@divbyzero@mathstodon.xyz

Dave Richeson @divbyzero@mathstodon.xyz · 4d ago

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 and a car.

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Dave Richeson @divbyzero@mathstodon.xyz · Jul 13, 2026

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. You can play it on a desktop or phone. Click to add a hedge, right-click (or long-press on a phone) to remove the segment. You can choose different board sizes and different difficulty levels. It is tricky when you begin playing it. But once you get the hang of it and start recognizing certain patterns, it is not too difficult to complete on the normal level. Mathy details: My idea for this puzzle was to create the "dual" puzzle to Loopy, one of my favorite puzzles in Simon Tatham's Portable Puzzle Collection https://www.chiark.greenend.org.uk/~sgtatham/puzzles/js/loopy.html In Loopy, squares contain numbers indicating how many edges surround them. In my game, Loopy's squares became vertices, vertices became squares, and edges became orthogonal edges. I added the exits to build the labyrinth theme. I found out later that Loopy was based on a puzzle called Slitherlink. I'm happy to hear any feedback. Enjoy!

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Dave Richeson @divbyzero@mathstodon.xyz · Apr 14, 2026

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

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Dave Richeson @divbyzero@mathstodon.xyz · Mar 29, 2026

Replying to @divbyzero@mathstodon.xyz

This isn't my area of expertise, but Franks was my PhD advisor, and he proved this result not long before I entered grad school. I thought it was really cool. Here are links to the Franks and Bangert articles: https://link.springer.com/article/10.1007/BF02100612 https://www.worldscientific.com/doi/10.1142/S0129167X93000029

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Dave Richeson @divbyzero@mathstodon.xyz · Mar 29, 2026

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 x²/a²+y²/b²+z²/c²=1, think of the ellipses where the ellipsoid intersects the coordinate planes.

In 1929, Lyusternik and Schnirelmann proved the conjecture. Although the idea of the proof was correct, it contained a flaw that was fixed in the 1980s by Grayson.

Their result was about simple closed geodesics. What about closed geodesics in general? You head off in some direction, you may cross your path multiple times, but you end up back where you started from, heading in the same direction. What can we say about those?

In the early 1990s, Franks and Bangert each published an article covering a special case: the sphere has positive Gaussian curvature (Franks) and all other spheres (Bangert). Together, their work proved the remarkable theorem:

Every Riemannian manifold that is topologically a sphere has infinitely many closed geodesics!

I was telling my topology class about this theorem because we were discussing the annulus, and the key to Franks's result was to show that a certain map on the annulus had infinitely many periodic points.

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Dave Richeson @divbyzero@mathstodon.xyz · Mar 13, 2026

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 differentiable there.

You can download the files here: https://www.thingiverse.com/thing:7314735

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Dave Richeson @divbyzero@mathstodon.xyz · Mar 13, 2026

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, whom I didn't know but who was my "academic uncle," approached me to chat about my work. He asked what I was doing the next year. I said I had nothing yet. He said to FedEx a copy of my application to him at Michigan State. I did, and I was hired for a 2-year position. That kept me in academia, it was where I met my wife, and it was surely helpful when I was on the job market again. He was so kind to me. My thoughts are with his wife, Patti Lamm, and the rest of his family.
https://www.dignitymemorial.com/obituaries/san-diego-ca/sheldon-newhouse-12777624

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Dave Richeson @divbyzero@mathstodon.xyz · Mar 12, 2026

Replying to @liuyao@mathstodon.xyz

@liuyao@mathstodon.xyz Neat idea. I'll have to think about it. Right now, the pencils are quite tight in the holes. So, I'd have to increase the hole sizes to slide the faces off. Also, the pencils would just fall apart without the structure, so you'd have to take off one face at a time and apply glue to the contact points before moving on to the next face.

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Dave Richeson @divbyzero@mathstodon.xyz · Mar 11, 2026

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 pattern. Thus, it requires 76 pencils.

