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: The Heptagram


A while back, I built myself a VLC. VLC stands for “Very Large Compass” and it’s a very different tool than the typical compass one buys in an art supply store. Most of those compasses can’t draw circles larger than a few inches in radius. I needed one that could draw really big circles. For really big geometry. On foamcore or posterboard.

A compass like this:

My VLC: very large compass

The VLC: very large compass

But of course, it’s hard with this kind of photograph to get a sense of what a VLC can do for your artwork, or your classwork, or for your students in Geometry class. Just seeing a point or pin, and a flexible radius (as indicated by the wooden dowel), isn’t really enough to tell you just what you can do with a Very Large Compass.

You could choose to produce a mandala of geomancy, as I did, and link it to alchemy.

But that doesn’t really give anyone a sense of why a VLC is necessary.  And so it becomes necessary to put the VLC to work on other projects.

Like the Heptagram.

Now, there’s actually more than one Heptagram. There’s a skip-1 heptagram, which isn’t actually a seven-pointed star, but more of a seven-sided regular polygon.  And that can be done fairly easily.

There’s also the skip-2 heptagram.  The skip-2 simply means, start at one corner of a heptagon (or seven-sided regular polygon), and draw a line from one point directly to the vertex after the next vertex.  The result is sort of a thing that your brain almost registers as a six-pointed star, except the angles are wrong.

There’s also the skip-3 heptagram. This time, instead of skipping one vertex and going to the next vertex, one skips two vertices, and goes to the third.  This time, one gets a very sharply angled star with a tiny irregular heptagon in the middle, instead of a big blocky one like the skip-2, or the very regular skip-1.

The three types of heptagons, when laid in a circle and assigned their traditional planetary associations with the points, looks something like this:


the three heptagrams

And I must say, I think it’s beautiful.

It’s also big.  Here, the circle (produced by the Very Large Compass) is perhaps two feet across.  I must admit, this is a heptagram produced by neusis rather than accurately measured angles… but there’s only so much that anyone can really do with just a VLC and a VLSE (Very Large Straight Edge— in this case, a long bit of masonite with a precisely cut and shaved edge).

This is eventually going to be a piece of artwork for my classroom, and maybe an auction item for my school’s annual auction — let me know if you’d like to buy it in May? — but for now it’s a elegant reminder of something I’ve said often:

You can’t think with tools you don’t have and have never used.

If you don’t have a VLC of your own, it’s time to build one.  If you don’t have a VLSE for your geometry class, you need one.  Get yourself some foamcore, and some pencils and sharpie markers, and start dreaming big. Start geometrizing big.  Because there are things that you learn on big paper and big surfaces that you cannot learn any other way — about tiling, about shortcuts, about tessellations, about design — that can’t be achieved with the usual tools.

A kid who never picks up a compass in geometry class hasn’t really learned any geometry at all.