Highly Unlikely Triangles and Other
Impossible Figures in Bead Weaving, Gwen L. Fisher
Summary
In her article, Gwen Fisher explores the intersection
of geometry, visual perception, and art by transforming 'impossible' optical
illusions into physical 3D sculptures. She focuses on the Penrose Triangle, a
paradox that cannot exist with straight beams and right angles. Fisher resolves
this paradox by using Cubic Right-Angle Weave (CRAW), a flexible bead-weaving
technique that allows her to introduce a 90-degree quarter twist into each
beam. This structural adjustment substitutes subtle curves for straight lines,
creating 'highly unlikely' objects that function as topological equivalents to
the illusions. These sculptures exhibit Möbius-like properties, where a single
path of beads must travel around the entire structure multiple times (four
times in the case of the triangle) to return to its start. Fisher successfully
extends these principles to more complex 3D polyhedra, such as tetrahedra and
dodecahedra, creating 'strange loops' that challenge our traditional
understanding of surfaces and boundaries."
Stop 1
“The twist in the beadwork allows the impossible triangle to be constructed in 3D. The twist in the beaded version also destroys the optical illusion due to the curvature it introduces to the edges that appear straight in the 2D drawing” pg. 100
Fisher’s description of the bead 'destroying the
optical illusion' hits on a powerful truth. It makes us see that something only feels impossible
because of the 'rules' we’ve trapped ourselves in. For instance, if I tell Grade
7 students to connect four dots in a square with only three lines, they’ll say
it’s impossible, but until they can fold the paper or use a thick brush to
change the 'straightness' of the line, they may not connect. That means that
just like Fisher’s quarter twist, these aren't just tricks; they show that flexibility
is a mathematical tool. Another thing that made me pause and reflect is that
just like how the beads were used to make the impossible triangle clearer, students
can use hand on manipulatives to gain a concrete understanding of abstract ideas.
Stop 2
Constructing a quarter twist in each beam takes some thought. Looking closely at the bottom beam, you will see that I accidentally omitted a quarter twist; the bottom beam is straight.pg 101
One aspect of the article that struck me was Fisher's admission of her 'very first attempt' failure. This connects deeply to what we see in the teaching and learning environment, where students often give up when a problem becomes difficult. Fisher’s mistake wasn't a sign of inability, but a productive struggle that allowed her to see the logic behind the shape better than if she had accidentally gotten it right. As a teacher, I emphasize that the 'understanding of why they failed' is more valuable than the final answer because they should be able to learn from their mistakes. Fisher's omitted twist serves as a perfect mathematical metaphor: sometimes it is our errors that finally reveal the hidden rules of the system we are trying to understand.
Considering Gwen Fisher’s 'impossible' structures that need a 90-degree twist and four passes to close, how can we expect a linear lesson plan to effectively teach complex math? Have you ever taken shortcuts in your teaching, only to realize that this resulted in students creating flawed mental frameworks that couldn’t support true understanding without hands-on exploration and new perspectives?
Hi Clementina,
ReplyDeleteYour reflection on Gwen Fisher’s work is such a powerful critique of how we view "impossibility" and "error" in the math classroom. I especially like your first stop about how the quarter twist "destroys" the illusion - it’s a perfect metaphor for how hands-on manipulation can break down the mental barriers students face when they are stuck in a 2D mindset.
I also see a strong link here to Andrea Hawksley’s (2015) layered beverages. Just as Fisher uses beads to make the "impossible" tangible, Hawksley uses liquid density to make abstract ratios "tastable". Both authors are essentially arguing for an embodied mathematics, and the idea that our physical senses (touch for Fisher, taste for Hawksley) are legitimate tools for resolving mathematical paradoxes.
I think "linear" lesson plans are often the very thing that creates the "zombie" learners Karaali warns us about. I have definitely taken shortcuts before, like giving a formula for volume instead of letting students build the shapes. What I realized was that without that "twist", without the hands-on struggle, the students were just "reading" my math rather than "authoring" their own. Fisher’s work proves that sometimes you have to go around the structure "four times" before you truly understand where you started.
Hi Clementina, thanks for your summary and reflections. I noticed that your second stop was similar to my own, where the author/artist encountered moments of difficulty in the process. I completely agree with you that, as teachers, we need to honour the opportunities we have to recognize and learn from difficult moments and mistakes.
ReplyDeleteTo answer your question, I am not sure that a strictly linear lesson plan is viable in general. While pre-service teachers are often required to write lengthy lesson plans based on an imagined ideal classroom (I have written some extremely long ones myself), the moment they step into a real classroom, those plans can quickly be “jeopardized.” This also brings to mind questions of power and control. Traditional education, and its undeniable intertwining with systems of power, has led us to believe that a linear lesson plan is feasible and desirable.
That said, I do think there are necessary building blocks in mathematics that develop over time. For example, we must first understand what a triangle is before identifying different types of triangles, and only then can we determine which trigonometric functions are applicable. However, in terms of a day-to-day lesson plan, I do not think strict linearity is realistic. It is important to have key concepts and material that the teacher intends to cover—especially given the prescribed curriculum—but now more than ever, students are restless, and we must expect that lessons will not proceed perfectly or flawlessly.In doing so, we disrupt entrenched power dynamics that often shape traditional classroom structures.