https://docs.google.com/presentation/d/1aKo1NBUWyKRRUJFRtsMAY4ITfhQYytLp/edit?usp=sharing&ouid=110902289334019534432&rtpof=true&sd=true

 

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?