Geometry book: end of prep 

Leave a comment

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.

Komebukuro variant

Leave a comment

The Komebukuro form lends itself well to a lot of variation. The squares can be made into rectangles, as here, to create a longer or rather taller bag. As shown here, the Japanese rice bag is simply two sets of vie squares — a base and four sides. The bag sides are sewn to the base, four straight stitches. Then you sew the four sides to each other, one edge at a time. the result is sort of a box or five-sixths of a cube; you could add a zipper and a lid fairly easily to this design, really.

In the photographs here, I’ve shown as best I can what I’m talking about. The gray fabric in the middle is the BASE of the bag, while the floral print in gray are the sides of the bag. I’ve laid out the fabric of the liner in all floral print, while the outside of the bag has a single white panel where I can write my name, or the name of the person the bag is for.  Embroidery could be done here, for someone who was particularly ambitious.

Seven inches appears to be a good size for the Komebukuro. You can get a lot larger than that, of course.  You can also get a lot smaller, but there’s a point of diminishing returns under about five inches on a side for the squares that make up the base and sides of the bag.  I also don’t tank I’d want to go much larger than a foot on a side.  More than that would be unwieldy, and you’d be better off with two or more bags.

Back to construction…

Once the two boxes of the inner and outer bag are made, they are nested, and the top edges are folded down and in between the two bags. We then top-stitch the seam between them. As ive discussed elsewhere, the last step as the sewing of the eight buttonholes.

There’s a picture, here, of the outer shell of the bag already assembled, but still inside-out. This is to show inside-out construction. When building a bag, the fabrics are sewn right sides together. This puts the seam on what will eventually be the inside or in-between space of the bag, between the liner and the shell.

Then you sew in the button holes.  Each side of the bag is now two panels of fabric, the shell and the liner. Each of those panels takes two buttonholes, which are maybe 3/4″ down from the top edge of the bag, and evenly spaced on the bag’s walls, about a try of the way in from the corner.  The corners of the bag’s open top should be fairly visible.  Threading a cord or a ribbon through the buttonholes creates the closure mechanism, but also creates a carrying strap.

My sense of this is that it’s fairly easy to vary the size of the squares into rectangles. But the square that forms the bottom or base is fairly rigid. You can’t alter that from a square too much without unbalancing the bag as a whole, I think.

This would look stunning in indigo-dyed fabrics, or with Japanese embroidery patterns done on the outside of the bag (doing them on the liner would create all sorts of things for your keys or other objects inside to get hung up on; stick to the outside).  Many of those patterns are based in triangular geometry, so there is some real potential for elaborate, hands-on mathematics here.

My mother has made several of these bags, without the button holes or cording, to use as trash cans for her art studio. Paper and beads and parts that can be recycled go in one of the bags; while trash goes in another. They’re prettier than regular trash cans, and collapsible. She can fold them up and put them away when they’re not in use.

I may have to make some of my own for that.

All in all, I think I’m going to make a lot of these, both with with and without cording, in a number of sizes.  They’re a good size for kids’ lunch bags, for example, or for an art kit for the car, or for portable storage of related items while camping. I think I’m going to try making some in 10″ and maybe 14″ sizes, but I think that a shoulder bag or something like that will work better as another project for teaching sewing for school books.


Creativity vs. Imagination: Moon Mansion Diptychs


Moon Mansion Diptychs
Originally uploaded by anselm23

The difference between creativity and imagination has been much on my mind lately.

Imagination appears as the ability to become ‘dreamy’, for lack of a better word, and visualize or ‘see’ things within the mental realm. It’s the capacity to form new ideas, or bring forth ideas based on things not presently or currently seen. Copying someone else’s vision — as I effectively did in these two drawings, largely based on the work of artist Nigel Jackson, from the book by Chris Warnock on the Mansions— is one thing.

But creativity is not really the same thing. I mean, we might think of them as the same thing, but they’re two different capacities. This, to me, is the power to call forth something from the imagination, and make it real or sensible or visible to someone else. I know plenty of imaginative kids, for example, but I know a lot of imaginative kids who aren’t very creative — they’re sort of lost in a fantasy realm where they are capable of dreaming themselves the heads of corporations or the most amazing rock guitarists. But those same kids don’t actually do the work that gets them moving forward toward that dream.

Likewise, I know plenty of creative kids who aren’t very imaginative. They do all sorts of little drawings, and they’re very productive — these kids wouldn’t dream of not doing their homework, because they’re actually eager to ‘create’ something, to bring something into being. But they’re not very good about bringing forth something new or unique to themselves.

There’s of course a third category, which is people who are both imaginative and creative. I wish that I fit consistently into this category, although most of the time I think I’m only one or the other; it takes a lot of time and effort to be both, and some days it’s just very hard to get anywhere near that combination of powers. It requires an incredible amount of practice to build up to the point where one can be both productive, and capable of summoning forth a vision of “things not seen” so that others can also participate in that vision.

So imagination is largely a mental skill, but creativity is largely the skill of taking mental-to-material. Where one is largely a matter of dream or day-dream, the other is a matter of tool use — whether memory or imagination or skill, or the use of actual physical tools, be they knives or drills or scissors or glue or word processors or graphics software programs or t-squares…

And I’m not at all sure that anyone would agree with these definitions, which only makes the problem harder. But I think in general that our culture makes much of imagination, without making an equal fuss over creativity. And yet, without creativity as I’ve defined it here, all the imagination in the world won’t actually get anything done.

