Art and Math · design · Drawings

Lines and Circles Workshop

Linear Mod Design by Carly
Linear Mod Design by Carly, tenth grade student

Last night 15 people showed up at the library for a couple of hours to make patterns based on lines and circles. I don’t think anyone knew quite what to expect but that didn’t keep them from showing up. A brave bunch. The participants were tweens, teens, and adults. There was at least a 50 year gap between the youngest and oldest: it was quite wonderful to have this group all in one room, as each generation brings their own aesthetic, energy, and reflective questions with them.

by Cordelll,  a ninth grade student
by Cordelll, a ninth grade student

I demonstrated two ways of making designs: using lines (which I wrote about in my previous post) and intersecting circles, which people have been exploring for many centuries. I had originally thought I would show just the circles geometry, but then considered that some people might be really uncomfortable using a compass.  A few people just worked with just the lines, a few people worked with just with circles, and the rest did both.


Some people were curious about the math that went into the number patterns that I gave them, and I explained it to those who asked.


At the next workshop I will bring in my laptop and show Dan Anderson’s linear mod open processing page, as well as the tables in to people who want to know more.

Siri's, in progress
Siri’s, in progress

There was a great mix of approaches to this way of working.


I embarrassed myself by not having reviewed the process for making the circle patterns right before the class. I had made many samples of the “seed of life” circles patterns, but then I had done other designs, and when I started demonstrating I got quite confused. I had to go sequester myself for a bit to reconstruct the pattern.

Quinn's designs
Quinn’s designs

One young man didn’t have any interest in coloring anything in. Not only that, but he decided to try out his own pattern of lines. Actually, he tried out everything he could think of, with both the circles and the lines, and ended up with a pile of papers filled with all sorts of designs. It was delightful to see him working out his own templates and number sequences.

Quinn's Crane and Siri's Drawing
Quinn’s Crane and Siri’s Drawing

By the end of the workshop this young man had started doing some origami, which he graciously gifted to me. I photographed (above) his crane with the work of an adult, because I so enjoyed seeing having all these young people and adults in one room, all together making art.

Here are some links for anyone who is interested:

The PDF to use with the line designs

Dan Anderson’s Open Processing Linear Mod page to see what’s going on with the line designs

Dearing Wang’s Circle Drawings

Circle Geometry by Paula Beardell Krieg

Next Tuesday I will do doing another one of these workshops! I’m looking forward to it.


Line Designs

6x + 5

It’s handy to have a few methods for creating designs at one’s fingertips. Three Tuesday evenings during the month of July at my local library I’ll have a chance to work with people on producing images that are part recipe, part personal. I’m describing the workshops as being focused on making patterned images based on curves and lines. Go ahead and click on the link in the first sentence here if you want too see some of the curves that we’ll be looking at. This post is about the lines

Starting with w parallel # lines
Starting with w parallel # lines Click HERE for PDF of this page

I’ve been working on this system of connecting dots. People will get this nearly blank paper and will choose a pattern of numbers to write across the top. In the image below the pattern 5,3,1,9,7 repeats 4 times along the top, then the numbers are connected with a straight edge to the corresponding numbers on the bottom. What results is a pattern of intersecting lines that can be colored in an infinite number of ways. Here’s just one of those ways (oops, note that the paper, and thus the numbers, has been turned upside down as I like the image better this way):


The drawing above shows all the criss-crossing lines, but if I zoom in on just one area the resulting image has a different sort of look.  In other words, something that looks like this (whose repeat pattern was  5,7,9,1,3) :

linear mod 6

….can be cropped to something like this:

Starting with two parallel line
Starting with two parallel line

My thought is that it’s possible to make many cropped images from the same “master” image, and thus end up with numerous designs that can stand alone, but that still go together.

I’ve been trying out different mediums to color these in with. Pencil, colored pencil, and markers all seem fine. I’m not having much luck with crayons or watercolors, but that may just be me. Here’s one that’s all pencil (the repeat pattern here is 5,3,1,9,7 : these numbers are visible at the bottom of the drawing, which I’ve turned upside down)

Pencil drawing
Pencil drawing

The image below is done entirely with markers. It differs from the others in that the spacing of the lines is twice as wide all the others here, and the pattern of numbers written across the top only repeats twice (3,1,5,1,7)


I’ve made so many of these, but they are all so different that I don’t feel like I’ve made enough. I’m interested to see what the participants in these upcoming classes do with this way of working.

number pattern 3,6,9,2,5,8,1,4,7,0
number pattern 3,6,9,2,5,8,1,4,7,0 turned upside down. To me. this image looks like two butterflies or hummingbirds on opposite sides of a flower.

