Starting Stars: Hexagon/Dodecagon Edition

I was recently inspired by Paul Salomon's thinking about stars and mathematics.  From our conversations I am excited about trying to understand stars better myself, and also about what they might have to offer in terms of elementary math explorations with my seven year old daughter.

Paul mentioned to me that stars have connections not only to geometry, but also to number theory and group theory (a branch of modern abstract algebra).  The visual interest provided by different kinds of stars is really what pulled me and my daughter in, with both of us wondering aloud: What's going on there? 

I started my inquiry last night by constructing a bunch of different 12 stars.  Sure I could have gone straight to the cool star applet Paul recommended, but I somehow found myself drawn to the classic geometer tools: pencil, straightedge and compass.  I wasn't sure how far I'd get, but I wanted to at least understand how a dodecagon (twelve sided polygon) was constructed so my kid and I could make one using floor tape in the morning.

I used an illustration from the book Quadrivium: The Four Classical Liberal Arts of Number, Geometry, Music and Cosmology as a reference.  I started with a circle 2" in diameter and then made six more circles with their centers evenly spaced around the circumference of the first circle.  From there it was pretty easy to connect the dots to make a hexagon, and use the outer circles to create the points/vertices of the dodecagon....

...and that's as far as the book illustration got me.

Then, looking at a photo of twelve pointed stars Paul had designed and cut out of plexiglass (I ordered my own set, it's coming soon!) I tried drafting my own versions.  Using the basic framework for the dodecagon I made two stars -- one made of two hexagons, the other out of three squares. It looks pretty straight forward, but at one point I really struggled with how to create the points of both stars -- you can see erroneous red lines in both pictures.  

And here are two more stars, below; one made from four triangles, and another from a six line asterisk.  There was one more that I couldn't figure out, but I was still pretty satisfied with my efforts.  My lingering confusion is the system and language Paul uses to classify stars; his notation looks vaguely like Cartesian coordinates.  I've read over his explanations to me, but I really wish he were across a table from me illustrating it in real life. What I notice in the picture above, for example, is that the green side of a square intersects with two points, one on either side of the red and orange points.  I see it but I don't know how to talk about it yet.

I did all that last night.  This morning I really wanted to see if we could at least tape down a hexagon on the floor.  Despite the fact that many of my posts seem like my kid and I are in some sort of harmonious learning nirvana, she is actually sometimes quite a resistant learner which was the case today -- she really wanted nothing to do with me and my math.

However, I went ahead and started measuring out 12" edges for the floor hexagon. I had one already measured and cut but had to leave the room for some reason, and when I came back I found her doing this:

She was measuring, taping and cutting as she went and was doing an excellent job at eyeing the angles.  Her fourth angle was a little too big, so I pulled in a hexagon from our pattern block set to show her how to use it to make each interior angle the same.  Resistance flared again, but I had the immutable laws of geometry on my side.  "Well," said I, "it won't really be a hexagon unless all the inside angles are the same."  Silence.

I sat quietly by after that and at some point I noticed she had taken the yardstick to make sure all the sides were the same length.  I heard her mutter something about "...these two are thirteen inches..."  and then I saw she had decided to trim them.  Her hexagon is below -- pretty good! Then she made sure to let me know she was making a square around it so no one would touch it.  It was hers, not the math mommy's.  She even went so far to install a burglar alarm in the form of percussion instruments so she would know if someone crossed over the line.
 

Given the mood around here today, I'm pretty happy with what got accomplished.  She did a bunch of independently initiated measuring (length and angles), and applied her understanding of equal sides in an analysis of the hexagon she created. And all of that pretty much without me. Later, when we were out and about, she applied the activity to other areas of her life.  One moment found her running up the zig-zaggy library ramp to the exit and exclaiming: "All you have to do is move your feet at an angle and you can run really fast up the ramp!"  In a restroom she saw that the sink's drain were made of small circular holes that formed a hexagon, and congratulated herself that she wasn't calling it an octagon anymore.

