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.

Sewing: Viking Bag 2 

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Remember that cute viking fabric that I made into those little drawstring bags?

I found another strip of it.  But it wasn’t enough to make into a bag, unfortunately.  I was going to have to combine it with another kind of fabric? What goes with vikings, though? How do you combine viking warriors with anything else? Floral prints?

But how about a brick wall?  If I make the bag tall enough, it will look like warriors peeking over the battlements of a tower, and that conveys the image that we’re looking for — not a bag, but a tower, not a small purse but a fortress.  It becomes a thing of the imagination, as much as a physical object.

I’d rather it was a stone wall, or maybe spiked logs, like on a rough-and-ready motte and bailey castle.  That would make sense, after all.  The vikings didn’t build too much in brick (they also didn’t wear horned helmets — not very practical in warfare, really).

No matter. I found some brick fabric, and it matches pretty nicely with the vikings.  And then I found some other fabric that sort of resembled the Lord Baltimore colors in the Maryland flag… somewhat heraldic, though not TOO heraldic… not shields with lions and snakes and so on.  That would have been a seriously lucky find, though, in an American fabric store.  In general, though, we don’t really understand heraldry’s rules, so they often get used against us — in advertising, in snobbery and class warfare, and other ways, as well.  That’s not really at issue here, though.
What is at issue, for me, is how much of Makery in schools seems to be “making for the sake of making” — that students should simply be allowed to make whatever it is they want to make, full-blown from their imaginations. I don’t know that there’s anything wrong with just letting people make what they want to make, mind you — there’s benefit to that, and real results can emerge from allowing that spark of creativity.

But I think there’s a place in Making for making with a purpose.  There’s a lot to be gleaned from making a quilt to keep a baby warm, or from making a bag like a tower for holding a bottle of wine or something similarly fragile and precious, or a bag that looks like a pencil case. There’s a place for unbridled creativity. But there’s also benefit to knowing how to do something the right way.

I mean, think about it.  In a tailor’s shop 500 years ago, an apprentice would have to work on a variety of tedious projects — sharpening scissors, ironing fabric (I can’t imagine how difficult that work was before electric irons), measuring clients (and then measuring them again when the measurements proved wrong), learning to sew straight seams. They would have made a variety of things that no one would care if they were slightly off — bags with drawstrings, bags with handles, awnings for market stalls, aprons for the shopkeepers, tool rolls for traveling workmen, sacks for flour, and similar projects.  These are a vital and necessary part of the learning of any artisanal technology, be it sewing or woodworking — the cruddy projects that no one really wants to do but that are genuinely vital to the good functioning of that sort of society.

Thirty years after my own first sewing experiences in a Home Economics class in 8th or 9th grade, I find that these skills are returning to me with some rapidity.  I’m much more skilled at whipping up a bag or an apron than I used to be, in part because I’ve trained those skills to a level of complexity and skill where it’s easier to just do it.  I’ve learned a good deal of the apprentice work of this business of sewing (maybe not everything, because every so often I encounter a challenge that I have to go to a YouTube video to solve).

But that’s what the Walking Foot was about.  That’s what making potholders was about. These aren’t “stupid projects” but the foundations of the craft.  In carpentry it’s probably smoothing boards; in bookbinding it’s folding and punching pages; in fabric it’s making potholders and bags.

What are the foundations of your craft?  When did you feel like you went from being an apprentice to being a journeyman or journey woman? When did you become a master of your art form?

Wood: trestles

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On Sunday I borrowed a friend’s woodworking shop to produce a couple of trestles for a table. They sort of look like sawhorses, but they’re not. They’re intended for lighter duty than that. I still have to produce the top of the table, which will have two bars or cleats on the bottom to help lock the table and the trestle together.

These are based on a medieval design produced by the St. Thomas Guild, a medieval reconstruction group in the Netherlands. They are pretty modern, though. I’m trying to decide if I want to add in the fancy carving and tracery work to them. I know how to do that work, I just don’t know if it will be worth it in DIY shop pine.

Some things to think about —

  • Adding a second board across the bottom for stability would help these trestles be less wobbly.
  • Adding a wedged mortise to the top cross-board would also make them less wobbly
  • Adding a cut-out to make the triangles more like a pair of legs would add stability, as well.
  • The table-top will have to have two cleats or bars on them, to slot into spaces at the top of each trestle.
  • Adding some pegs to the bottom of the table top; or to the top board, that slot into the table top, would also improve stability, generally.

There’s a lot of things to think about.

In general, though, I like this idea.  The trestle table has some serious advantages for me, in that I can take the table up or down as needed, and have the flat surface or not as I need.

Additionally, in reviewing the St. Thomas Guild website, I see that I can design this table top to do many things that may help it be quite portable or adjustable.

Table Top DesignMy initial thinking resembles something like this — a kind of construction known as “frame and panel” (which I’d like to learn), with four types of members:

  1. Dark green outer frames, with one groove and three mortises.
  2. Internal ribs, with a tenon on each end and a groove on each side (purple)
  3. four panels (mottled blue) with a tongue carved all the way around them.
  4. Two internal frames, with three slots and a groove (light green)
  5. Four outer frames (yellow) with tenons on each end and a groove on one side.

The blue panels thus fit into the groove on each side.  A quartet of hinges join the two inner frames to one another, so the table can fold flat and store more easily;  or be arranged to provide a wide or narrow table as needed.

I may have to rebuild the trestles to accommodate the larger table surface. But my understanding is that panel and frame construction is fairly lightweight, and this might do quite well for my general needs.

Dimensions of the surface still need to be worked out.

 

Quilts: walking foot

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Ivy at Circle Thrice responded beautifully to my comment with a post about the magic of making things ourselves, and particularly for people that we care about.

