Week 4: The Road to Helsinki

 

So, I have decided that I have to get to the Bridges conference in the summer somehow. It sounds like the best conference ever: music, dance, art, drama, math. The only issue I have with going is that I feel like I am not enough of a math nerd. I would be a fraud: a career K-8 educator with lots of foundational math, but very little memory of the really awesome stuff. I could do it once...that means I can do it again, right?  I am currently helping my son with Math 10 (and then grade 11 and then grade 12) with the express purpose of being a nice mom AND remembering how to do the stuff. I'll get there.

I read Thomas & Schattschneider (2011), Fenyvesis (2016), and Capozucca & Fermani (2019). They were manageable and I find the topics fascinating. The Thomas & Schattschneider paper was an interesting window into the development of a young artist (Dylan Thomas) as he brought together Escher's tessellations and Coast Salish art elements through mathematical explorations. What was particularly satisfying was his developing sense of himself as both a Coast Salish artist and mathematician. It was the perfect example of art and mathematics supporting and elevating each other. 

This theme continues in the The Fenyvesis (2016) history of the Bridges conference.  I love the Persian cultural-historical perspective Reza Sarhangi brings to the table. I also want to know more about Solomon Markus' work and the latest research on the connections between mathematics and poetry. Poetry and music are linked...even free verse. When I compose poetry, there is something about it that seems right. Something about the rhythm and way the words sound as I as say them, even without an obvious meter and rhyming scheme. A good segue, I think, to the Capozucca & Fermani paper.

I don't mean to gush, but it can't be helped: I want to participate in this workshop (Make Music Visible, Play Mathematics)! It was really helpful to think of the chromatic scale as a circle. I've fiddled with a linear music scale and geometric transformations before. Basically, I took a roll of overhead projector film and made a loop around a glockenspiel. Students in groups of 4 used whiteboards marker to record a sequence of notes numerically. They then slid (translated) the sequence forward or back on the glockenspiel to create harmonic patterns. Cool idea, but I like idea of exploring triangles of notes within the chromatic circle better. I am very curious about the sounds produced by different kinds of triangles and what happens when you "rotate" them. I also like the boomwhacker greeting idea. Basically, you give everyone a whacker and let them test to see whether their whacker "plays well" with another whacker. The other parts of the workshop (choosing a song and analyzing its the geometry; creating a composition collectively) also sound fun. Anyway, hopefully they have this going on in Helsinki this summer. I'll sign up for sure :)

Ok! Onward to the Bridges art. I chose Pahoehoe by Eve Torrence (click here to see it). It a gorgeous

piece of geometrical origami (origamical geometry?). I chose it because it reminded me of the the beading work I am engaged in right now with Cynthia and the folks who signed up for the workshop led by Nico Williams. If you read my last post, you'll know we are creating hyper-squares and all kinds of interesting geometrical entities. Diving into this piece of art is a way to meditate on complex geometric figures. I also liked the idea of folding my very own hyperbolic paraboloid. 

Eva Torrence had to fiddle with starting shape of her chosen hyperbolic paraboloid until she found one that allowed her to create the final geometric shape. The starting shape most choose is the square. This is the starting shape I chose as a beginner to the craft of hyperbolic paraboidal origami. Why? YouTube how-to's proliferate on the subject. Torrence experimented until she found a particular rhomboid shape with a specific angle ratio that worked (specifically 26:31). Not a very friendly-sounding ratio. I decided to begin with the standard square and then experiment with rhomboid shapes with various angles. 

I decided that the folding process for the hyperbolic paraboloid based on the square was straight-forward enough to do it from memory (after having seen it once on a YouTube video). Here is how it "unfolded":

*Note my despondent dog under the table. I hadn't taken him for a walk yet. This I finally did, thinking all the while about how I might approach the non-square rhombus. When I returned, I had a plan. This is what I did:

My thought was that I needed to simply pay attention to the dimensions of the starting rectangle. I could make a perfect rhombus by folding from the top of the vertical centre fold line to the tip of the horizontal fold line in each quadrant. The result is a perfect rhombus:

Next comes the pleating, which I decided had to be parallel with opposing sides of the rhombus:

Then comes the folding into the hyperbolic paraboloid (HP):

And now here are my second and first HP's side by side:

Next, I decided to adjust my rhombus by folding and trimming a bit off of my starting rectangle like so:

And after pleating and refolding, here are all three of my HP's (front and back views):

The only bit that I'm not sure about now is how the artist determined the ratio that describes the rhombus she used for her creation. At first, I thought it was based on angles. But the more I worked with the shapes, the more it made sense to determine the ratio comparing the length and width of the diagonals of starting rhombus. The square is a simple 1:1 ratio. The diagonals of the second rhombus are based on the dimensions of a standard piece of letter-sized paper (27:35) and the third are a based on the dimensions of a standard piece of paper with a quarter of the width trimmed off (81:140). If I am right about what is being compared in the ratio, I can directly compare my shapes to Eve Torrence's. In that case, my second rhombus is closest to Eve Torrence's rhombus (the ratio of her rhombus diagonals is 0.8387 compared to 0.771 for my second rhombus). There you go!