I think I'll let it sit for a while, maybe long enough to order some more floor tape in a huge amount of colors.  Perhaps a pile of new tape and some scissors left lying around will be enough to inspire a more collegial attitude around here!

Like Bees to Honey

I'm at my final program site up in the city for this summer.  Every day we work with Jump Patterns for a while, learn some clogging and then it's time to sit down for a while and explore other kinds of patterns.  Nature's patterns, to be exact!

A couple weeks ago, at a different site, the kids got so excited about finding a 'nature's number' (Fibonacci!) inside an apple that I had to go out and buy a bunch of lunchbox apples and cut them in half so everyone could have their own apple star the next day.  I've never seen kids so excited about numbers, or apples for that matter. 

I did my apple 'magic trick' again yesterday, but for today I decided to focus on shape and design in nature.  First I taped out a pretty respectable hexagon on the floor in the middle of our dance space.

When it was time, I asked the kids if they knew what shape it was.  The older kids knew, but not the youngers.  (At this site I have five to twelve year olds in the same class.) 

I asked how many sides a hexagon has, but before I'd allow an answer I started asking for volunteers to stand at each of the sides.  "One child, one side," I said, "Who wants to stand on the next side?  Okay, two children, two sides..." and counted up from there. 

I picked the little kids on purpose since they had been the most challenged with the dancing. 

"Okay," I said, "Everyone put your arm in."  I guess I wanted there to be some sense of division of the space.  Tomorrow I'll got back and put down more tape to show some of the internal structure of this hexagon.

"How many corners does a hexagon have?  One child per corner...Oh look!  Six sides, six corners!"

Then each kid got their own pattern block hexagon, or two trapezoids, to add to the tiling I started.

"What can you find in nature that looks like this?" I wondered aloud.  Only the last class (with more older kids in it) knew right off the bat.

Then they got their very own taste of local honey.  It was a bit too strong a taste for most kids, but still a good new thing to try.

And then, much to my delight, something wonderful happened.  My first class was waiting to go on to their next workshop and were hanging around in the space.  They naturally found their way to the hexagon on the floor, made a circle around it and started playing song and chant games.  This is exactly the kind of thing that always happens when I put down tape where its never been before.  It changes the space and kids notice.

Another example from earlier in the morning: I was creating a set of parallel tape lines as I measured out the sides for the hexagon.  A little boy came over and jumped over the width and then the length.  I wish everyone would put tape down on the floor and then start the camera rolling to record all the awesome things that happen when kids discover this restructuring of their world.

This much I know: kids are to tape as bees are to honey.  Or maybe that's bears to honey?  Anyhow, here's the video of the girls playing 'Little Sally Walker' around the hexagon waiting for their next class:

Hexagon Poetry

Can't remember why I got out the pattern blocks; it's been a while.  'How many different ways can I make a hexagon?' I asked myself.  The kid joined in at some point and then we left it, on to other things.

Later in the day I found this. 

It says: "Each ray is different with the days."

This post is linked to Saturday's Artist at Ordinary Life Magic. :-)

A Work in Progress: Paper Quilts & Hexagonal Rotation Designs

I am in the process of developing a paper quilt project for kids using a hexagonal design built from triangles.  Here's a picture of what my kid and I have done with paper quilts in the past:

The design incorporates lots of little triangles, but the end result is pretty square.  Not that there's anything wrong with that, but wouldn't it be cool if a beginner paper quilter could create something more circular using triangles? 

Yes!

Here's the book that was the inspiration for my hexagon-based project, as detailed in a recent post:

It turns out snowflakes are hexagonal!  But they look so different from what we usually think of as a hexagon...

source

There are endless tessellating design possibilities here which would be fun to explore with paper in the future, but I've got something else on my mind at the moment....

In her snowflake book, Paula Nadelstern shows how to design snowflakes using a "60˚ triangle" unit.  She doesn't say equilateral, but I think that's what she means. I read through her book and tried to get the gist of how she makes these beautiful fabric images; it appears the whole thing revolves around one single, accurate triangle template. 

