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• Post #4499121

  At Bridges, I gave a talk about how to make "impossible objects" from curves defined in terms of functions of x, parametric, polar, and implicit equations. Matt Parker asked if a similar technique could be used for a curve given by an SVG file. Yes! Here are some proofs of concept. I hope to make more complicated examples. If people have ideas that are interesting, clever, contradictory, funny, etc. Let me know! In case it isn't clear, these are a chicken and an egg, and a sneaker...

• Post #3787006

  I created this logic/grid deduction puzzle several months ago just to try out vibe coding. I created it for myself and have been enjoying playing it, so I thought I'd share it more widely. I call it Daedalus' Labyrinth. https://divisbyzero.github.io/Daedalus-Labyrinth/ Each segment in the grid must either become a wall (a "hedge") or be removed. Numbers at the intersections indicate how many hedges must meet there. Ultimately, you create a labyrinth from one exit to the other...

• Post #1967911

  This is so cool! Given only the "exponential minus log" function, elm(x,y)=exp(x)-ln(y), and the constant 1, you can perform +, -, x, ÷, exp, ln, trig, powers, roots, etc. You can also obtain e and π! See https://arxiv.org/pdf/2603.21852 and https://arxiv.org/src/2603.21852v2/anc/SupplementaryInformation.pdf

• Post #1209972

  I designed and 3D printed a couple of models of Riemann sums for multivariable functions—the monkey saddle and the sombrero function. (I also got to try out this cool filament.) The files are on Thingiverse. https://www.thingiverse.com/thing:7019158 https://www.thingiverse.com/thing:7019184

• Post #926402

  Given a Riemannian manifold that is topologically a sphere, we can talk about "geodesics"—paths as close to straight as possible on the surface. On a true sphere, geodesics are great circles, but on topological spheres, the geodesics may roam around the sphere forever without closing up. In 1905, Poincaré conjectured that any such sphere must have at least three simple closed geodesics. That is, they are closed curves without self-intersections. For example, for an ellipsoid...

• Post #676538

  Still playing with these impossible cylinders. Here I have an ellipse with the "Batman curve" in the reflection.

• Post #676537

  One more, which I call "I ❤️ the Cardioid." (A cardioid from one direction and a heart in the other.)

• Post #676536

• Post #676535

  Gathering 4 Gardner is this week. I'll be running an activity where we'll make one of these giant cardioids (and a giant nephroid). Today, I'm taking advantage of the sunny, above-freezing day to laser-cut the pieces outside.

• Post #676533

  Here's the first paragraph of the preface to Euler's pre-calculus textbook (Introductio ad analysin infinitorum, 1748) as translated from Latin by ChatGPT. The more things change, the more they stay the same! "I have very often observed that by far the greatest part of the difficulties which hinder students of mathematics in approaching the analysis of the infinite arises from this source: although they scarcely grasp common algebra, they nevertheless attempt to apply t...

• Post #676532

  At Gathering 4 Gardner, Colm Mulcahy suggested I try the following 3D-printing project. Recall the "napkin ring problem": Take a sphere of radius r and drill out a hole along a diameter so the remaining shape has height h. Then the volume, V = πh³/6, does not depend on r. He thought that if we print these shapes as solids (i.e., with 100% fill), they should all weigh the same. See the following post for the results.

• Post #676531

  Another 3D-printing suggestion from Colm Mulcahy: Archimedes proved that if you slice a sphere of radius r with two planes a distance h apart, the surface area is 2πrh—the same as a cylinder of the same radius and height. So, it doesn't depend on where the slicing occurs. 1/2

• Post #676530

  I just made this design, based on the "72 Pencils" sculptures by George Hart: https://www.georgehart.com/sculpture/pencils.html. George's design consists of 72 pencils arranged in a triangular lattice. His bundles of 18 pencils are arranged in a hollow hexagon pattern and are glued together. In this version, I create a 3D-printable polyhedron (a truncated octahedron) to hold the standard-sized hexagonal pencils. Each bundle consists of 19 pencils in a filled hexagonal p...

• Post #676529

  I was sad to learn about the passing of Sheldon Newhouse. Even though we were not close, I can honestly say he changed the course of my life. I was on the job market in anticipation of finishing my PhD in 1998. It was a brutal market. I gave a talk at "Smalefest" during my last year of grad school. I had recently found out that my "sure thing" postdoc (in my mind) fell through, and I was not having much luck elsewhere in my job search. After my talk, Sheldon...

• Post #636173

  In Multivariable Calculus, we're going to talk about limits, continuity, and differentiability of functions of two variables. I just printed these three examples of functions that do not have a limit at the origin. Their equations are: f₁(x, y) = ((x² – y²)/(x² + y²))² f₂(x, y) = x²/(x² + y⁴) f₃(x, y) = xy/(x² + y²) Also, if we define f₁(0, 0) = 1 and f₃(0, 0) = 0, then these two functions also have the property that both partial derivatives exist at the origin, but the function is not...