21 and September 21. Of course, two of those months are now gone. Which means that it’s crunch time, people. Are you in, or are you out? I’ve more or less finished my own challenge, but I’m curious if you’re still working on your ten, or if you finished a few of them and haven’t told me.  I’ve made books, worked my way throughton of origami, built a machine, sewed a tunic and wrote a couple of knitting reports, sewing reports, and carpentry reports  about my efforts. There was even a progress report. What did you do this summer?

This continues my painting projects. It’s been a long time since I’ve been home long enough to work on one of these paintings.  Given that my habit is to mix a batch of color, or two batches of two different colors, and apply those one or two colors to six or ten canvases at once, not being home is challenging.

Painting progress shotsLet me explain further. When I first started painting, my habit was to buy individual mini-tubes of various colors from the acrylic paint aisle at Michael’s Arts and Crafts.  The result was that I tended to expand the palette of colors hugely, because I ‘needed’ dozens of paints instead of just four or five. Then my mother the artist explained a few things, and an artist colleague of mine explained a few more.  The base of color theory, as most of my readers know, is that there are three primary colors — red, yellow, and blue.  There are also three principal tinctures, namely white, black, and gray.  By combining the primary colors with each other, one gets the secondary colors: purple, orange, and green.  By combining the primary colors with the tinctures, one gets lighter or darker versions of the primary and secondary colors.  These primar and secondary colors, and there various hues from very dark red up to very light purple, are generated from the same five (!) tubes of paint: a yellow, a blue, a red, a white, and a black.   To which I have added, at my mother’s suggestion, a seventh and eighth paint, namely a gold and a silver wash or varnish.  This has been a substantial improvement over what I had before.  One can also use water as an additive or thinner, but I haven’t had much luck with that yet as a painter. As you can see, this method is yielding some results.  I was trying to explain to my lady what I liked about this method, of applying one or two primary paints to a range of canvases, mixing them, and then applying the secondary paint to the same range of canvases.  Painting progress shots

The core of it is this: when I work through my painting process like this, I’m learning — in my hands, in my brain — how to mix colors.  I’m learning how to manage the creation of different shades by a very deliberate process.  Different kinds of paint produce different kinds of effects when combined; phthalate red and cadmium yellow do different things than a crimson red and an aquamarine blue; but they would also do different things if I was using a powdered paint vs. a watercolor vs. this acrylic vs. some other acrylic vs. oil.  I’m teaching myself a system of color, apparently, but not the only system of color.

The results so far have been deeply interesting to me.  You can see this at work particularly in the third photograph in this entry, namely the one of the red pentagram against a varied blue background.

Pentagram PaintingThis painting uses a primary red exclusively; and a primary yellow, exclusively (although in retrospect I should have made it more white, and therefore more opaque).

The blue background, and the purple ground for the whole image, are produced using exactly the same shade of blue. Yet in each case, it’s mixed (or in one case, unmixed) with another color.  In the case of this design, the red of the star has been mixed in unequal proportions with the blue, and with some white.  The unmixed blue fills the left part of the yellow ring.  The center part of the yellow ring is filled with the blue + white mix, and the third blue — the one on the right — is more or less equal proportions of white and blue.

So it’s kind of weird.  I’m teaching myself color theory by being a painter.  This is very different than teaching yourself color theory by reading a book — which is what we’d normally do in schools, actually.

But training the eye and mind to see color requires a different set of tools than books and websites. Apparently it requires a painter’s eye, and a palette.

Tai Chi Y3D315: Go Deep

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Tai Chi this morning took me a little bit more than a half-hour.  I was able to slow down quite a bit with my practice, largely by taking a great deal of care to move on each inward breath, and to pause on the outward breath.  There’s a visual image going around on some of the social media boards of a series of polygons opening up from triangle to square, from square to pentagon, and so on up to an octagon. Hang on. I should find it.

Polygon breathing exercise

Polygon Breathing Process

Yep. This is the speed at which I was trying to breathe each breath.  I don’t know if the animation is going to work yet; it’s pretty clunky at the moment, but maybe that will improve once the page is saved and published.  Is it helping that I’m using the memory of this image to improve my breathwork in the tai chi form?  I hope so.  It’s been valuable so far, and I’ve gained quite a bit of extra time on each performance of the tai chi form while using this to time my breathwork.

Tai Chi Y3D124: Work


Book of geometry: square orthogonal to a line

A square set orthogonal to a line.

I’ve spent a fair bit of time on my geometry book over the last few days.  I’m at my parents’ house, and my mother has been very encouraging of my work as an artist.  She works on her art, and I work on mine. Thanks to some new tools (namely a transparent plastic ruler with lines running parallel to the straight edge), I’ve been able to shorten the amount of time it takes to make one page, from about two hours to about half-an-hour.  I’ve produced ten pages in the amount of time it used to take me to do four.

Today, as I did tai chi, I was reflecting on this combination of challenges. Some work we want to do fast, like completing an art project such as this book of geometry that I’ve been working on for months (I’m three pages away from finishing side one, and maybe 15 hours from finishing side two… although side two has a lot more complicated geometry, so maybe it will take longer).

Some work, like tai chi, we want to do more slowly.  I get that.

Book of geometry

Doubling squares and halving them by means of geometry.

The trick is in not mistaking fast work for slow work, and vice versa. My father, tender loving guy that he is, told me today how proud he is of the work I’m doing, and how proud he is of me, and of the way that so many aspects of my life are in my command and under my control.  And yet, as he pointed out (because with Dad, there’s always a yet), I don’t have mastery of my weight.  I’m not sure, after two years and a third of tai chi, that it’s actually shifting my weight at all.  I mean, I probably have denser bones and stronger ligaments, but the push-ups have done more to bulk up my musculature than the tai chi has.  And, further, I haven’t really changed weight at all — I’m still a pretty solid 300# even after two years. Maybe I’ve shifted some weight from my gut to my bones, or from my gut to my biceps… but I don’t think so.

My doc says my cholesterol is up.  My good cholesterol is rock-solid good; my bad cholesterol is up more than it should be. This could be diet, this could be genetics, this could be the beginning of health issues. Every body is an experiment, as one of my doctors used to say.

Maybe it should say that Every life is an experiment.

Today is my birthday. Happy birthday, me.  I’ve begun to change my diet (again).  I’ve begun to be an artist (again). I’ve begun to reconnect with old friends (again). I’ve begun again so many times, that the experiment feels new and different every time.  Now we begin again, again.

But there’s an underlying order to the work.  In tai chi, as in geometry, each line and each angle and each ligament and each muscle has a sense of what it wants to be, and what it wants to do.  When joints creak in pain, we listen to them. When we over-extend a line or an arc, we listen to them.  When we discover truths about ourselves that others have made before us, we listen to them.  When we find a movement with power and grace, we listen to that. These are the building-blocks of our reality.

Today’s tai chi was much like yesterday’s tai chi; and tomorrow’s presumably will be similar to today’s.  We build successes and power a little bit at a time, by slow degrees and by slight changes — and eventually we come to a place where further changes are both commanded and needed.  We master the basics so that we can move on to the advanced work.  We return to the basics when the advanced work becomes too hard.  We re-discover how advanced the basics are when return to them.

May the year ahead be full of wonder.

Book of geometry

the book in all of its fold-out magnificence, so far.

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