Fisher, "Highly Unlikely Triangles and Other Impossible Figures in Bead Weaving" (2015)
This short article begins by describing "impossible triangle and other impossible figures" as "optical illusions that have inspired much artwork since their discovery by Oscar Reutersvard in the 1930s" (p. 99). According to the article, the triangles were also designed "independently" (p. 99) by psychiatrist Lionel Penrose in the 1950s. As the author explains, "An impossible triangle and other similar impossible figures are only impossible to construct in 3D if we assume the edges are straight and the connections are right angles. In contrast, there is nothing impossible about a two-dimensional drawing of an impossible triangle, and if you believe in Euclidean 3-manifolds, then you can make the topological equivalent of a 'real' impossible triangle with them [8]. However, a true three-dimensional interpretation of an impossible object really is impossible" (p. 100).
With that last sentence, the "optical illusion" appears to become something like a "paradox" for the author, who then proposes, using beaded triangular sculptures, a "novel three-dimensional representation of impossible figures that eliminates this distinction and resolves the paradox in its own way" (p. 100). The paper goes on to develop similarly "highly unlikely squares and hexagons and other figures" (p. 102). The article does not provide historical context or background for this "novel" idea - or the "discovery" attributed to Reutersvard - and without such context, the approach may indeed appear quite unprecedented and "novel."
I want to be cautious regarding the use of terms such as "discovery" and "novel," particularly in light of the history of academic claiming and extractivism in the narratives of origins and developments of particular disciplines (Linda T. Smith's book, Decolonizing Methodologies comes to mind, among other critiques). This concern also connects to the critical question shared previously regarding the academic article read in class, specifically the claim of coming up with something "new" (as in, "This article outlines a new method of mathematical discourse analysis..." [Staats, 2021, p. 2]).
In the same spirit, a couple of talking points are shared below:
Stop 1: Elements related to optical illusions can be observed in ancient and medieval art - for example, entasis used in structures such as the Parthenon. In the 19th century, F. C. Penrose studied entasis with much interest, describing it as a subtle optical correction intended to counteract 'disagreeable optical illusion.' From this vantage point, the broader fascination with optical illusions as well as their incorporation into sculptural and architectural design does not appear entirely new and could problematize the use of the term "discovery."
Stop 2: The idea of three-dimensional representations of impossible figures through curved structures also reminded me of the long-standing mathematical tradition of spherical trigonometry, which studies triangles drawn on the surface of a sphere. Historically, spherical trigonometry has been used for a wide range of purposes, including modeling celestial motion, maritime navigation, determining the Qibla, calculating prayer times, mapmaking, and related applications. From this vantage point, one could scrutinize how "novel" the approach itself was to construct curved 3D triangles.
Hi Aun, thank you for your summary and thoughtful reflections. Your points highlight how Fisher’s work might be better understood as a creative continuation of existing mathematical and artistic traditions rather than as an entirely new discovery. This perspective is especially meaningful because it reminds us that many mathematical ideas have long and complex histories.
ReplyDeleteI also found the terms like highly unlikely squares and hexagons very interesting. They made me think about how this article highlights the creativity, imagination, and even playful aspects of mathematics. Even if the idea may not necessarily be a completely new “discovery,” it still shows how mathematical thinking can emerge through experimenting with shapes and exploring possibilities in creative ways.
Hi Aun, thanks for your reflection! I really appreciate in Stop #2 the way you highlight the real-world applications of spherical trigonometry/curved 3D triangles. While often academics claim ideas may be new or original, we often fail to recognize the ways various communities may have been using that knowledge practically for multiple generations without credit. This really highlights the hierarchy of academia versus community and experiential knowledge.
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