I hope to be posting photos of images made by workshop participants during the course of July.

My plan, by the way, is to basically hand out the number patterns that I’ve come up with, so it really will be a connect the dots kind of activity, at least until the coloring begins. If anyone is interested in where these numbers come from and likes reading about linear equations, I put a post up on Google Plus to explain all.

Wish me luck in running a fun workshop!

Flip Book

Flip-Book by Committee?

Flip books in progress, holding things together with binder's clips
Flip books in progress, holding things together with binder’s clips

This first set of flip-books, showing some transformations of the graph of a line, have been keeping me occupied. I have a small circle of math-friends to send these to, for comments, but I would like to expand this circle. It has taken me by surprise how many revisions this book has gone through already.



Flip Book: Cutting the pages down to size
Cutting the pages down to size

There’s been revisions in sequence, color, line weight and paper weight. It feels like this tweaking can go on endlessly. First, though, I would like to get this in the hands of some more teachers who are willing to put a copy of this flip book into student’s hands, and then tell me how it goes. I am aiming at something like a template to use as base for his book as well other books like it.

Flip=Book cover pieces
Flip-Book cover pieces

Is there anyone that I can interest in playing with me and my books? If you teach the equation of a line, and would like to have one of these books in exchange for giving me some feedback and suggestions, please let me know. The contents of what’s in the book is similar to the GIF in my last post. As I said there, the GIF can show the concept, but it seems to me that  it ‘s more valuable if students hold the book in their hands, as this then allows them to slow the animation down, so that they can work out for themselves the secret of what’s going on between the equation and the picture.

Three Flip-Books and gold envelopes
Three Flip-Books sliding into gold envelopes

So who wants to play? Leave a comment below, or send an email to me in the address that’s listed under my About tab. Later this week I am hoping that three teachers will receive a little book in a gold envelope.

Math and Book Arts

I am so embarrassed. I want to do math.

drawing 1
Pencil on Layers of Tracing Paper

My daughter’s math class is working on logarithms. I have a special enthusiasm for logarithms. Every single thing about them appears to be overwhelmingly opaque and indecipherable.  Everything. And the most awesome thing about them being so completely crushingly incomprehensible is that Mr. John Napier (1550 – 1617) invented this system was to make life exponentially easier for us.  And he succeeded.

Now here’s another cool thing about logarithms. The spelling. No one confidently remembers how to spell this word. But there’s a trick to remembering.

How to spell logarithm
How to spell logarithm

The trick is to spelling logarithm is to notice that it starts with L O G (that’s the easy part) and ends with the most of arithmetic. No pun intended.

squares on page
Altered Graphs

I’ve been experiencing something that I mistook for an internal tug-of-war: I like blogging about book arts, but my mind of late has been drawn to playing with ideas that seem to have more to do with math than with books. It’s been a dilemma, how to keep writing about book arts when my mind is elsewhere. Finally I’ve had an ah-ha moment: I had forgotten that what brought me to book arts in the first place was wanting to make visual sequences of images that were related to a simple equation.

The fifth and the eighth page of a doubling book

The equation that drew me into making books is the one which starts with the number 2 and doubles, then doubles again, then doubles again and again.  It takes eight pages of doubling to get from 2 to 256. I’m infatuated by the slow measured way the numbers increase until there’s this tipping point, when  the quantities then erupt into unmanageable largeness.  I had created maybe a dozen of these books, experimenting with using lines, circles, overlapping lines, droplets of paint ect. I bound these books in a most inefficient and cumbersome way. Eventually a friend pointed me in the direction of The Center for Book Arts in NYC and new part of my education began. I found the  geometry of  constructing  books to be a satisfying, even sublime experience. And, since I didn’t really know any other intoxicating mathematical equations I just kept making books.

Solve for x                click here for the answer

Now, many years later, my daughter is coming home with problems like the one pictured above.  My son offered an insight on this kind of problem, one that I hadn’t thought of before. He said that the answer to this problem made no more sense to him that the problem itself.  It’s tough to remember that these functions have a look to them, and that solving for x looks like something.  A day or two after having this problem as part of a long, mind numbing homework assignment, my daughter came home and bemoaned that her teacher had just that day told them that the point of logarithms was to find exponents. She wanted to know why they weren’t told that in the first place, and what was the point of doing all those numerical gymnastics? We had quite the discussion about that. And it keeps me thinking about pictures.