Geometry: finished

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

Eight pages of geometry

​​​The geometry book is finished.

I’ve been working on this project on and off since 2013. It’s a Japanese Album moleskine, sometimes called an accordion-fold book, of about 50 leaves or panels.  At this point, both sides of every sheet of paper in the book are inked with 111 geometry problems in both diagram and text.  (technically, there’s space for about three more on the inside of the front cover).

I began this project when I came to the realization that geometry was part of the underpinnings of good design work — that if you could see the key elements of the geometry underlying a project or a design, that the quality of your work would improve because you understood how different relationships were managed between various parts of the project.  Because ultimately, geometry is about relationships.

More than that, this project has been about self-discipline.  I started the book at one scale, then shifted to a tighter scale after about fifteen panels.  Then I stopped for a good long while, somehow afraid that I’d ‘ruined’ it by the change in scale.  Then, I picked it up again, and worked all the way to the end of one side of the book, and about seven panels into the other side. There was another long pause, maybe as long as a year.  About six weeks ago, I picked the project up again.  I couldn’t find the right marker pens that I’d used to start the book. I shifted to different pens (Prismacolor was the first brand, at 0.5mm; Staedtler triplus fineliners the second, at 0.3mm). I got better results, especially on the more complex constructions, with the new tools.

And this morning, I finished.  I woke up early, and I came downstairs. I finished the page on ellipses that I began yesterday, and then began work on the page on spirals.  The spirals page went far more easily than I expected.  It was simple to turn from that to the final page, which is in part about the Golden Section, and the process of laying out multiple proportions.  I thought that page was going to be difficult, too, but it wasn’t.

And then I was done.

It doesn’t actually feel like it’s done; that may take a little while to sink in. But the project (except for maybe some unusual problem not covered by the earlier work, that can go on the inside front cover), is now complete.

​​More than any other project I’ve done, though, I feel like this is the one that lets me say, “I am an artist and a designer.”

Hello.  My name is Andrew Watt. I’m an artist and a designer.

Yarn: Untangling

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skein is a unit of yarn, in which an elaborate amount of yarn (often 100-200 yards) is first looped around an object (usually a yarn-swift) numerous times, until there are several dozen loops or spirals of yarn around the arms.  This is then twisted around several times, and then twisted around itself several times, before being tucked into a compact shape that fits easily onto the shelf of a yarn shop. Skeins do not as a general rule, roll or run away from the knitter.IMG_5287.JPG

If you are not careful, though, they easily become a tangled mess. Like this one did.

The only solution is patience, and time.  If you don’t want to give up the yarn, then you have to sit patiently, picking apart the knots and tangles.  This can take a long time; some people don’t believe it’s worth the trouble.  Some people would rather take scissors to the whole thing and churn out two piles of yarn:

  • “bits long enough to work with”
  • “string too short to save”

I belong to the third category of yarn-workers, which demands patience and time while the skein is brought to a new category of order, the yarn ball.

IMG_5288.JPG The yarn sometimes loses a good deal of the sheen and luster that attracted you to it in the yarn store as you do this patient work of untangling. Your partner will roll eyes at you as you do this work, and even tut-tut at you as you nearly scream in frustration at it.  But sooner or later — given enough time, and enough patience — all of the knots and tangles will be removed, and you will have a yarn ball.

It is imperative that all skeins be turned into yarn balls before you start knitting with them.  Under no circumstances should you attempt to knit from a skein, not even “for a few stitches” or “for a few lines of purl” or what-have-you.  ALWAYS take the time to untwist the skein before you knit with it.  Your patience will thank you.

Knitting: Second Hat

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I had some time this weekend, and I was in the mountains where it was cold and rainy over the weekend.  So I spent a fair number of hours working on my second hat.  I finished it on Sunday. And a good thing, too, because I needed it on Monday, when it was again cold and rainy and ugly.

The hat is a little bit on the large side for me. I was trying to scale it up from the “Adult L” size to my extra-large head, and I made it a little too big, I guess.

All the same, there’s a couple of things here that I managed to get right:

  • Ribbing to create a frame for the hat
  • knitting in the round on a circular needle
  • knitting in the round on four double-pointed needles
  • managing decreases (knit2 together)

So, all in all, a successful second hat was made. By me. To wear. Right away. I’m eager to make another one, but this time I think I’ll keep it at the Adult L size, rather than trying to add in another 18 or so stitches to make it conform to what I ‘think’ is the correct size.  This kind of thing only gets easier with practice.

The next challenges?  Socks and mittens.  Then gloves.

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.

A Talk on Memory Palaces

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Yesterday, at the District 53 Toastmasters Spring Conference (part of Toastmasters International), I delivered a talk on the Palace of Memory technique. These were my working notes and my slideshows.

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Geometry Book

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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 

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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.

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