Via Flickr:
I’ve noted in the past that paper doesn’t seem to hold a magical charge for very long…. and yet it turns out that you can make quite an interesting power simply by folding the paper in half. A friend of mine is having difficulty with her health, so this evening I made a pair of the Mansions of the Moon for her — Egibiel to drive away the bugs that make her ill, and Amutiel to bring her health. These two mansions are not normally used for matters related to health, particularly not lung-health (which is her particular issue), but she wanted something immediate. This, plus some good cold-care tea, seemed to be a good combination.

It’s worth saying that a Moon Mansion, or any sort of tool like this, is not a useful substitute for actual health care. This only serves the purpose of bringing spiritual forces to bear on a physical problem; but the realms of being are discrete and not continuous. Simply having a pair of angels watching out for your health in no way obviates the need for genuine health care.

Papercraft: The Boxes

1 Comment

The Boxes

Originally uploaded by anselm23

I’m teaching a class on paper-craft and in particular pop-up books during summer school this year, and I wanted to start working on my skills so that I can teach my students some new skills when it comes time. There’s a colleague of mine, as well, who’d like to teach her students some pop-up structures, for making cards and mini-books about Native American peoples they’ve studied this year.

I figured, it was time to teach myself some skills. So, I brought home Carol Barton’s book, and I made the first six of her designs: a straight box (purple and yellow in the upper left of the photo), a stepped box (purple, white and yellow in the center back left), a freestanding box-support (back right), a weird “carved box” shape (lower left), a modified box (the shield shape in lower center), a heart, (right hand side, in red), and a scallop shell (center, and hard to see).

About two hours of work. Taught me a lot about following directions, about learning to see possibilities and potentials. I’ve already decided that I want to make a mini book for someone, detailing the Five Elements, the Seven Planets, and the signs of the Zodiac. Call it a mini-kavad in book form. Not sure when I’ll get to it. It’s clear that knowing the structures is one thing — having a clear sense of the book you might produce with such a thing is another. The technology and the vision are separate from one another; learning the methods will not help you come up with creative ideas of how to use the construction techniques. You need the mysteries, or access to the imaginal realm, or the ability to travel astrally, to get access to those sorts of things.

Via Flickr:
Carol Barton’s “boxes” from her book The Pocket Paper Engineer: Vol. 1.Am I getting ideas for the kavad? Of course. Are all of them practical? Of course not.

This is about two hours of work. I learned a great deal in the process about design and structure of pop-ups, and how challenging its going to be to teach some of this in a class this summer. Knives and rulers and protractors and pencils oh my!

One of my aphorisms for design is my friend Mark’s saying, Tools dictate solutions. If all we give students is lined paper, graph paper, three ring binders and pencils and pens… All of their solutions start to look like that. Even the addition of a knife or a pair of scissors is something.

I look forward to tackling triangles soon.

Taiji Day 308: Warming Up


The house is cold, cold, cold this morning.  Even though it’s in the 30°± F range outside, I have the heat turned way down in my apartment — I woke this morning to a bedroom in the mid-50s, which is not a typical “American” temperature to be sleeping in.  And I was shivering in bed. I had to walk backward and forward through the house to warm up, and put on a sweatshirt to keep myself up and out of bed even so.  I may have found the lower bound of what I’m personally comfortable with, based on the “turn down the thermostat” exercise from last week’s Archdruid Report.

Today is the 22nd Mansion of the Moon.  If you want to see how far I’ve come as an artist, check out yesterday’s drawing of the 21st Mansion, and then check out the drawing I did for the 22nd Mansion of the Moon, a month ago. And one I drew in November 2011… Progress, I’d say.  And yet not as much as one would think.

Plateaus. We all have them.  I think I’ve hit one in my tai chi practice.  Sure, I’m slowing down my form, with some difficulty, and moving through the positions with greater dignity and less speed.  It’s hard, and some of the forms are causing me to tremble a lot.  But I’ve yet to encounter a movement or a posture that makes me want to give up or give in.  I think that’s important.  But I haven’t yet found which rules I can break in this practice.

At the same time, I haven’t uncovered or unlocked any great secrets of practice.  Even in sixty-ish days, I’m not going to be a tai chi master.  That’s 20 years down the road, if ever.  I may never be anything more than an enthusiast, really.  It’s hard to tell.   But 80% competence is pretty good.  And 80% competence in many different things will take you far: a lot farther than almost anything at 100%.  Apparently I’m now 80%-ish competent as an artist in a few materials (notebooks, pen and ink), and 80% competent (though a long way from 100% competent) in tai chi.  That’s enough to keep my health from deteriorating rapidly, or shrug off colds, or keep myself from injuring myself.  I’m 80% competent as a writer.

Competent is pretty good.  Competent in more than one thing, that’s better. Here in this cold house, even with the shivering that I woke to — I started the day without a trace of the tiny hint of a cold or the flu that was dogging yesterday morning’s practice.  With 80% competence — good enough — I shed yesterday’s illness.  I’ve shifted how I carry my wallet, and added a psoas stretch to my tai chi routine, and that nagging grabber in my lower left side has gone away.  And I’m working on a new poetry sequence, which pleases me.  I’m doing work at the end of my hands, to.

What are your projects?

Older Entries