My apologies for the post and responses being late. We went north to dig my mother-in-law out after a significant snowfall and cell service and internet were both down. It would have been enjoyable to unplug if it weren't for things to do!
I looked at the artworks and chose several to examine more closely. Despite a relatively strong background in math, many of the principles behind the pieces were outside of my linear algebra and calculus area of experience. I noticed that most of the pieces I gravitated towards had a lot of symmetry. The one I chose to recreate is called "The Stijlish Seed of Life" by Emanuela Ughi. I was curious why six circles perfectly overlapped the central circle so I used a compass to start drawing it. I drew one circle, then added a second centered at a random point on the first circle's circumference. When I added a third circle centered on the intersection between the first two, I found that I could draw an equilateral triangle with side lengths equal to the radius of the circles. As a result, the arc length is pi*r/3 so six circles would be 2*pi*r, equalling the circumference of the circle.
I was also fascinated by the fact that a single circle has continuous rotational symmetry and infinite axes of reflection. If you ignore the different colours, the entire piece has six fold rotational symmetry and six axes of reflection symmetry. Once the colours are considered the piece no longer has any rotational or reflection symmetry because the four pieces of each colour are different shapes and sizes. Yet, at the same time the puzzle can be put together many, many different ways because the curvature is the same in all pieces. Any convex edge can be matched with any concave edge. Below are just a few examples!
I appreciate the goal of bridging mathematics with art and culture. I have found that people assume I am not interested in things like art or literature because I am a "math/science person". I do have interests beyond math and science, but will admit that I do view the world through a lens that is a bit more technical than artistic. When reading Spinning Arms in Motion: Exploring Mathematics within the Art of Figure Skating I was honestly more interested in how biomechanics and physics engaged with the geometry. The authors explored the geometry of a figure skater's arms in relation to their body while performing an upright spin. They presented an elementary and secondary version, including appropriate mathematical questions. The article referenced modeling, but when I followed the links the privacy settings did not allow me to view the videos. I felt like the link between artistry and geometry was a little forced to be honest and I am not sure that I would use this particular idea.
I think it would be interesting to look more closely at things like sacred geometry, how math can be used to create depth and perspective, or looking for the Golden Ratio in art and architecture. But perhaps that is my overly analytical brain must watching to know how things work? My mother does quilting and she told me that if the entire quilt is perfectly symmetrical the brain decides it is boring so you have to add one subtle thing to break the pattern, like a colour that doesn't quite match. Then your brain finds it interesting and subconsciously tries to figure out why it is not quite right.
I also found many interesting art pieces in our Bridges year but found that my mathematical background in many of them was too lacking. I liked your selection for the ability to apply circle geometry and the ability for it to have symmetry. Plus the colours were beautiful! We actually both chose puzzles for our choices but I didn't try and recreate the puzzle itself. I like how you did!
ReplyDeleteThe conference art connects art and mathematics in so many ways that break down the stereotype of "math people" and "art people". I have always enjoyed art but lean toward things with symmetry and defined lines. Too bold of colours or "messy" art is not my favourite. So maybe that is the math part of me coming out.
I appreciate your reflection on your article selection and that the connection between the geometry and the activity seemed forced. There is often a balance between curricular goals and needing an activity to do that. Perhaps the activity of figure skating would have been better suited to the mathematics in physics? I do still like the idea of having students think about things that they do (i.e., hockey, skiing, swimming) and the mathematics behind it.
I agree that it is good to involve students' interests and I can certainly see the artistry in figure skating appealing to many people. It is just not an example that I would choose myself which is more of a reflection on my own interests than the validity of the example. I do appreciate it as an example that you can find math pretty much anywhere if you are willing to look!
DeleteWow Danielle! You did an amazing job at recreating the art. I truly appreciate your showing that there was so much math in the construction. The question is, how do we translate this connection of the math and art in our classrooms?
ReplyDeleteAs I mentioned in my blog, my students are creating tessellations currently in FOM12. I am struggling to help them see the math. They see tessellation as art and cannot seem to connect the math to the art. They keep asking me, is that all we are going to do, or when will we do the math. I'm afraid that we have compartmentalized the subject areas so well that overlaps seem to be a violation of sorts. Like Andrea, I too appreciate your sentiments about "forcing" connections between activities and curricular content. While tessellation is in the curriculum of FOM12, it is difficult to elicit authentic mathematical thinking from it. Many students just did the bare minimum and kept asking me if that was an "A". I kept asking them to explain the math to me and most could not do it. I am thinking that if the teacher did not have a genuine interest in geometry and art, this could be another case similar to the use of the algebra tiles wherein it could hinder rather than help mathematical understanding.
Thanks for your thoughts and questions. Considering this conundrum, I wonder if something like "Show Me Your Math" could be a template for connecting the arts and math. Art and culture being so closely linked!
Deletehttps://showmeyourmath.ca/
Thanks Danielle and group for a thought-provoking discussion! This is at the heart of what we are looking at in Week 5, and for the course project: how to integrate embodied, arts-based and outdoor pedagogies into our math teaching, rather than just having it as a 'fun' add-on that doesn't connect to anything. That IS the challenge, and if the teacher feels that the artistic activity is fluffy, embarrassing or simply unconnected, the students will not be convinced otherwise either!
ReplyDeleteBut in Danielle's exploration of the Bridges art piece she chose, I see plenty of mathematical connection and exploration! The interdisciplinary activities need to be focused on aspects of mathematical ideas and thinking (like, for example, why six circles completely fill the circumference of the original circle, and how you can show that those triangles produced are equilateral). The point of doing the art is not just to create something pretty, but to explore and extend mathematical thinking in ways that you can only do through the art. If it feels like 'forcing' things to you as a teacher, you ought to find another representation that doesn't feel forced. There is plenty of mathematics in tessellations (check out the wikipedia article on tessellations for starters https://en.wikipedia.org/wiki/Tessellation), and I'm very inspired by Islamic tilings and their beautiful tessellations -- but find something that really speaks to you, and that you have fun exploring, and that might be the thing to start with in your classes!