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

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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.

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.)

The Dodecahedron

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I haven’t done a Maker’s Grimoire exercise in a while, and it’s sort of time.  I’m a big fan of paper-prototyping, first of all, and second of all I’m fascinated by the Platonic Solids.  Once upon a time, a student’s understanding of geometry would be rooted in the study of the polygons first of all, and then in the polyhedrons which could result from those shapes… including this one.  Plus, I’m fascinated by Rosicrucian-Vault’s wooden dodecahedron how-to.  But I don’t want to build one in wood before I’m really confident of my construction abilities in wood.  On the other hand, I understand paper quite well. So much so that I drew up a Dodecahedron template in Pages (which is a word processor. No way should it be used for graphic design. Really!)  And then I added some tabs to that pentagon, and made a template. Print out twelve templates on colored cardstock, and you too can build a dodecahedron… and you can do it right, with three yellow faces, three blue faces, three green faces and three red faces, and label them with the months of the year or the Zodiac signs and so on.

Dodecagon — construction Process

First, of course, you’re going to take three of the cut-out pentagons, and assemble them. As you do so, you’re going to wind up with a structure rather like this — a trio of weird flaps glued together in a weird star formation, like this.  Don’t worry.  It gets better, just not right away.  Once you add in a fourth pentagon, you have something that looks more like a university chorus stage performance, with those odd sound baffles behind the risers so that all the choristers can see the conductor.

Dodecagon — construction Process

You keep adding pentagonal panels to your model, and glue them in place (or use tape, but tape is wonky — use glue.) Gradually, your structure begins to be oddly sphere-shaped, but not sphere-shaped.   And at this point, things begin to get tricky.

See, the template I made has three flaps attached to the sides of the pentagon.  Which is awesome, really.  And most of the time, it’s fine.  But once things reach this point in the construction,  sometimes one of those flaps has to be cut off. But it has to be the right flap.  Which means doing a bit of fitting and second-guessing before it all gets assembled correctly.

Dodecagon — construction Process

So… this is how the pieces look as you’re fitting them together, and figuring out which foldable tabs need to stay attached, and which ones need to be cut off.  As a general rule, only cut one tab off at a time — because they’re hard to re-attached, and relatively easy to leave on until the last minute.  Gluing and fitting the last two or three panels into place is a tricky job… Do the pre-fitting first.

Dodecagon — construction Process

You will get a fairly large dodecagon out of this. It isn’t a small sphere — it’s large enough for some children to throw back and forth as if it were a dodge-ball… though of course it isn’t one of those. Here’s one of them with several common objects alongside to show scale.

Dodecagon — construction Process

 

And that’s how you build  a dodecahedron big enough to play dodge-ball with.  Of course, there are real mathematical advantages which come from a study of the Platonic solids. They raise questions about area and volume, about points and vertices, and edges and all kinds of mysterious questions which are not easily answerable.  It’s practical usefulness is less clear.  I suppose the inside of this thing could have been taped, and the thing filled with sand as a doorstop… but it would need some rigidity to accomplish that, I think.

But there’s also the benefit of teaching kids to work with sharp tools like scissors or knives, and cut out precise shapes and glue them together. They’re going to want to build clear, obvious and beautiful models… and they can’t do that with those dull, unsharpened scissors with blunted tips. These are not the sort of models that can be assembled slapdash.  There’s an art here…

The Five Platonic Solids

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The Five Platonic Solids

Originally uploaded by anselm23

Via Flickr:
So here they are. They’re built. In Ancient Greek elemental theory, the cube represented Earth; the tetrahedron represented Fire. The dodecagon was Universe. The icosahedron was Water. The octahedron was Air.

Mathematically, a cube has six sides (the Doctor says "seven — including the INSide"). The tetrahedron has 4 (five, says Doctor Who). The dodecagon has 12 (13!). The icosahedron has 20 (21! Why must I keep saying this??), and the octahedron has eight ("nine! Nine sides muhahahahahaha… Oh, wait… Wrong tv personality.")

Hot geometry in planar solids.

Four Astrolabes, Prototyping and Discovery

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Cardboard Navigation
Originally uploaded by anselm23

Today was supposed to be for grading papers. But it turned out to be a day for building astrolabes. I built four, using PDF kits I found online, cardboard left over from the Makedo challenge, and a a few sheets of paper (and an almost nicked thumb from a dull Xacto blade… curiously enough, scissors almost work better.

Why? My history classes are learning about the American Revolution. My Latin classes are studying uses of the Infinitive, and the formation of adjectives. Astrolabes don’t fit into either part of the curriculum.

Ah… but they do fit into the design lab. I think. I mean, here’s a complex medieval instrument for discerning the passage of time, the movement of the heavens, and uncovering the geometry of the worlds above the earth. It’s eminently practical — Columbus probably used one to cross the Atlantic and estimate his position day-to-day. And yet it’s a relic of an ancient age: the first was invented by Hipparchus in ancient Greece almost two thousand years ago, back when people thought planets were Gods. Building one of these even as recently as eight hundred years ago required days of a geometer’s time and the labor of several skilled craftsmen, to make sure the numbers were correct and that the lines were in the correct position… too many things could go wrong. They were slide rules and calendars, calculators and protractors, elaborate devices for knowing one’s place in the world. To use one, you had to know how to use one; to know how to use one, you had to practice with one. A bit of a catch-22, but right in line with the designer’s maxim, “build to learn.”

Here’s a video about using the Astrolabe, using a TED talk:

Fourth Astrolabe

fourth astrolabe

Although each design that I built was more complicated than the last, by the end I had a pretty good idea how each device worked, and how I could use it in the classroom, or convince a mathematics or science teacher to use it in the classroom. They’re not fantastically complex tools, but they enable one to do trigonometry fairly easily — provided you know trigonometry. Don’t know trigonometry? Don’t worry… you can be taught, with the help of an astrolabe.

Hmm. Is this a way to open up advanced mathematics to younger students?

Via Flickr:
I tried really hard during my free periods today to grade papers. But the design lab called: I’ve been thinking about stars for a whole lot of time now, and how to integrate the study of astronomy into a school program that only meets during the day. And, of course, how to include mathematics in a history class. And, of course, how to include history in a math class.

The answer is astrolabes. Of course they’re outdated technology. They were the great grandparents of slide rules when slide rules were invented. They were practically second cousins of the abacus.

Today I downloaded three basic models of them and made them.

Here they are:

You paste a printout onto cardboard (cereal boxes work great), and with an hour’s work you have an astrolabe similar to the one Geoffrey Chaucer described in the first technical manual in English, “treatise on the astrolabe”, written in (I think) 1391 AD.

These two are simple. They require glue, printouts, and a knife. In a classroom environment they’d need to be spread over two days — one day to glue the paper sheets down to the cardboard, and then some drying time; followed by a day to do the cutouts. The one on the left is a little more elaborate. The third one, above, is a quadrant rather than a true astrolabe… but the principle is the same. Teach a class the second day on sun sighting, or calculating the height of a flagpole. Extrapolate to sighting stars on the deck of a ship in the midst of the heaving Atlantic Ocean, and you have a seriously cool lesson about the Age of Exploration.

But only if you have willing teachers in Mathematics and history classes. You can combine it with a lesson about globes, using an icosahedron made of paper plates and a Buckminster Fuller projection of the Earth. But only if you have willing students and instructors.

Today they’re only mathematical toys. But in the late Renaissance, they were the tools of empire. And they could be the tools of a student’s modern day empire of learning if we taught students to build and use them.

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