It’s not a post I wrote myself, alas.  I shoulda.

Because this blog, for better or worse, is a blog about magic.  In large measure it’s a blog about the power of making things; it’s blog about the power of co-creation — the act of pulling the materials of the world to you, shaping them and changing them according to a particular vision, and producing something on the other end which is more than the sum of its parts.   Something of your self is woven into the finished product, for better or worse.  Yet in many ways, it is better than something bought through anonymous channels, through the mercantile trade of your hours for abstract concepts, and the trade of abstract concepts for the physical goods of your life.

Today I broke down and went to the local fabric store. It has a name that sort of rhymes with pollyanna, but not really.   I’d heard there was this magical tool there, something akin to a wand of fire or a disk of earth — a walking foot.  (I also needed binding for the quilt I’m holding in these pictures. The binding is the pale green band around the outside of the quilt — it’s better to make that by hand from a 100% cotton fabric, but I am not good at making bias tape or binding — I could spend as long making enough bias tape to make this quilt, as I spent making the quilt).

A Walking Foot? I hear you ask.  Don’t most normal feet walk? Not on a sewing machine, they don’t.  A walking foot attaches to the low shank of a sewing machine; and it has a mechanically-driven lever arm that hooks on to the needle’s drive shaft. The result is that the walking foot engages a second pair of dog-legs (those are the feet on the face of a sewing machine, that advance the fabric through the needle’s binding mechanism.

This particular quilt had been giving me trouble. Truthfully, all of my quilts give me trouble in the same way. The first part of the work involves stitching together squares. I’ve gotten a LOT better at that work. The second part involves quilting three layers together — in this case, the topper of green and blue squares; the batting, basically a thick layer of felt; and the backing fabric (the dark blue in the first picture).  The three layers can be seen in the third picture.

Today I broke down after doing the first half dozen straight stitches of this quilt. I needed quilt binding anyway. So I went to the fabric store for binding for this quilt, and got the walking foot.

What a difference the right tool makes! The walking foot guides the fabric sandwich into the needle-space with great diligence and accuracy. It’s a supremely powerful focus for the sewing machine, like a lens focusing a powerful beam of light into a laser — swiftly went the work of sewing the quilting stitches.

I was able to finish the rows of the quilting, and then the columns, with great accuracy. And then the binding around the edges went quite easily as well. Before the Walking Foot, this might have taken me the better part of two days of work with my old sewing machine. With the Walking Foot, a few hours at most.

This is my seventh quilt, I think.  It might be my ninth, but I think it’s probably the seventh. It’s taken me seven quilts to learn how to do this well enough, and effectively enough, that I learned what sort of problems I was having, and was able to do the research necessary to find the solutions.  Every single one of seven quilts, I’ve gotten better at this.  They may not be Etsy quality, but they’re a lot better than they were.  In a short while, they’ll be a lot better than they are now.

All aspects of Making are like this — you have to be an apprentice before you can be a journeyman; and you have to do the journey-man or journey-woman’s work work before you can be a master of a craft.  It doesn’t matter if you’re talking poetry or quilting or fashion design or musicianship or painting or carpentry or engineering or basket-weaving — or even magic and re-enchanting the world: Start somewhere near the beginning. Get Better. Keep Going.  To Walk the Path Of Power, the Work Is On You.

Quilts: finished

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Two quilts finished in three-ish half-days. The top one is the penguins from earlier, the bottom one in pink is the owls quilt.  Both are already spoken for, you can’t buy them.

Really, it’s three quilts but I’m having my doubts about the arrangement of materials for the third quilt. I think I could get the third done if I could get some feedback on whether or not people think it’s a good layout.

Here’s the third quilt. Tentatively, anyway.  I’m worried that the dark blue of the starry swirls doesn’t really match the green or the blue of the top of the quilt; and that the baby-blue trim doesn’t match either.  I suppose I could tie it all together with some thread of the right color… but it still makes me a little nervous.


What do you think? Leave me a comment.

Quilt: penguins 

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I find that I’m enjoying a great deal the process of laying out a quilt, and then sewing the rows and columns together to make the quilt.  These squares are 5″ to a side for this quilt with a penguin theme. The quilt is going to be much wider than a typical crib quilt, but about the same length.

Unfortunately, the dark blue fabric is polyester and slippery.  I don’t know if this is going to work.  I’ve found conflicting opinions about quilting with polyester fabrics — some people love them, some people hate them.  I’ve decided on a 100% cotton backing fabric, though, so if the baby winds up being sensitive to poly they can always flip it over and display the quilt top to the world and wrap the baby in two layers of cotton away from the artificiality.

Why use poly at all? Do you know how hard it is to find penguin fabric to begin with?  I also didn’t choose the fabric, in this case. This is a custom order, and I’m not sure that we knew it was poly when we ordered it.

In any case, there’s this delightful process that you can see in the third photo, where the rug gradually vanishes behind the fabric as the quilt takes shape. This one should be done later today, or at least it should be done later today.

There’s another thing that I quite like about quilting with these sorts of prints.  When you look at the whole fabric, it’s very hard to admire it — it’s the same pattern repeated over and over again.  It’s mind-numbing in its regularity.  And it’s often dull to look at.

But then something happens when you cut it up.  As the fabric is sliced in two directions, the pattern becomes more randomized. Sometimes it’s the father and mother penguin in the foreground, sometimes it’s in the background, sometimes it’s the large line of penguins in the middle ground that becomes prominent.  The pattern’s regularity becomes irregular, as the rotary blade cuts and slices the repetitive imagery into squares that don’t respect the pattern’s repeat mode.  And so something new emerges.  It’s the original cut-and-paste, in some ways.  Except that with quilting, it’s cut-and-baste.

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