Here's what happened when I thought I could get away with a random black-line triangle pulled of a google search, one that had every appearance of being an accurate equilateral triangle:

Not good!  Seems to be missing a few degrees.  So, I tried again.  In her book, Paula gives a formula that was helpful for constructing an accurate triangle.  The smallest one had a vertical length along the center axis of 2 3/8" and a horizontal length, to both the right and the left of the center axis of 1 3/8".  It worked!  Here's a picture of how I marked it out.  I cut out the triangle template using an exacto knife.

I made the template out of cardstock because it was all I had on hand.  I made it what I'll call the 'reverse' of a traditional-type template so I could easily see where I was putting the triangle on the patterned paper. 

As you can see, below, I was pretty successful at getting the same part of the repeating design in the same place on each triangle using this 'see through' template. 

Here it is, starting to come together:

And here is what it looked like when all six triangles had found their places:

Yay! I thought it looked really good at this point, but I wanted to see if I could give it just a little more shape.  

What I noticed in Paula's designs is that the snowflakes get their form by focusing the design at the edges where the triangles meet; one edge has half the design, the adjacent edge has the reflection.  Focusing on the 'spokes' (for lack of a better term) that are created where the edges meet (and create lines that radiate out of the center of the hexagon and through each vertex) ultimately creates the snowflake's design.  I think!

I had just enough time to try a small experiment to see if I could highlight the 'spoke' portion of the hexagon. After making a couple more measurements and lines I cut a new template out of the leftover 'insides' of the triangle template.

I think the final design looks more like a flower or a star, but I'm still pretty pleased.   

I think I'll try stripes next time and see what happens! 

While I was in the process of cutting out and placing triangles, my daughter got really excited about the intermediate shapes I was creating.  For the K- 2 set I think a great start to the project would be to give them some patterned paper triangles and some glue sticks and let them explore what kinds of designs they can make, observe others' work, and help them find ways to describe what they see.  Later, you could bring in some inspiring hexagonal designs (both your own and others') to serve as models for the rotational design element.  Once you spend some time deconstructing the examples it would be time to give the kids more materials and start the design/observation/reflection process all over again. 

I believe that, even with the younger elementary kids, the experience of putting six duplicate, congruently patterned triangles together to make a hexagon has the potential to reveal the complexity of this shape in a new and inspiring way.  I, for one, am seeing hexagons in a whole new light.

For middle and high school students the challenge of building the perfect template for such a project could easily catapult this whole investigation into some really cool mathematical inquiry.  (Plus, I like the idea of giving them a choice, at first, about whether to use a 'pretty good' black-line template or the measuring method and let them compare the results!) 

Finally, to make this a viable project for multiple children I really need to find a easier way to prep a large amount of paper triangles.  Does anyone out there know what kind of cutting tool might be able to do this?   I'm planning to play around with this some more, so stay tuned!

Inspiration Strikes: Hexagonal Rotation Designs

Inspiration can strike at any moment.  Even mundane moments like flying in and out of the library to pick up a book on hold and then running home to make dinner.

Here's what I found on my way to the 'hold' shelf:

These are quilted snowflakes.  Snowflakes are hexagonal don't you know and just LOOK at these beauties.  I've always thought of a hexagon as a bit cumbersome, but these take my breath away.

I snatched up the book and devoured it.  I have no intention of quilting anything, but I am handy with paper and I think that somewhere within all this complexity is a (much simpler) project for younger kids, and possibly a fun construction challenge for older ones.

In my next post I'll share my first attempts (one failure and one more successful) at finding bilateral symmetry within a paper/fabric print and using it to create the rotational symmetry of something resembling a snowflake, or a star, or...err, well it looked nice, whatever you want to call it.

Until the next post when I share my modest efforts, here's a quick peek at the glorious visions of Paula Nadelstern's kaleidoscope quilts.