Here's the 3D-printable file: https://www.thingiverse.com/thing:7313100

My first attempt was to make the faces with the hexagonal holes. The holes had to be offset in a certain way so that, when joined into a polyhedron, the pencils would line up correctly inside it. I kept making errors. Then, I decided to completely restart the project. I made the pencils and the polyhedron in OpenSCAD, then "subtracted" the pencils from the polyhedron to create the holes.

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Dave Richeson @divbyzero@mathstodon.xyz · Feb 28, 2026

Replying to @divbyzero@mathstodon.xyz

I just put these files on Thingiverse for anyone to download and print https://www.thingiverse.com/thing:7305536

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Dave Richeson @divbyzero@mathstodon.xyz · Feb 28, 2026

Replying to @divbyzero@mathstodon.xyz

I just put these files on Thingiverse for anyone to download and print https://www.thingiverse.com/thing:7305545

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Dave Richeson @divbyzero@mathstodon.xyz · Feb 28, 2026

Replying to @sibrosan@mastodon.social

@sibrosan@mastodon.social Ah! I misuderstood your original comment. I thought you just wanted me to make the model larger. You are 100% correct. In fact, this is version 2.0. For version 1.0, the sphere was smaller with the same thickness walls, and when I sliced the sphere straight across, the difference in weight was measurable. So, in this version 2.0, I made the sphere larger and the slices are sliced along cones converging at the origin rather than along flat planes. That way, the "height" of each radial shell in the thin solid object is the same. (It has the added benefit that one ring sits slightly inside its neighbor, which is nice.)

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Dave Richeson @divbyzero@mathstodon.xyz · Feb 28, 2026

Replying to @sibrosan@mastodon.social

@sibrosan@mastodon.social Did you see the second of the two posts? That's essentially what I did (but with different dimensions).

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Dave Richeson @divbyzero@mathstodon.xyz · Feb 28, 2026

Replying to @divbyzero@mathstodon.xyz

Idea: Produce 3D-printed slices of a (hollow) sphere, all of the same height. Print with 100% fill. They should all weigh the same. Bam!

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Dave Richeson @divbyzero@mathstodon.xyz · Feb 28, 2026

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.
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Dave Richeson @divbyzero@mathstodon.xyz · Feb 28, 2026

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.

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Dave Richeson @divbyzero@mathstodon.xyz · Feb 28, 2026

Replying to @divbyzero@mathstodon.xyz

Ta da! (Equal ±ε)

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Dave Richeson @divbyzero@mathstodon.xyz · Feb 25, 2026

Replying to @divbyzero@mathstodon.xyz

"by the methods of ordinary algebra, which are usually treated within the analysis of the infinite, so that the connection between the two methods may henceforth be made more clearly apparent." 2/2

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Dave Richeson @divbyzero@mathstodon.xyz · Feb 25, 2026

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 their minds to that more sublime art. From this it comes about that they not only remain standing, as it were, at the threshold, but even form distorted notions of that infinite, whose understanding is here brought to their aid. Now, although the analysis of the infinite requires not only a thorough knowledge of algebra but also an acquaintance with all the methods thus far discovered, there nevertheless exist many questions whose development is capable of preparing the minds of learners for that higher science—yet these are either omitted from the usual elements of algebra, or are not treated with sufficient care. For this reason, I do not doubt that the material I have assembled in these books will abundantly supply this deficiency. For I have taken care not only to set forth more fully and distinctly those matters which the analysis of the infinite strictly requires than is commonly done, but also to work through quite a number of problems by which readers may gradually—and almost without realizing it—render the idea of the infinite familiar to themselves. Moreover, I have here resolved many problems
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Dave Richeson @divbyzero@mathstodon.xyz · Feb 16, 2026

Replying to @divbyzero@mathstodon.xyz

I printed a new version—a little stronger construction with an added leg so it stands on its own.

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Dave Richeson @divbyzero@mathstodon.xyz · Feb 14, 2026

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.

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Dave Richeson @divbyzero@mathstodon.xyz · Feb 13, 2026

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Dave Richeson @divbyzero@mathstodon.xyz · Feb 12, 2026

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

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Dave Richeson @divbyzero@mathstodon.xyz · Feb 11, 2026

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

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Dave Richeson @divbyzero@mathstodon.xyz · Apr 24, 2025

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

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