Video: Circle Center

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Once you finish a book on geometry, everyone wants help. 🙂

I wound up making this short video on how to find the center of a circle for my dad.  It’s not ideal; I need a better set-up for making videos at my desk.  But the essence of it are these steps:

  1. On a given circle, find three points about 1/3 of the way around the circle, A, B, and C.
  2. Arc the distance AB center A, and BA center B, so the arcs overlap one another at two points, D and E.  Draw a line between D and E — the center will be on that line.
  3. Arc the distance AC center A, and CA center C, so the arcs overlap one another at points F and G.  Draw the line between D and E — the center has to be on that line.
  4. Point O is the place where lines DE and FG cross. That’s the center of the circle.

I hope this helps!

Geometry: finished


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.

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.

Tai Chi Y3D124: Work


Book of geometry: square orthogonal to a line

A square set orthogonal to a line.

I’ve spent a fair bit of time on my geometry book over the last few days.  I’m at my parents’ house, and my mother has been very encouraging of my work as an artist.  She works on her art, and I work on mine. Thanks to some new tools (namely a transparent plastic ruler with lines running parallel to the straight edge), I’ve been able to shorten the amount of time it takes to make one page, from about two hours to about half-an-hour.  I’ve produced ten pages in the amount of time it used to take me to do four.

Today, as I did tai chi, I was reflecting on this combination of challenges. Some work we want to do fast, like completing an art project such as this book of geometry that I’ve been working on for months (I’m three pages away from finishing side one, and maybe 15 hours from finishing side two… although side two has a lot more complicated geometry, so maybe it will take longer).

Some work, like tai chi, we want to do more slowly.  I get that.

Book of geometry

Doubling squares and halving them by means of geometry.

The trick is in not mistaking fast work for slow work, and vice versa. My father, tender loving guy that he is, told me today how proud he is of the work I’m doing, and how proud he is of me, and of the way that so many aspects of my life are in my command and under my control.  And yet, as he pointed out (because with Dad, there’s always a yet), I don’t have mastery of my weight.  I’m not sure, after two years and a third of tai chi, that it’s actually shifting my weight at all.  I mean, I probably have denser bones and stronger ligaments, but the push-ups have done more to bulk up my musculature than the tai chi has.  And, further, I haven’t really changed weight at all — I’m still a pretty solid 300# even after two years. Maybe I’ve shifted some weight from my gut to my bones, or from my gut to my biceps… but I don’t think so.

My doc says my cholesterol is up.  My good cholesterol is rock-solid good; my bad cholesterol is up more than it should be. This could be diet, this could be genetics, this could be the beginning of health issues. Every body is an experiment, as one of my doctors used to say.

Maybe it should say that Every life is an experiment.

Today is my birthday. Happy birthday, me.  I’ve begun to change my diet (again).  I’ve begun to be an artist (again). I’ve begun to reconnect with old friends (again). I’ve begun again so many times, that the experiment feels new and different every time.  Now we begin again, again.

But there’s an underlying order to the work.  In tai chi, as in geometry, each line and each angle and each ligament and each muscle has a sense of what it wants to be, and what it wants to do.  When joints creak in pain, we listen to them. When we over-extend a line or an arc, we listen to them.  When we discover truths about ourselves that others have made before us, we listen to them.  When we find a movement with power and grace, we listen to that. These are the building-blocks of our reality.

Today’s tai chi was much like yesterday’s tai chi; and tomorrow’s presumably will be similar to today’s.  We build successes and power a little bit at a time, by slow degrees and by slight changes — and eventually we come to a place where further changes are both commanded and needed.  We master the basics so that we can move on to the advanced work.  We return to the basics when the advanced work becomes too hard.  We re-discover how advanced the basics are when return to them.

May the year ahead be full of wonder.

Book of geometry

the book in all of its fold-out magnificence, so far.