Week 9: Bead Weaving and Triangles
Highly Unlikely Triangles and Other Impossible Figures in Bead Weaving – Gwen L. Fisher
In this paper, Gwen Fisher explores how mathematical ideas can be expressed through bead weaving by creating physical versions of “impossible figures,” such as the Penrose triangle. Impossible figures appear coherent in two-dimensional drawings but cannot exist in three-dimensional space if their edges remain straight and rigid. Fisher shows that by using bead weaving techniques, particularly cubic right angle weave (CRAW), these shapes can be constructed as sculptures because the flexibility of the beads allows the beams to twist and curve. In doing so, the paradox of the impossible triangle is resolved through small structural changes such as quarter twists in the beams. The project demonstrates how mathematical structures, visual perception, and artistic craft can come together to produce new forms of mathematical expression.
Stop 1: Mathematics as Material and Craft
I stopped reading when Fisher explained how the impossible triangle becomes possible through beadwork. The key insight is that the illusion only exists when we assume straight beams and rigid angles. Once beadwork introduces flexibility and curvature, the paradox disappears.
This made me think about mathematics as something that is often treated as abstract and fixed, but here it becomes tactile and material. Instead of proving something on paper, Fisher is exploring mathematical ideas through making. The beads, threads, twists, and paths become a way of thinking through geometry. This reminded me that mathematical understanding can emerge through craft, movement, and experimentation, not only through symbolic reasoning.
Stop 2: Paths, Loops, and Unexpected Structures
Another moment that stood out to me was when Fisher describes how the surfaces of the beaded triangle form paths that loop around the object multiple times, similar to a Möbius strip. This idea of paths and loops resonated with me because it shows how a simple geometric object can contain multiple layers of structure depending on how we follow it. It reminded me that mathematics is often about noticing patterns that are not immediately obvious. When we move around the object, what initially appears simple becomes complex. That process feels very similar to learning mathematics itself. At first a concept may seem straightforward, but deeper exploration reveals unexpected relationships.
While the paper beautifully demonstrates how art and mathematics can intersect, it focuses mostly on the aesthetic and structural aspects of the objects rather than on how these ideas might translate into classroom practice. The sculptures show how mathematical creativity can emerge through making, but the paper leaves open the question of how educators might bring similar experiences into everyday learning environments. For students who already feel distant from mathematics, hands-on creative approaches like this could potentially help them see mathematics as something exploratory and expressive rather than purely technical.
Reading this paper also made me think about my own research (as always) on Funds of Knowledge and early STEM learning. In many households, mathematical thinking appears through everyday activities such as weaving, cooking, measuring, or building things. These practices are often not recognized as “mathematics” in formal schooling. Fisher’s beadwork highlights a similar idea: mathematical thinking can live inside craft, art, and material practice. Recognizing these forms of knowledge could help make mathematics more inclusive and connected to people’s lived experiences.
I wonder if mathematics can be explored through artistic practices such as bead weaving, poetry, or visual design, how might this change the way students see mathematics?
In this paper, Gwen Fisher explores how mathematical ideas can be expressed through bead weaving by creating physical versions of “impossible figures,” such as the Penrose triangle. Impossible figures appear coherent in two-dimensional drawings but cannot exist in three-dimensional space if their edges remain straight and rigid. Fisher shows that by using bead weaving techniques, particularly cubic right angle weave (CRAW), these shapes can be constructed as sculptures because the flexibility of the beads allows the beams to twist and curve. In doing so, the paradox of the impossible triangle is resolved through small structural changes such as quarter twists in the beams. The project demonstrates how mathematical structures, visual perception, and artistic craft can come together to produce new forms of mathematical expression.
Stop 1: Mathematics as Material and Craft
I stopped reading when Fisher explained how the impossible triangle becomes possible through beadwork. The key insight is that the illusion only exists when we assume straight beams and rigid angles. Once beadwork introduces flexibility and curvature, the paradox disappears.
This made me think about mathematics as something that is often treated as abstract and fixed, but here it becomes tactile and material. Instead of proving something on paper, Fisher is exploring mathematical ideas through making. The beads, threads, twists, and paths become a way of thinking through geometry. This reminded me that mathematical understanding can emerge through craft, movement, and experimentation, not only through symbolic reasoning.
Stop 2: Paths, Loops, and Unexpected Structures
Another moment that stood out to me was when Fisher describes how the surfaces of the beaded triangle form paths that loop around the object multiple times, similar to a Möbius strip. This idea of paths and loops resonated with me because it shows how a simple geometric object can contain multiple layers of structure depending on how we follow it. It reminded me that mathematics is often about noticing patterns that are not immediately obvious. When we move around the object, what initially appears simple becomes complex. That process feels very similar to learning mathematics itself. At first a concept may seem straightforward, but deeper exploration reveals unexpected relationships.
While the paper beautifully demonstrates how art and mathematics can intersect, it focuses mostly on the aesthetic and structural aspects of the objects rather than on how these ideas might translate into classroom practice. The sculptures show how mathematical creativity can emerge through making, but the paper leaves open the question of how educators might bring similar experiences into everyday learning environments. For students who already feel distant from mathematics, hands-on creative approaches like this could potentially help them see mathematics as something exploratory and expressive rather than purely technical.
Reading this paper also made me think about my own research (as always) on Funds of Knowledge and early STEM learning. In many households, mathematical thinking appears through everyday activities such as weaving, cooking, measuring, or building things. These practices are often not recognized as “mathematics” in formal schooling. Fisher’s beadwork highlights a similar idea: mathematical thinking can live inside craft, art, and material practice. Recognizing these forms of knowledge could help make mathematics more inclusive and connected to people’s lived experiences.
I wonder if mathematics can be explored through artistic practices such as bead weaving, poetry, or visual design, how might this change the way students see mathematics?
Thank you, Sushi. Your writing encourages me to think about how visual design can shift my understanding of mathematics—even when I’m already confident that I “get” the concept. What struck me most is how design can reveal logic that stays hidden in symbolic form. Sometimes I think I fully understand an idea, but when it becomes visual, spatial, or embodied, I suddenly see a new layer of reasoning that wasn’t accessible before.
ReplyDeleteVisual design has a way of slowing me down, making me notice structure, symmetry, and relationships that I might gloss over when I’m working purely abstractly. Even simple changes in perspective—rotating a diagram, tracing a path, or imagining how a shape behaves in space—can create surprising clarity. It reminds me that mathematical logic isn’t just something we think; it’s also something we perceive.
This shift in perspective matters because it opens mathematics to more ways of knowing. When ideas become visual or tactile, they invite curiosity rather than compliance. They show that understanding can arise from creativity, play, and aesthetic intuition. And for learners who feel disconnected from traditional math, visual design offers another doorway—one that can illuminate the beauty and internal coherence of mathematical thinking in unexpected ways.
Thank you Sushi, I really enjoyed reading your reflection on the article. The concept of an "impossible" figure, such as the Penrose triangle, being created through bead weaving is very interesting. It demonstrates that when mathematics is transferred from paper to physical materials, new possibilities emerge. The flexibility of the materials allows the figure to twist slightly, thus resolving the paradox. It made me think of how math can be explored through creation and experimentation rather than just through formulas or diagrams.
ReplyDeleteYour discussion of paths and loops also caught my attention. When we walk around the object and follow its surface, we start to notice patterns and relationships that are not immediately obvious. I am reminded of how learning and understanding in mathematics develop in this way.
I also agree with you that creative practices can help students develop a new perspective on mathematics. Activities that allow students to design, build, or craft things can help them notice mathematical thinking in a more interesting and relevant way.