Organizer's Email: adam.cartisano@belmont.edu
Imagine a father handing his three-year-old daughter a cereal spoon to dig a swimming pool—not because he cannot finish the job with an excavator in an afternoon, but because his daughter would miss the entire experience of working on the project freely, failing safely, and noticing the small wonders: how dirt sparkles in sunlight, how grass holds clumps together, and what creatures make their home in the soil. Mathematics can be explored in this same spirit: slowly, by hand, noticing the texture of ideas rather than rushing toward results. While our discipline possesses powerful abstractions and theorems—our “excavators”—this special session celebrates the down-to-earth experience of engaging with mathematics more granularly. Doing math “with a spoon” means slowing down, exploring with elementary tools, and savouring the texture of the problem itself. We invite presentations that sidestep heavy machinery and instead explore interesting mathematical questions using accessible approaches, discovering beautiful curiosities along the way. Join us to get your hands dirty and experience the joy of mathematics at a human scale.
This project extends the concept of grationality from regular polygons to regular polyhedra. In two dimensions, grationality arises naturally from area scaling and geometric manipulations. In three dimensions, we introduce structural and arithmetic constraints, since only five regular polyhedra exist and volume scales differently than area. By defining a nice polyhedron as a regular polyhedron with natural-number side lengths, and calling an integer 𝑚 > 3 3D‑Grational when a polyhedron with 𝑚 vertices can be broken into 𝑚 smaller congruent copies, we analyze how these solids behave. These definitions and constraints highlight the fundamental differences between 2D and 3D behavior, reveal new geometric handicaps, and creates conjectures about which vertex counts can support grationality in three dimensions.
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Stanley Tennenbaum proved the irrationality of $\sqrt{2}$ using Carpets Theorem with square carpets, and others (myself, Conway, and Miller) have tried to generalize this proof by using triangular and pentagonal carpets. Although those families of proofs used simple tools, a.k.a. “spoons,” in this talk, I am going to try to “spoonify” this even more. I’ll prove the irrationality of $\sqrt{3}$ and $\sqrt{5}$ using only square-shaped carpets. This work extends the previous findings for grationality and opens up the possibility of more research into families of irrationality proofs.
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Advanced physics utilizes mathematical concepts which are often inaccessible to both undergraduate and unprepared graduate students. In keeping with the theme of teaching with spoons, this talk will illustrate examples of presenting advanced modern physics concepts with only using low-level tools accessible to typical math or science majors. In particular, we will derive gravitational time dilation without using the differential geometry of general relativity, we will solve the quantum harmonic oscillator without using Schrödinger's equation, and we will present a conceptual proof of Bell's theorem using only Venn diagrams. The intent of this talk is to inspire physics teachers to try utilizing spoons in their classrooms to illustrate otherwise unavailable content using more accessible intuitive tools.
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In our previous work, we explored the higher dimensional analogues for Primitive Pythagorean Triples (PPTs) both algebraically and geometrically. In this session, we take a curiosity driven exploration of primitive Pythagorean results in two dimensions, emphasizing the recognition of patterns within differently defined collections of PPTs and observing predictable frequencies and occurrences of non-PPTs. What possible connections exist between this creative exploration and the higher dimensional analogues?
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Mathematician Stanley Tennenbaum demonstrated a geometric proof of the irrationality of the square root of 2 in the 1950s using a lesser known theorem called Carpets Theorem. Using a “spoon”, I plan to show Tennenbaum’s proof in an easy to understand light. However, to avoid steering away from the more mathematical aspects of such a proof, I will show the core algebra and math as well. Both of these points of view will be viewed from a geometric lens just as Stanley Tennenbaum did.
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Hilbert's Third Problem, presented in 1900, asked: given two polyhedra of equal volume, are they scissors-congruent? That is, can one always be cut into a finite number of pieces and be reassembled into the other? In two dimensions, it is clear that two polygons have equal area if they are scissors congruent, but the converse was proved in 1807. The parallels between flat origami and polygonal decomposition suggest a common framework, which motivates us to define a notion of fold-congruence. We pose the question: are two polygons of equal area always fold-congruent? In this talk, we discuss preliminary investigations into this seemingly difficult question.
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